EUROPEAN ORGANISATION FOR THE SAFETY OF AIR NAVIGATION EUROCONTROL EUROCONTROL EXPERIMENTAL CENTRE. EEC Technical Report No

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1 EUROPEAN ORGANISATION FOR THE SAFETY OF AIR NAVIGATION EUROCONTROL EUROCONTROL EXPERIMENTAL CENTRE REVISION OF ATMOSPHERE MODEL IN BADA AIRCRAFT PERFORMANCE MODEL EEC Technical Report No Project: BADA Public Issued: February 2010 European Organisation for the Safety of Air Navigation EUROCONTROL 2010 This document is published by EUROCONTROL in the interest of the exchange of information. It may be copied in whole or in part providing that the copyright notice and disclaimer are included. The information contained in this document may not be modified without prior written permission from EUROCONTROL. EUROCONTROL makes no warranty, either implied or express, for the information contained in this document, neither does it assume any legal liability or responsibility for the accuracy, completeness or usefulness of this information.

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3 REPORT DOCUMENTATION PAGE Reference EEC Note/Report No. 2010/001 Originator: CND/VIF/ACP Sponsor EUROCONTROL Security Classification Unclassified Originator (Corporate Author) Name/Location: EUROCONTROL Experimental Centre B.P.15 F Brétigny-sur-Orge CEDEX FRANCE Telephone : Internet : Sponsor (Contract Authority) Name/Location EUROCONTROL Agency Rue de la Fusée, 96 B 1130 BRUXELLES Telephone : Internet : TITLE : REVISION OF ATMOSPHERE MODEL IN BADA AIRCRAFT PERFORMANCE MODEL Author D. Poles Date 02/2010 Pages xvi+146 Figures 3 Tables 172 Annexes 2 References 7 Project BADA Distribution Statement: (a) Controlled by: Head of Section (b) Distribution : Public Restricted Confidential (c) Copy to NTIS: YES / NO Descriptors (keywords) : Task no. sponsor CND/VIR/ACP Period 07/09 to 12/09 BADA, aircraft performance model, BADA atmosphere model, International Standard Atmosphere (ISA) Abstract : This document provides details about the newly developed BADA atmosphere model. It presents the main differences compared to the atmosphere model currently implemented and provides information on impact of introducing the changes in the BADA family 3 model algorithms. Benefits in terms of improved aircraft performance parameters accuracy and ensured consistency between the atmosphere models used by aircraft performance models and aircraft Flight Management System are also identified. The new BADA atmosphere model has been used in the identification process of the BADA revision 3.7 and it will be used in all subsequent revisions of BADA family 3. The aircraft performance coefficients of the BADA 3.7 and onwards are compatible and may be used with both atmosphere models. Although the implementation decision is left to the users of the BADA family 3 users, the implementation of the new model is highly recommended by the BADA project team at EUROCONTROL.

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5 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL FOREWORD This document provides details about the newly developed BADA atmosphere model. It presents the main differences compared to the atmosphere model currently implemented and provides information on impact of introducing the changes in the BADA family 3 model algorithms. Benefits in terms of improved aircraft performance parameters accuracy and ensured consistency between the atmosphere models used by aircraft performance models and aircraft Flight Management System are also identified. The new BADA atmosphere model has been used in the identification process of the BADA revision 3.7 and it will be used in all subsequent revisions of BADA family 3. The aircraft performance coefficients of the BADA 3.7 and onwards are compatible and may be used with both atmosphere models. Although the implementation decision is left to the users of the BADA family 3 users, the implementation of the new model is highly recommended by the BADA project team at EUROCONTROL. Project: BADA EEC Technical/Scientific Report No v

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7 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL TABLE OF CONTENTS FOREWORD... V LIST OF APPENDICES... VIII LIST OF FIGURES... VIII LIST OF TABLES... VIII LIST OF ABBREVIATIONS... XIII REFERENCES... XV 1. INTRODUCTION INTERNATIONAL STANDARD ATMOSPHERE (ISA) MODEL THE EQUATION OF THE STATIC ATMOSPHERE AND THE PERFECT GAS LAW GEOPOTENTIAL AND GEODETIC ALTITUDES PHYSICAL CHARACTERISTICS OF THE ATMOSPHERE AT THE MEAN SEA LEVEL TEMPERATURE AND VERTICAL TEMPERATURE GRADIENT PRESSURE DENSITY SPEED OF SOUND NEW BADA ATMOSPHERE MODEL DEFINITIONS HYPOTHESES RELATIONSHIP BETWEEN GEOPOTENTIAL AND GEOPOTENTIAL PRESSURE ALTITUDES EXPRESSIONS KEY DIFFERENCES BETWEEN THE OLD AND NEW ATMOSPHERE MODEL IMPACT OF CHANGES ON BADA FAMILY BENEFITS OF IMPLEMENTING THE NEW ATMOSPHERE MODEL CONCLUSIONS AND RECOMMENDATIONS Project: BADA EEC Technical/Scientific Report No vii

8 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model LIST OF APPENDICES APPENDIX A: REVISION 3.7 NON-ISA ACCURACY TABLES APPENDIX B: THE NEW BADA ATMOSPHERE MODEL TABLE DATA LIST OF FIGURES Figure 4-1: The relationship between tropopause altitude and temperature for the old model Figure 4-2: The relationship between geopotential pressure altitude and temperature for the new model Figure 4-3: The relationship between geopotential altitude and temperature for the new model LIST OF TABLES Table 2-1: Temperatures and vertical gradients... 6 Table 5-1: New BADA atmosphere model impact Table A-1: A30B with the new atmosphere model Table A-2: A30B with the old atmosphere model Table A-3: A310 with the new atmosphere model Table A-4: A310 with the old atmosphere model Table A-5: A319 with the new atmosphere model Table A-6: A319 with the old atmosphere model Table A-7: A320 with the new atmosphere model Table A-8: A320 with the old atmosphere model Table A-9: A321 with the new atmosphere model Table A-10: A321 with the old atmosphere model Table A-11: A332 with the new atmosphere model Table A-12: A332 with the old atmosphere model Table A-13: A333 with the new atmosphere model Table A-14: A333 with the old atmosphere model Table A-15: A343 with the new atmosphere model Table A-16: A343 with the old atmosphere model Table A-17: A346 with the new atmosphere model Table A-18: A346 with the old atmosphere model Table A-19: A388 with the new atmosphere model viii Project: BADA EEC Technical/Scientific Report No

9 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL Table A-20: A388 with the old atmosphere model Table A-21: AT43 with the new atmosphere model Table A-22: AT43 with the old atmosphere model Table A-23: AT45 with the new atmosphere model Table A-24: AT45 with the old atmosphere model Table A-25: AT72 with the new atmosphere model Table A-26: AT72 with the old atmosphere model Table A-27: ATP with the new atmosphere model Table A-28: ATP with the old atmosphere model Table A-29: B462 with the new atmosphere model Table A-30: B462 with the old atmosphere model Table A-31: B712 with the new atmosphere model Table A-32: B712 with the old atmosphere model Table A-33: B722 with the new atmosphere model Table A-34: B722 with the old atmosphere model Table A-35: B732 with the new atmosphere model Table A-36: B732 with the old atmosphere model Table A-37: B733 with the new atmosphere model Table A-38: B733 with the old atmosphere model Table A-39: B734 with the new atmosphere model Table A-40: B734 with the old atmosphere model Table A-41: B735 with the new atmosphere model Table A-42: B735 with the old atmosphere model Table A-43: B736 with the new atmosphere model Table A-44: B736 with the old atmosphere model Table A-45: B737 with the new atmosphere model Table A-46: B737 with the old atmosphere model Table A-47: B738 with the new atmosphere model Table A-48: B738 with the old atmosphere model Table A-49: B742 with the new atmosphere model Table A-50: B742 with the old atmosphere model Table A-51: B743 with the new atmosphere model Table A-52: B743 with the old atmosphere model Table A-53: B744 with the new atmosphere model Table A-54: B744 with the old atmosphere model Table A-55: B752 with the new atmosphere model Table A-56: B752 with the old atmosphere model Table A-57: B753 with the new atmosphere model Project: BADA EEC Technical/Scientific Report No ix

10 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model Table A-58: B753 with the old atmosphere model Table A-59: B762 with the new atmosphere model Table A-60: B762 with the old atmosphere model Table A-61: B763 with the new atmosphere model Table A-62: B763 with the old atmosphere model Table A-63: B764 with the new atmosphere model Table A-64: B764 with the old atmosphere model Table A-65: B772 with the new atmosphere model Table A-66: B772 with the old atmosphere model Table A-67: B773 with the new atmosphere model Table A-68: B773 with the old atmosphere model Table A-69: BE9L with the new atmosphere model Table A-70: BE9L with the old atmosphere model Table A-71: BE20 with the new atmosphere model Table A-72: BE20 with the old atmosphere model Table A-73: BE58 with the new atmosphere model Table A-74: BE58 with the old atmosphere model Table A-75: C130 with the new atmosphere model Table A-76: C130 with the old atmosphere model Table A-77: C510 with the new atmosphere model Table A-78: C510 with the old atmosphere model Table A-79: C550 with the new atmosphere model Table A-80: C550 with the old atmosphere model Table A-81: C560 with the new atmosphere model Table A-82: C560 with the old atmosphere model Table A-83: C750 with the new atmosphere model Table A-84: C750 with the old atmosphere model Table A-85: CL60 with the new atmosphere model Table A-86: CL60 with the old atmosphere model Table A-87: CRJ1 with the new atmosphere model Table A-88: CRJ1 with the old atmosphere model Table A-89: CRJ2 with the new atmosphere model Table A-90: CRJ2 with the old atmosphere model Table A-91: CRJ9 with the new atmosphere model Table A-92: CRJ9 with the old atmosphere model Table A-93: D228 with the new atmosphere model Table A-94: D228 with the old atmosphere model Table A-95: D328 with the new atmosphere model Table A-96: D328 with the old atmosphere model x Project: BADA EEC Technical/Scientific Report No

11 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL Table A-97: DA42 with the new atmosphere model Table A-98: DA42 with the old atmosphere model Table A-99: DH8A with the new atmosphere model Table A-100: DH8A with the old atmosphere model Table A-101: DH8C with the new atmosphere model Table A-102: DH8C with the old atmosphere model Table A-103: DH8D with the new atmosphere model Table A-104: DH8D with the old atmosphere model Table A-105: E120 with the new atmosphere model Table A-106: E120 with the old atmosphere model Table A-107: E135 with the new atmosphere model Table A-108: E135 with the old atmosphere model Table A-109: E145 with the new atmosphere model Table A-110: E145 with the old atmosphere model Table A-111: E170 with the new atmosphere model Table A-112: E170 with the old atmosphere model Table A-113: E190 with the new atmosphere model Table A-114: E190 with the old atmosphere model Table A-115: EA50 with the new atmosphere model Table A-116: EA50 with the old atmosphere model Table A-117: F27 with the new atmosphere model Table A-118: F27 with the old atmosphere model Table A-119: F50 with the new atmosphere model Table A-120: F50 with the old atmosphere model Table A-121: F70 with the new atmosphere model Table A-122: F70 with the old atmosphere model Table A-123: F100 with the new atmosphere model Table A-124: F100 with the old atmosphere model Table A-125: F900 with the new atmosphere model Table A-126: F900 with the old atmosphere model Table A-127: FA50 with the new atmosphere model Table A-128: FA50 with the old atmosphere model Table A-129: H25A with the new atmosphere model Table A-130: H25A with the old atmosphere model Table A-131: JS31 with the new atmosphere model Table A-132: JS31 with the old atmosphere model Table A-133: JS41 with the new atmosphere model Table A-134: JS41 with the old atmosphere model Project: BADA EEC Technical/Scientific Report No xi

12 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model Table A-135: LJ35 with the new atmosphere model Table A-136: LJ35 with the old atmosphere model Table A-137: LJ45 with the new atmosphere model Table A-138: LJ45 with the old atmosphere model Table A-139: MD11 with the new atmosphere model Table A-140: MD11 with the old atmosphere model Table A-141: MD82 with the new atmosphere model Table A-142: MD82 with the old atmosphere model Table A-143: MD83 with the new atmosphere model Table A-144: MD83 with the old atmosphere model Table A-145: P28A with the new atmosphere model Table A-146: P28A with the old atmosphere model Table A-147: PA34 with the new atmosphere model Table A-148: PA34 with the old atmosphere model Table A-149: RJ85 with the new atmosphere model Table A-150: RJ85 with the old atmosphere model Table A-151: SB20 with the new atmosphere model Table A-152: SB20 with the old atmosphere model Table A-153: SF34 with the new atmosphere model Table A-154: SF34 with the old atmosphere model Table A-155: SH36 with the new atmosphere model Table A-156: SH36 with the old atmosphere model Table A-157: SW4 with the new atmosphere model Table A-158: SW4 with the old atmosphere model Table A-159: T134 with the new atmosphere model Table A-160: T134 with the old atmosphere model Table A-161: T154 with the new atmosphere model Table A-162: T154 with the old atmosphere model Table A-163: TRIN with the new atmosphere model Table A-164: TRIN with the old atmosphere model Table B-1: ISA -20 table data Table B-2: ISA -10 table data Table B-3: ISA table data Table B-4: ISA+10 table data Table B-5: ISA+20 table data Table B-6: ISA+30 table data xii Project: BADA EEC Technical/Scientific Report No

13 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL LIST OF ABBREVIATIONS APM BADA CAS CMB CRZ DES FMS ICAO ISA ISO MCMB MSL RMS STD TRJ TROP Aircraft Performance Model Base of Aircraft Data calibrated airspeed climb cruise descent Flight Management System International Civil Aviation Organisation International Standard Atmosphere International Organisation for Standardisation maximum climb mean sea level root mean square standard deviation trajectory tropopause Project: BADA EEC Technical/Scientific Report No xiii

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15 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL REFERENCES [1] Base of aircraft data (BADA) aircraft performance modelling report - Revision 3.7, EEC Technical Report No , EUROCONTROL EEC, [2] AMEBA - Concept Document, EEC Technical Report, EUROCONTROL EEC, [3] User manual for the base of aircraft data (BADA) Revision 3.7, EEC Technical Report No , EUROCONTROL EEC, [4] International standard ISO 2533, Standard atmosphere, Ref. No (E), International Organization for Standardization, [5] Manual of the ICAO standard atmosphere, Doc 7488/3, [6] AMEBA Improvement report, EEC Technical Report, [7] BADA 4.0 Aircraft Performance Model Toolbox, User Manual v1.3, ECC Note No. 06/09, Project: BADA EEC Technical/Scientific Report No xv

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17 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 1. INTRODUCTION Enhancements to the BADA aircraft performance modelling capabilities have been the subject of research efforts over the past years. The research was based on exploiting today s aircraft performance resources, data and software, which were not available in the past when BADA was initially developed 1. Encouraging results that demonstrate benefits of using new data resources and provide improvements in various aspects ([2], [6]) of aircraft performance modelling capabilities have been obtained. One of the findings which contribute to the improvements is related to the way the International Standard Atmosphere (ISA) model is applied in the BADA aircraft performance model. Namely, by using detailed aircraft manufacturers performance reference data, some discrepancies were observed between atmospheric properties at non-isa conditions provided by the atmosphere model currently used by BADA family 3 [1] and the one used by aircraft manufacturers. After having analysed the discrepancies, it was found out that both of the atmosphere models use the same basic principles of the International Standard Atmosphere (ISA), but with different definition of tropopause altitude and temperature gradient. This causes differences in the calculation of some aircraft performance parameters (energy share factor, rate of climb, etc) with a consequent impact on aircraft performance models accuracy. Since the aircraft model identification process in BADA family 3 [1] and 4 ([2], [7]) is based on aircraft manufacturers performance reference data for different atmospheric conditions, it is deemed important to ensure consistency between the atmosphere models used by aircraft performance models and aircraft manufacturers. Furthermore, this ensures consistency in calculation of some aircraft performance parameters between simulation tools and aircraft Flight Management System (FMS) for given atmosphere conditions. For this reason, a new BADA atmosphere model was developed in a way that best fits the aircraft performance reference data by including precise modelling of atmosphere for ISA and non-isa initial conditions (temperature and pressure deviations). Benefits in implementing the new atmosphere model in terms of aircraft horizontal speed, vertical speed and fuel flow were assessed and demonstrated during development of the aircraft models in BADA 3.7. The objectives of this document are to describe the newly developed BADA atmosphere model; identify the main differences compared to the atmosphere model currently implemented (referred to as the old one); provide information on cost (impact of introducing the changes in the BADA family 3 model algorithms) and benefits (improved aircraft performance parameters accuracy). 1 Nowadays, aircraft manufacturers develop aircraft performance engineering software that can provide high quality aircraft performance reference data including other relevant parameters that facilitate aircraft performance models development and validation. Project: BADA EEC Technical/Scientific Report No

18 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model The document is organised in 7 sections including Section 1, Introduction. Section 2 contains the ISA atmosphere descriptions on which the new BADA atmosphere model is based. Section 3 describes the new BADA atmosphere model in detail. Key differences between the old and new model are given in Section 4. The impact of the new BADA atmosphere model on BADA operations performance model is described in Section 5. The benefits are discussed in Section 6, while conclusions and recommendations for use are given in section 7. Accuracy tables for all aircraft types developed for BADA revision 3.7, for current and new BADA atmosphere models, are given in Appendix A. In Appendix B, for the new BADA atmosphere model, table data for different conditions are given. 2 Project: BADA EEC Technical/Scientific Report No

19 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 2. INTERNATIONAL STANDARD ATMOSPHERE (ISA) MODEL The International Standard Atmosphere (ISA) is an atmospheric model of how the pressure, temperature, density and viscosity of the Earth s atmosphere change over a wide range of altitude. The International Organisation for Standardisation (ISO) publishes the ISA as an international standard [4]. Another organisations, the International Civil Aviation Organisation (ICAO) publishes an ICAO standard atmosphere [5]. In this section, the basic principles and calculation formulas used in both cases and relevant for aircraft performance modelling activity are presented THE EQUATION OF THE STATIC ATMOSPHERE AND THE PERFECT GAS LAW Being static with respect to the Earth, the atmosphere is subject to gravity and the conditions of fluidostatic (air static) equilibrium are determined by the following equation: -dp = ρ g dh (2-1) where p is air pressure, ρ density, g acceleration due to gravity and h geodetic altitude. The perfect gas law relates air pressure, p, to density, ρ, and temperature, T, as follows: p = ρ R T (2-2) where R is the specific gas constant, R = 287,05287 [m 2 /Ks 2 ] for dry air GEOPOTENTIAL AND GEODETIC ALTITUDES In order to characterize pressure distribution in the atmosphere, the gravity potential or geopotential, Φ, should be introduced. The gravity potential characterizes the potential energy of an air particle at a given point, (x, y, z). The equation: Φ(x, y, z) = const. (2-3) defines a surface of the same potential. Moving along on external normal from any point the change in the gravity potential will be equal to the work performed: dφ = g (h) dh (2-4) Project: BADA EEC Technical/Scientific Report No

20 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model and finally, integrating and dividing by g 0, standard acceleration due to gravity, g 0 = 9,80665 [m/s 2 ]: Φ H = g 0 1 = g 0 h 0 g ( h) dh (2-5) where H is numerically equal to the geopotential altitude and h is geodetic altitude. The mean sea level is taken as a reference for readings for both geopotential and geodetic altitude. It can be seen, from (2-5), that in order to relate geopotential and geodetic altitudes it is necessary to find a relationship between acceleration due to gravity, g, and geodetic altitude. It is known that gravity is a vectorial summation of the gravitational attraction and the centrifugal force induced by the earth rotation and it is a complex function of latitude and radial distance from earth s centre. In general, it is complex and unpractical for use. However, acceleration may be obtained with sufficient accuracy by neglecting centrifugal acceleration and using only Newton s gravitational law, in which case: 2 r 0 g = g (2-6) r + h where r is the nominal earth s radius, r = [m] (the earth is considered as a sphere). Combining (2-5) and (2-6) gives the following relationship between geopotential and geodetic altitudes: r h H =, r + h r H h = (2-7) r H Note that values of g calculated using the equation (2-6) for the altitude of [m] do not differ by more than 0,001 percent from the values calculated using the more accurate equations (2-5) ([4], [5]). 4 Project: BADA EEC Technical/Scientific Report No

21 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 2.3. PHYSICAL CHARACTERISTICS OF THE ATMOSPHERE AT THE MEAN SEA LEVEL For the calculation of the ISA atmosphere, the mean sea level is defined as zero altitude with the following initial values for the most important characteristics: standard acceleration due to gravity, g 0 = [m/s 2 ]; atmospheric pressure, p 0 = [Pa]; atmospheric density, ρ 0 = 1,225 [kg/m 3 ]; temperature, T 0 = 288,15 [K]; speed of sound, a 0 = 340,294 [m/s]. The remaining characteristics are calculated using initial values as a basis. By definition, at mean sea level geodetic and geoptential altitudes are the same: h = H; (2-8) and acceleration due to gravity, g, is equal to the standard acceleration due to gravity, g 0 : g = g 0. (2-9) 2.4. TEMPERATURE AND VERTICAL TEMPERATURE GRADIENT According to the temperature variations with altitude, the atmosphere is divided into several layers: troposphere, stratosphere, mesosphere and thermosphere, whose respective boundaries are the tropopause, stratopause and mesopause. For calculating a standard atmosphere, the temperature of each layer is taken as a linear function of geopotential altitude: T = T b + β T (H H b ) (2-10) where point b is the point of lower limit of the layer concerned, β T is the vertical temperature gradient and T b and H b the temperature and the geopotential altitude of the lower limit, respectively. Project: BADA EEC Technical/Scientific Report No

22 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model Table 2-1: Temperatures and vertical gradients ([4], [5]) Geopotential altitude, H, [km] Temperature, T, [K] Temp gradient, β T, [K/km] Layers and boundaries ,65-6,5 troposphere 0 288, troposphere ,65 tropopause ,65 1,0 stratosphere ,65 2,8 stratosphere ,65 0,0 stratosphere ,65 stratopause -2,8 mesosphere ,65-2,8 mesosphere , PRESSURE Combining hydrostatic equation (2-1) with perfect gas law equation (2-2) and taking into account temperature variations given in (2-10), the following solution is obtained for pressure: p g0 β βt R T = pb 1 + ( H H b ) (2-11) Tb for β T different from zero and: g 0 p = pb exp ( H H b ) (2-12) RT for β T equal to zero. 6 Project: BADA EEC Technical/Scientific Report No

23 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL For example, taking into account just troposphere and stratosphere, pressure is calculated as follows: below tropopause o β T,< = -6,5 [K/km]; o lower limit is mean sea level, H b = 0 [m] and T b = 288,15 [K]; o (2-11) equation is used; above tropopause; o β T,> = 0 [K/km]; o lower limit point is the tropopause, H b = [m] and T b = 216,65 [K]; o (2-12) equation is used DENSITY The density is calculated from the pressure and the temperature using the perfect gas law (2-2): p ρ = (2-13) RT 2.7. SPEED OF SOUND The speed of sound, a, is the speed at which the pressure waves travel through a fluid and it is given by the expression: a = κ RT (2-14) where κ = 1,4 is the adiabatic index of air. Project: BADA EEC Technical/Scientific Report No

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25 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 3. NEW BADA ATMOSPHERE MODEL The objective of this section is to provide analytical expressions for the atmospheric properties (pressure, temperature, density and speed of sound) as a function of position. These expressions are necessary to obtain the aircraft performances and movement, as both of them depend on the atmospheric properties. Although atmospheric models custom fitted for specific regions and seasons are used for certain applications, the International Standard Atmosphere (ISA) and its variations (referred to as non-isa standard atmospheres in this document) are widely accepted as standards for computing and evaluating aircraft performances. These are the atmospheric models treated in this section. The development of the new BADA atmosphere model is presented in detail in [2] 2. The new BADA atmosphere model supports altitudes up to 20 km which is considered to be sufficient for aviation purposes DEFINITIONS The following definitions are necessary to properly comprehend and derive an atmospheric model: i) Speed of sound. The definition is given in section 2.7. It is a function of the fluid properties (air in this case) and its temperature (2-14) ii) A generic atmospheric model is a set of relationships providing the atmospheric pressure, temperature, and density as a function of position, usually defined by its geodetic reference system coordinates: p, T, ρ = f (λ, φ, h) (3-1) where λ is the longitude angle, φ the latitude angle and h the geodetic altitude above the surface. Generic atmospheric models usually provide very realistic results for a small geographic area, as they are based on experimental measurements or equations custom fitted to the local atmospheric phenomena. 2 It is worth mentioning that the ISA model describes the atmosphere s behaviour just for specific initial conditions, for standard conditions at MSL (2.3). The hypotheses and definitions in the case where initial conditions differ from ISA, with temperature and pressure deviations at the mean sea level, are not specifically provided by the ISA model. Since BADA APM provides means of calculating aircraft performances for ISA and non-isa conditions, the new atmosphere model provides means for that. Project: BADA EEC Technical/Scientific Report No

26 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model Although (3-1) represents the general case, the influence of latitude and longitude is usually small and can be neglected. The insertion of the relationship between geodetic and geopotential altitudes derived in section 2.2 results in the following expression: p, T, ρ = f (H) (3-2) iii) ISA atmosphere, defined in section 2, is widely used as the standard for computing and evaluating aircraft performances throughout the world. It provides expressions for the standard atmospheric pressure, temperature, and density as functions of the geopotential altitude H, i.e. as functions of the geopotential pressure altitude H P since they are the same for ISA atmosphere. p, T ISA, ρ = f (H P ) (3-3) iv) Standard temperature T ISA is the atmospheric temperature that occurs in the ISA atmosphere. It is a function of the geopotential pressure altitude H P. v) Geopotential pressure altitude H P is the geopotential altitude that occurs in the ISA atmospheric conditions. vi) Mean sea level standard conditions are those that occur in the ISA atmosphere at the point where the geopotential pressure altitude H P is zero. They are denoted as T 0, p 0, ρ 0, and a 0 with the values defined in section 2.3. vii) Mean sea level conditions are those that occur in a non-isa atmosphere. They are identified by the sub index MSL and differ from (T 0, p 0, ρ 0, a 0 ) in non-isa conditions. viii) Non-ISA atmospheres are those that follow the same hypotheses as the ISA atmosphere but differ from it in that one or both of the following parameters is not zero: 1. ΔT. Temperature differential at mean sea level. It is the difference in atmospheric temperature between a given non standard atmosphere and ISA at mean sea level. 2. Δp. Pressure differential at mean sea level. It is the difference in atmospheric pressure at mean sea level between a given atmosphere and ISA. The values of these two parameters uniquely identify any non ISA atmosphere. Thus, a non ISA atmosphere provides expressions for the atmospheric pressure, temperature, and density as functions of the geopotential altitude H and its two differentials. p, T, ρ = f (H, ΔT, Δp) (3-4) 10 Project: BADA EEC Technical/Scientific Report No

27 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL ix) Atmospheric ratios are dimensionless variables containing the ratios between the atmospheric properties at a given point and those found in standard mean sea level conditions. δ = p / p 0 (3-5) θ = T / T 0 (3-6) σ = ρ / ρ 0 (3-7) α = a / a 0 (3-8) 3.2. HYPOTHESES Any atmospheric model that does not comply with any of below listed hypotheses would belong to the category of generic atmospheres, as described in section 3.1, and not comply with the expressions that appear in section 3.4. i) The atmosphere is composed of air, which is a perfect gas. Its thermodynamic properties (pressure, temperature, and density) at any point are thus related by the law of perfect gases given in (2-2). ii) The atmosphere is static in relation to the Earth, so the fluidostatic equilibrium given by (2-1) must be used. The insertion of expression (2-5) in (2-1) provides the pressure differential in terms of geopotential altitude: dp = ρ g 0 dh (3-9) iii) The tropopause is the separation between two different layers: the troposphere, which stands below it, and the stratosphere, which is placed above. Its altitude H P,trop is constant when expressed in terms of geopotential pressure altitude: H P,trop = [m] (3-10) Note that subindex < denotes values below and equal to the tropopause and subindex > denotes values above the tropopause. Project: BADA EEC Technical/Scientific Report No

28 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model iv) Temperature changes with altitude according to a given gradient β T, which is constant for each atmospheric layer when expressed in terms of geopotential pressure altitude: 3 [ K / m] ; H P [ m] [ K / m] ; H > [ m] 6,5 10 dt = β T dh P, where βt = (3-11) 0 P For more details see Table 2-1. v) The air humidity R H does not affect the value of any other atmospheric property, and hence is not taken into account RELATIONSHIP BETWEEN GEOPOTENTIAL AND GEOPOTENTIAL PRESSURE ALTITUDES Although both types of altitudes have been defined in section 3.1, the obtention of the mathematical expression linking them is of particular importance, as the performances of an aircraft are usually provided in terms of geopotential pressure altitude H P, while its movement must be expressed in geodetic altitude h, which is directly linked with the geopotential altitude H per (2-7). The combination of the law of perfect gases (2-2) and the fluidostatic vertical equilibrium (2-1) expressions results in: dp = (p/rt) * g 0 dh (3-12) In the case of the ISA atmosphere, section 2, and according to the definitions of section 3.1, the atmospheric temperature becomes the standard temperature (T = T ISA ), and the geopotential altitude turns into the geopotential pressure altitude (H = H P ). Inserting these results in the expression above results in: dp = (p/rt ISA ) * g 0 dh P (3-13) The ratio between the differentials of both types of altitudes is obtained by dividing expression (3-12) by (3-13): dh / dh P = T / T ISA (3-14) The ratio between incremental changes of geopotential altitude and geopotential pressure altitude is thus the same as that between the atmospheric temperature at that point and the temperature that would occur in the standard atmospheric conditions at the same point. 12 Project: BADA EEC Technical/Scientific Report No

29 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 3.4. EXPRESSIONS This section contains the relationships linking the following five variables for any non ISA atmosphere identified by ΔT and Δp as defined in section 3.1. The expressions are also applicable to the case of the standard atmosphere, by replacing both parameters with zeros. These five variables are listed below: i. Atmospheric pressure p. ii. Atmospheric temperature T. iii. Atmospheric standard temperature T ISA. iv. Geopotential pressure altitude H P. v. Geopotential altitude H. The atmospheric density ρ and the speed of sound a can be easily obtained by using expressions (2-2) and (2-14), respectively. The atmospheric ratios (pressure δ, temperature θ, density σ, and speed of sound α) defined in section 3.1 can then be determined by dividing the atmospheric variables by their values at mean sea level standard conditions. The expressions are the result of the integration of differential equations, so their values at certain points are needed to obtain the complete expressions. These reference points are the following: i. Standard mean sea level (subindex H P = 0). Its definition in section 3.1 includes the values for the geopotential pressure altitude H P, pressure p, and standard temperature T ISA. The temperature differential ΔT sets the value of the real temperature T in non standard atmospheres. H P,HP = 0 = 0 (3-15) T ISA,HP=0 = T 0 (3-16) p HP=0 = p 0 (3-17) T HP=0 = T 0 + ΔT (3-18) H HP=0 = 1/β T, < (T 0 T ISA,MSL + ΔT Ln (T 0 / T ISA,MSL )) (3-19) Project: BADA EEC Technical/Scientific Report No

30 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model ii. Mean sea level (subindex MSL). Equation (2-8) shows that the values of the geopotential and geodetic altitudes (H and h) coincide at mean sea level. In order to simplify the expressions, this document assumes that the geopotential altitude at mean sea level is always zero, which repercutes in the atmospheric expressions provided in this section. They should be amended in case very detailed geopotential models are used. The pressure differential Δp sets the value of the atmospheric pressure p. H MSL = 0 (3-20) p MSL = p 0 + Δp (3-21) H P, MSL T0 = β T, < p p MSL 0 βt, < R g 0 1 (3-22) T ISA,MSL = T 0 + β T,< H P,MSL (3-23) T MSL = T 0 + ΔT + β T,< H P,MSL (3-24) iii. Tropopause (subindex trop). Equation (3-10) sets its geopotential pressure altitude H P. The other values are obtained throughout this section. According to the definitions of section 3.1, the standard temperature T ISA and the geopotential pressure altitude H P are not necessary to define an atmospheric model. However, they are included as they simplify the resulting expressions and facilitate their understanding. i. T ISA = f (H P ) T ISA,< = T 0 + β T,< H P,< T ISA,trop = T 0 + β T,< H P,trop (3-25) T ISA,> = T ISA,trop 14 Project: BADA EEC Technical/Scientific Report No

31 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL ii. p = f (H P ) g0 T R T H < < P <, = p 1 β β +,, 0 p< T 0 p g0 T R T H < < P trop trop = p 1 β β,,, + T 0 0 (3-26) g exp R T ( H H ) 0 p > = ptrop P, > P, trop ISA, trop iii. p = f (T ISA ) TISA, < p< p T = 0 0 g0 βt, < R p trop T = p 0 T0 ISA, trop g0 βt, < R (3-27) T ISA, = T ISA, trop >, so p > does not directly depend on temperature T ISA>. iv. T = f (H P, ΔT) T < = T 0 + ΔT + β T,< H P,< T trop = T 0 + ΔT + β T,< H P,trop (3-28) T > = T trop v. T = f (T ISA, ΔT) T = T ISA + ΔT (3-29) Project: BADA EEC Technical/Scientific Report No

32 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model vi. p = f (T, ΔT) T< T p p < T = 0 0 g0 βt, < R (3-30) p trop T = p 0 trop T T 0 g0 βt, < R T > = T trop, so p > does not directly depend on temperature T >. vii. H = f (H P, ΔT, Δp) H < = H P, < H P, MSL T + β T, < T0 + βt, Ln T, < ISA MSL H P, < H trop = H P, trop H P, MSL T + β T0 + βt Ln T, < H T, < ISA, MSL P, trop (3-31) T + T + β ( H H ) 0 T, < P, trop H > = H P, trop + P, > P, trop T0 + βt, < H P, trop H viii. H = f (T ISA, ΔT, Δp) H < 1 = β T, < T ISA, < T ISA, MSL T + T Ln T ISA, < ISA, MSL H trop 1 = β T T + T Ln T ISA, trop ISA, trop TISA, MSL T, < ISA, MSL T ISA,> = T ISA,trop, so H > does not directly depend on temperature T ISA>. (3-32) ix. H = f (T, ΔT, Δp) H < 1 T< = T< T TISA, MSL + T Ln βt, < T T, ISA MSL H trop 1 = β T T + T Ln T T, trop trop T TISA, MSL T, < ISA MSL T > = T trop, so H > does not directly depend on temperature T >. (3-33) 16 Project: BADA EEC Technical/Scientific Report No

33 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL Project: BADA EEC Technical/Scientific Report No x. H = f (p, ΔT, Δp) + = < < < < < < ISA MSL g R ISA MSL g R T T p p T Ln T T p p T H T T, 0 0, 0 0, 0, 0, 1 β β β + = < < < MSL ISA g R trop ISA MSL g R trop T trop T p p T Ln T T p p T H T T, 0 0, 0 0, 0, 0, 1 β β β (3-34) = > < < > trop ISA trop trop P T trop P T trop p p Ln g RT H T H T T H H 0,,, 0,, 0 β β

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35 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 4. KEY DIFFERENCES BETWEEN THE OLD AND NEW ATMOSPHERE MODELS For the standard ISA conditions with no temperature or pressure deviation the old and new models are the same. The differences appear in non-isa atmosphere conditions due to use of different altitude types (section 2.2 and 3.3) and the way the tropopause altitude is defined. Namely, the old model defines tropopause altitude in terms of the constant tropopause temperature: T trop = [K] ; (4-1) Below the tropopause the temperature is calculated as a function of geopotential altitude, ISA mean see level temperature (T 0 ) and temperature gradient β T : T = T 0 + β T *H ; (4-2) Figure 4-1 depicts the variation of the tropopause altitude in function of temperature deviation as used by the old model. H Tropopause level at To+ΔT m Tropopause level at To Tropopause level at To-ΔT ΔT ΔT T trop To-ΔT T 0 To+ΔT T Figure 4-1: The relationship between tropopause altitude and temperature for the old model The new model introduces the notion of geopotential pressure altitude and the definition of tropopause is given in terms of the constant tropopause geopotential pressure altitude: H ptrop = [m] ; (4-3) and below the tropopause the temperature is calculated as: T = T 0 + β T * H p ; (4-4) Project: BADA EEC Technical/Scientific Report No

36 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model as function of geopotential pressure altitude, which differs from geopotential altitude H, and where T 0 is ISA mean see level temperature (288,15 [K]). The relationship between geopotential pressure altitude and temperature for the new model is shown in Figure 4-2. Figure 4-2: The relationship between geopotential pressure altitude and temperature for the new model In Figure 4-3 the relationship between geoptential altitude and temperature for the new model is given together with tropopause geopotential pressure altitude (dashed line, H p = [m]). Figure 4-3: The relationship between geopotential altitude and temperature for the new model All subsequent differences between the old and the new BADA atmosphere models are based on the above mentioned basic differences. 20 Project: BADA EEC Technical/Scientific Report No

37 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 5. IMPACT OF CHANGES ON BADA FAMILY 3 Implementation of the new atmospheric model described in previous sections implies modifications in some of the algorithms of the BADA Aircraft Performance Model family 3 [1]. In principle these modifications are the result of the introduction and implementation of proper relationship between different altitude types: geodetic, geopotential and geopotential pressure described in section 2.2 and 3.3. As such, they can be considered more as clarifications, which would help in reducing ambiguity and alignment with standard definitions, rather then the changes. In line with this, the formulation of the total energy model remains the same, while though a clear distinction is made between the definition of vertical speed and rate-of-climb/descent. Vertical speed is defined as the variation with time of the aircraft geodetic altitude. The assumption of a standard constant gravity field derives in identical geodetic and geopotential altitudes. dh dt 1 (T D)V TAS VTAS dvtas = 1 + (5-1) mg g dh The rate-of-climb/descent is defined as the variation with time of the aircraft geopotential pressure altitude Hp. It is the preferred way of presenting the performances of an aircraft as it eliminates possible variations caused by the atmospheric conditions: 1 dh p TISA (T D)V TAS VTAS dvtas ROCD = = 1 + (5-2) dt T mg g dh The formulation of energy share factor is modified to take into account proper altitude definitions. Table 5-1 provides a summary of all the changes in the BADA operations performance model affected by the new atmosphere model. Project: BADA EEC Technical/Scientific Report No

38 EUROCONTROL Revision of Atmosphere Model in BADA Aircraft Performance Model Table 5-1: New BADA atmosphere model impact Change BADA User manual revision 3.7 New BADA atmosphere model impact Formula (3.1-6) Formula (3.1-7) γ R k f {M} = g f {M} = 1 + T M 2 1 γ R k f {M} = g T M 2 T T 1 1 γ 1 γ γ R k 1 1 T 2 γ -1 2 γ γ -1 2 γ 1 1 M + 1+ M 1+ M 1 γ R k T 2 T T γ -1 2 γ γ -1 2 γ 2 g 2 2 f {M} = g M T T M 1+ 2 M (.3048) (6.5) Formula (3.2-18) h = [ T ( θ )] Maximum speed and altitude h MO Maximum speed and trans 0 1 trans geopotential pressure altitude must be used, H P,trans geopotential pressure altitude must be used, H P,MO geopotential pressure altitude must be used, H P,max altitude H max Formula (3.5-1) h MIN[ max/act = h MO,h max + G t ( TISA CTc,4 ) + G w ( m max mact ) ] geopotential pressure altitudes must be used, H Pmax, act Section Altitude > Section Altitude < Formula (3.7-1) 2 ( T ) = + max climb C ISA Tc,1 1- CTc,3 h CTc,2 h geopotential pressure altitude must be used geopotential pressure altitude must be used gepotential pressure altitude must be used: HP 2 T max climb = CTc, C ISA Tc,3 HP CTc,2 ( ) 22 Project: BADA EEC Technical/Scientific Report No

39 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL Change BADA User manual revision 3.7 New BADA atmosphere model impact Formula (3.7-2) Formula (3.7-3) Formula (3.7-9) h ( Tmax climb ) = CTc,1 1 - V + CTc,3 ( ) ISA C Tc,2 h C T max climb = C ISA Tc, C Tc,2 T C T TAS Tc,3 V TAS gepotential pressure altitude must be used: ( Tmax climb ) C P = Tc,1 1 - V + C Tc,3 ISA H C Tc,2 TAS gepotential pressure altitude must be used: H C P Tc,3 ( T max climb ) = C ISA Tc, C Tc,2 V TAS des, high = Tdes,high max climb The threshold altitude value must be geopotential Section 3.8 h < (0.8*hmax) Formula (3.8-2) Formula (3.9-4) dh dt f min pressure altitude, H P,des Geopotential pressure altitudes must be used. (Tmax,climb D) VTAS C pow, red dh T - T (Tmax,climb D) VTAS C pow, red = f { M } = f { M } mg dt T mg = C f3 1- h C f4 geopotential pressure altitude must be used: H P f = min Cf3 1- Cf4 Project: BADA EEC Technical/Scientific Report No

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41 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 6. BENEFITS OF IMPLEMENTING THE NEW ATMOSPHERE MODEL The most evident benefit of implementing the new atmosphere model is ensured consistency between the atmosphere model used by aircraft performance model and the one used by the aircraft manufacturer. From the aircraft performance modelling point of view the major advantages of this consistency are: 1. miss-modelling issues during optimisation of coefficients of an aircraft performance model are avoided 2. improved results in terms of aircraft model accuracy are obtained To this end, an assessment was done during development of the aircraft models for BADA 3.7. All newly developed aircraft models in BADA 3.7 (82) were identified using the new atmosphere model. The aircraft performances in terms of aircraft vertical speed and fuel flow obtained from the BADA 3.7 coefficients together with old and new atmosphere models were compared. This showed improved accuracy for 82% (67 of 82) of aircraft models in BADA 3.7 for non-isa conditions. The results for ISA conditions remain the same. Accuracy tables for new and old BADA atmosphere models are provided in the Appendix A. By implementing the new atmosphere model, the aircraft trajectory prediction tools would gain the benefits from improved aircraft performance model accuracy in vertical and horizontal plane for off ISA atmosphere conditions. Project: BADA EEC Technical/Scientific Report No

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43 Revision of Atmosphere Model in BADA Aircraft Performance Model EUROCONTROL 7. CONCLUSIONS AND RECOMMENDATIONS The introduction of the new atmosphere model improves consistency between aircraft performance related applications on the ground and in the air. It also provides a window of opportunity to increase accuracy of the aircraft performance models and simulated or predicted aircraft trajectories. All newly developed aircraft models in BADA 3.7, which were identified using the new BADA atmosphere model, demonstrated the advantages in doing so. The new atmosphere model shall be therefore employed as a standard from BADA 3.8 onwards model developments. It is worth emphasising that the aircraft model coefficients of BADA 3.7 and the future BADA family 3 versions are back compatible with the current (old) atmosphere model described in BADA 3.7 User Manual [1]. The user is given the opportunity to evaluate benefits and decide if and when the upgrade to the new atmosphere model will be done. What is important remembering is that for ISA conditions, the use of both atmosphere models would provide the same results. In the case of non-isa atmosphere conditions, the use of the new atmosphere model in average should provide the better results. Despite the compatibility of the BADA family 3 coefficients with two atmosphere models, the implementation of the new model is highly recommended. Note that a proper integration of the new BADA atmosphere model requires the implementation of all changes. Project: BADA EEC Technical/Scientific Report No