NOMENCLATURE

a _i, b _j, c _k

aerodynamic model coefficients (i = 1 to 3, j = 1 to 4, and k = 1 to 2)

a _i, b _i

reference thrust-specific fuel consumption based on altitude (i = 1 to 2)

A _i, E _i

aircraft engine type group (i = 1 to 6)

AETG

aircraft and engine type group

AIDS

aircraft integrated data system

ALT_STDC

standard corrected altitude

ARPP

aircraft range performance parameter

ATM

air traffic management

BADA

base of aircraft data

c

thrust-specific fuel consumption model constant

C _D

drag coefficient

C _D0

zero-lift drag coefficient

C _fi

reference thrust-specific fuel consumption model (i = 1 to 2)

C _fcr

reference thrust-specific fuel consumption for cruise

C _L

lift coefficient

D

drag force

DESTINATION

destination aerodrome

EASA

European Aviation Safety Agency

FDAU

flight data acquisition unit

FDR

flight data recorder

FBURN

total amount of fuel burned

FBURN1

amount of fuel burned in engine-1

FBURN2

amount of fuel burned in engine-2

FF-i

fuel flow rate (i = 1 to 3)

FF1C

fuel flow of engine-1

FF2C

fuel flow of engine-2

FLT_PATH

flight-path angle

FF-QAR

real fuel flow rate

GWC

mass of the aircraft

h

altitude

IATA

International Air Transport Association

ICAO

International Civil Aviation Organization

K, K _i

induced drag coefficients (i = 1 to 2)

L

lift force

M

Mach number

M_Ref

reference Mach number

MAPE

mean absolute percentage error

MDO

multidisciplinary design optimisation

n

drag polar lift coefficient exponent

ORIGIN

origin aerodrome

PITCH

pitch angle

QAR

quick access recorder

R

range

R _BB

basic Breguet range equation

R _BH

basic Hale range equation

R _E

enhanced Breguet range equation

R _QAR

real cruise range

S

wing area

T

thrust

TAIL_1

country tail code

TAIL_2

company tail code

TAS

true airspeed

TIME

flight time

TOC

top of climb

TOD

top of descent

TSFC

thrust-specific fuel consumption

TSFC-i

thrust-specific fuel consumption model (i = 1 to 3)

TSFC^H

reference thrust-specific fuel consumption in Howe’s model

TSFC^*

reference thrust-specific fuel consumption in Martinez-Val’s model

VTAS

true airspeed

W, W _i

weight of aircraft (i = 0 to 1)

Greek Symbols

λ

bypass ratio

Θ

relative temperature

Θ^*

reference relative temperature

β

thrust-specific fuel consumption Mach number exponent

Є ₀

compressor pressure ratio

ρ

atmospheric density

2.0 PROBLEM DESCRIPTION

A typical commercial flight operation consists of five basic phases: take-off, climb, cruise, descent and landing (Fig. 1). The cruise phase starts when the aircraft reaches the Top of its Climb (TOC) and ends at the Top of its Descent (TOD)^{(Reference Saarlas33)}. The cruise performance is critical to evaluate and understand the environmental and economic impacts of all aircraft^{(Reference Corda34)}. The cruise phase is not only the phase where commercial aircraft consume the most fuel but also the phase where 80% of the CO₂ emissions are produced⁽³⁵⁾. This study focuses on the estimation of cruise range, which is the primary indicator of the cruise performance of a commercial flight operation.

Figure 1. Typical flight profile.

The distance travelled during the cruise flight is called the cruise range. The general range equation can be expressed as follows^{(Reference Hale29)}:

(1) \begin{equation}R= \int_0^1 - \frac{{{V_{TAS}}}}{{{\rm{TSFC}}}}\frac{L}{D}\frac{{dW}}{W}. \end{equation}

In Equation (1), ‘0’ indicates the initial conditions for the cruise flight while ‘1’ indicates the final conditions. The term V _TAS is the true airspeed, L/D is the lift-to-drag ratio and W is the weight of the aircraft. Considering a cruise at a constant airspeed and constant lift coefficient, the integration of Equation (1) yields

(2) \begin{equation}R = \frac{{{V_{TAS}}}}{{{\rm{TSFC}}}}\frac{{{C_L}}}{{{C_D}}}\ln \frac{{{W_0}}}{{{W_1}}} \end{equation}

Equation (2) is the most commonly used cruise range formula, referred to as the Breguet range equation. It provides a quicker and more practical estimate for the cruise range compared with other range equations such as the Hale range Equation (29) (constant altitude and constant airspeed) and is thus generally used in range estimations during preliminary aircraft design, especially for cruise fuel weight predictions.

The Hale^{(Reference Hale29)} range equation (constant altitude and constant airspeed) can be derived from the following formula for a simple parabolic drag model:

(3) \begin{equation}R=2\frac{{{V_{TAS}}}}{{{\rm{TSFC}}}}\frac{1}{{2\sqrt {{C_{D0}}K} }}{\tan ^{ - 1}}\left[ {\frac{{{C_{D0}}}}{K}\frac{{\rho {V_{TAS}}^2S}}{{2{W_0}}}\frac{{1 - \frac{{{W_1}}}{{{W_0}}}}}{{\frac{{{C_{D0}}}}{K}{{\left( {\frac{{\rho {V_{TAS}}^2S}}{{2{W_0}}}} \right)}^2} + \frac{{{W_1}}}{{{W_0}}}}}} \right]\!. \end{equation}

In Equation (3), the terms C _D0, K, S and ρ are the parasitic drag coefficient, induced drag coefficient wing planform area and air density, respectively. This equation is complex, but when the argument of tan⁻¹ is a rather small angle, one can use the approximations

(4) \begin{equation}\theta \cong \tan \theta , \end{equation}

(5) \begin{equation}\theta \cong {\tan ^{ - 1}}\theta . \end{equation}

based on which Equation (3) can be expressed as

(6) \begin{equation}R \cong 2\frac{{{V_{TAS}}}}{{{\rm{TSFC}}}}\frac{1}{{2\sqrt {{C_{D0}}K} }}\left[ {\frac{{{C_{D0}}}}{K}\frac{{\rho {V_{TAS}}^2S}}{{2{W_0}}}\frac{{1 - \frac{{{W_1}}}{{{W_0}}}}}{{\frac{{{C_{D0}}}}{K}{{\left( {\frac{{\rho {V_{TAS}}^2S}}{{2{W_0}}}} \right)}^2} + \frac{{{W_1}}}{{{W_0}}}}}} \right]. \end{equation}

The most important feature of the Breguet range equation is that it combines three main terms that capture the performance of an aircraft: TSFC, VL/D and ln(W ₀/W ₁), referring to the aircraft’s engine, aerodynamic and structural performance, respectively. Accurately calculating the range performance of an aircraft is critical to ensure flight economy and protect the environment; consequently, two of these main performance groups, namely the engine and aerodynamic performance in the range equation, are examined in this study.

The engine performance of an aircraft is typically described in terms of the total engine thrust (T) and fuel flow rate (FF). Both of these vary significantly according to the type and design configuration of the engines used in the aircraft. The TSFC can be considered to be the most critical performance metric because it combines both of these engine performance indicators. The TSFC is the amount of fuel spent per unit time per unit reaction force, being formulated as follows:

(7) \begin{equation}{\rm TSFC}=\frac{1}{T}\cdot {\rm FF} \end{equation}

Different TFSC models are used for each flight phase. It is possible to calculate TSFC values for the cruise flight phase by correctly selecting the thrust-specific fuel consumption models. The TSFC model proposed by How et al.^{(Reference Howe23)} is one of the important models used for determining the thrust-specific fuel consumption in the cruise phase. In this model, the TSFC is formulated as a function of the engine bypass ratio (λ), Mach number and temperature ratio (ɵ) as follows:

(8) \begin{equation}{\rm TSFC}={{\rm TSFC}}^H\left(1-0.015{\lambda }^{0.65}\right)[\left(1+0.28\left(1+0.063{\lambda }^2\right){\rm M}\right)\theta^{0.08}, \end{equation}

In Equation (8), TSFC^H is a factor which can be determined by reference to the real TSFC of a given engine at a datum condition. The values of TSFC^H are 0.95N/N/h for supersonic engines, 0.85N/N/h for low-bypass engines and 0.70N/N/h for high-bypass turbofan engines.

Martinez-Val et al.^{(Reference Martinez-Val and Perez26)} proposed a TSFC model depending on the ratios of the temperature and Mach number to reference values provided by the engine manufacturer.

(9) \begin{equation}{\rm{TSFC}} = {\rm{TSF}}{{\rm{C}}^*}\frac{{{{\rm{M}}^{\;\beta }}}}{{{{\rm{M}}^*}^{\;\beta }}}\sqrt {\frac{\theta }{{{\theta ^*}}}.} \end{equation}

The terms marked with an asterisk are the specific fuel consumption during the flight, the Mach number and the relative temperature given by the engine manufacturer. The value of β ranges from 0.2 to 0.4 for low-bypass turbofan engines, and from 0.4 to 0.7 for high-bypass turbofan engines.

Roux^{(Reference Roux24)} proposed a TSFC model including the altitude (h), Mach number (M), temperature ratio (ɵ), bypass ratio (λ) and compressor pressure ratio (${ \in _c}$):

(10) \begin{equation}{\rm TSFC}=\left[\left(a_1(h)\lambda +a_2(h)\right){\rm M}+\left(b_1(h)\lambda {\rm +}\ \ b_2(h)\right)\right]\sqrt{\uptheta}+c\left({\in }_c-30\right) \end{equation}

In Equation (10), the values of the coefficients a ₁, a ₂, b ₁ and b ₂ vary depending on the altitude, while c is a constant. The above-mentioned TSFC models represent immensely accurate and simple solutions for cruise range calculations. Apart from these models, Eurocontrol’s Base of Aircraft Data (BADA)⁽³⁶⁾ also includes a TSFC model. The BADA modelling system relies on historical data and statistical analysis to calculate the performance parameters for the climb, cruise and descent phases, including a coefficient, C _fcr, to calculate TSFC values in the cruise phase. The following expression is the TSFC model proposed by BADA, illustrating that it depends only on true airspeed⁽³⁶⁾:

(11) \begin{equation}{\rm{TSFC}} = {C_{fcr\;}}{C_{f1}}\left( {1 + \frac{{{V_{TAS}}}}{{{C_{f2}}}}} \right). \end{equation}

Regardless of which TSFC model is used, accurate range prediction also requires correct values of L/D to be determined, which depends on an accurate representation of the drag polar equation. The wing profile of the aircraft should be carefully examined while creating this drag polar. It is possible to divide the wing profiles used on aircraft into two types: symmetrical (uncambered) and cambered. In fact, the wing profiles of aircraft used in commercial transportation are cambered. The difference between the symmetrical and cambered wing profile is shown in Fig. 2.

Figure 2. Lift and drag coefficient curves for symmetrical and cambered wing profiles^{(Reference Brandt, Bertin, Stiles and Whitford37)}.

In cruise configurations at low subsonic speeds, the drag coefficient (C _D) is expressed as a function of the aerodynamic lift coefficient (C _L). When these effects are included in the model, the simple parabolic drag model of the aircraft can be written as

(12) \begin{equation}{C_D}\left( {{C_L}} \right) = {C_{D0}} + K{\left( {{C_L}} \right)^2}. \end{equation}

This equation can be considered to be valid for Mach numbers below 0.6. Consequently, the drag polar is almost exactly linear. In other words, Mach numbers of 0.6 and below can be ignored in the drag polar expression. The drag polar model used by BADA is a simple parabolic drag model⁽³⁶⁾. Most modern aircraft, however, fly faster than Mach 0.6. Therefore, it is not correct to ignore the effect of the Mach number on the drag polar. Drag polars for Mach numbers above 0.6 are represented by the following equation^{(Reference Young38)}:

(13) \begin{equation}{C_D}\left( {{C_L},{\rm{M}}} \right) = {C_{{D_0}}}\left( {\rm{M}} \right) + {K_1}\left( {\rm{M}} \right){C_L}^2 + {K_2}\left( {\rm{M}} \right){C_L}^n. \end{equation}

where C _D0, K ₁ and K ₂ vary with the Mach number. The exponent n in this equation can be found by curve fitting to experimental data. To illustrate this, the value of the exponent n is 4 for fighter aircraft. Curve fitting the drag data for a commercial aircraft results in a good solution when using the exponent n = 6^{(Reference Young38)}. As mentioned above, the effects of the Mach number (M) and cambered wing should be taken into account when constructing the drag polar equations for commercial aircraft. In this study, aircraft equipped with narrow-body, turbofan engines that are used for short-distance domestic subsonic flights and commercial transportation services were examined. The results confirm that such planes fly at speeds of Mach 0.6 and above during the cruise phase. In conclusion, modern commercial aircraft have cambered wings and turbofan engines with a high bypass ratio. Due to these characteristics, the use of a constant TSFC and a simple parabolic drag polar is not appropriate to calculate the cruise range and cruise flight conditions.

3.0 DATA PROPERTIES

The QAR data were obtained from various types of turbofan-engined narrow-body aircraft performing short-distance domestic subsonic flights. The data include six different aircraft and engine type groups referred as to A ₁E ₁, A ₂E ₂, A ₃E ₃, A ₄E ₄, A ₅E ₅ and A ₆E ₆ in this work. These groups belong to two different manufacturers, designated as group 1 and group 2 in Fig. 3. A total of 6,574 flights between 31 city pairs were considered in this analysis. The frequency of the aircraft and engine type groups and city pairs in the QAR data are shown Figs 3 and 4, respectively.

Figure 3. Frequency of aircraft and engine type groups and in the QAR data.

Figure 4. City pairs and frequency in the QAR data.

The QAR data include a total of 58 parameters regarding a single flight, but only 18 of them are used in the cruise range calculations. These parameters are classified as time, mass, engine (fuel flow rates and amount of fuel burned), kinematical and dynamic parameters (altitude, pitch and flight path angles), atmospheric (Mach number, true airspeed, static and total temperature) and other parameters (country and company tail codes, origin and destination aerodromes). The types, physical meanings, symbols, units and sampling rates of these parameters are presented in Table 1.

Table 1 The parameters in the QAR data and their descriptions

Descriptive statistics for the cruise range values obtained using the QAR data are presented for each aircraft and engine type group in Table 2. Depending on the stage length of the flight, the cruise range varies between 59 and 624nm. Cruise flights lasting less than 10min (600s) are usually very short level-offs before climbing to the final cruise altitude or before commencing a descent. Therefore, such flights were excluded from our study to perform an accurate analysis.

Table 2 Cruise range values in the QAR data

All cruise flights identified within the QAR data took place between altitudes of 20,026 and 41,040ft, depending on the flight stage length, aircraft performance, direction of the flight (i.e. eastbound or westbound) and air traffic density (Table 3). For all aircraft and engine type groups, the mean and median cruise altitude values were very close to standard flight levels (FL320 and FL330). The standard deviations also indicate that the majority of the cruise flights (about 70%) took place at three or four flight levels (3,000–4,000ft) above or below the mean cruise levels. Table 4 presents the distribution of the cruise Mach number of the flights in each aircraft and engine type group. Except for a few outliers, all flights took place within the compressible speed regime. The mean cruise Mach numbers were around 0.72, while the maximum Mach numbers reached up to 0.805, very close to the maximum operational Mach number (MMO) of the given aircraft types. The mean, standard deviation, minimum, median and maximum values of the final-to-initial weight ratio (W ₁/W ₀) are presented in Table 5 for all the aircraft types. As can be seen from these data, the weight ratio values are relatively high, with means ranging between 0.977 and 0.987. In other words, the cruise fuel ratios (W _{fuel_spent}/W ₀) are low for the QAR dataset considered. This is mainly because the data include short-range flights, thus the effect of W ₁/W ₀ is very low.

Table 3 Altitude distribution in the QAR data

Table 4 Cruise Mach number distribution in the QAR data

Table 5 Final to initial weight ratio (W ₁/W ₀) distribution in QAR data

Although angle-of-attack is not directly provided in the QAR dataset, it can be calculated as the difference between the flight path angle and pitch angle during flight. In the analysis of the cruise phase of the flights, no significant variation was observed in the angles of attack or L/D ratios. This is mainly because the cruise ranges considered in this study were relatively short, i.e. between 50 and 600nm. Therefore, variations in the L/D ratio were not considered in the analysis.

4.0 METHODOLOGY

All the flight data were examined and processed in three stages. First, the data were imported from the Excel files. Second, the data were processed, and mathematical models were generated for the drag polar using a code written in MATLAB. Corrupt or missing data were detected and filtered using this code. In the final stage, the processed data were exported to output files in Excel format to perform statistical analysis in MINITAB 19 software and generate tables and figures. The steps of the code used for the estimation of the cruise performance parameters are as follows:

Figure 5. Block diagram for selection between TSFC models (Steps 1–3.1).

• Step 1 Determine the cruise flight phase of each flight in the QAR data

• Step 2 Identify and classify the aircraft and engine types

• Step 3 Generate and calculate the cruise performance model:

  • Step 3.1 Calculate the thrust-specific fuel consumption values

    • Step 3.1.1 Calculate the thrust for simple drag polar and an uncambered wing

    • Step 3.1.2 Calculate the fuel flow rate (FF) for each thrust-specific fuel consumption model

    • Step 3.1.3 Select the thrust-specific fuel consumption model

  • Step 3.2 Generate the drag polar model:

    • Step 3.2.1 Calculate the thrust values using the selected thrust-specific fuel consumption model

    • Step 3.2.2 Calculate the drag polar values considering the effects of camber and compressibility

    • Step 3.2.3 Generate the drag model for the lift coefficient and Mach number as independent variables

• Step 4 Verify the aerodynamic performance model of the different aircraft and engine type groups

• Step 5 Calculate R _E (the Breguet cruise range based on the estimated drag and selected TSFC model using the QAR data), R _BB (the basic Breguet cruise range based on simple drag polar and the TSFC model from BADA) and R _BH (the basic Hale cruise range based on simple drag polar and the TSFC model from BADA).

4.1. Selection of the TSFC model

A flowchart describing Steps 1–3.1 is presented in Fig. 5. The code first checks the flight path angle, altitude, and flight time (FLT_PTH, ALT_STDC and TIME) parameters of each flight to determine the top of climb and top of descent times of the cruise. The code only assigns level flights (with zero flight path angle) at or above 20,000ft as the cruise phase.

Table 6 Performance characteristics of engines

Figure 6. Block diagram for the drag polar model (Step 3.2).

After determining the cruise phases, it is necessary to identify the aircraft and engine types in order to estimate the TSFC values accurately. These data are not available in the QAR data and were thus obtained indirectly using tail codes (TAIL_1) and company tail codes (TAIL_2) from Airfleets⁽³⁹⁾. Based on this aircraft and engine type groups information, type-specific parameters such as the reference TSFC and Mach numbers (TSFC^* and M^*), and the bypass ratio (λ) were obtained from manufacturers’ published data. Table 6 presents the reference engine characteristics provided by manufacturers^{(Reference Roux40)}. To calculate the thrust-specific fuel consumption values of the cruise flight phase, it is necessary to select thrust-specific fuel consumption models suitable for this flight phase. In Step 3, the code first selects the TSFC that most closely fits the QAR fuel flow rate (FF) data through Steps 3.1.1 to 3.1.3. For each cruise flight phase, three TSFC values are estimated in Step 3.1.1 using the models of Martinez-Val et al.^{(Reference Martinez-Val and Perez26)} (TSFC-1), How et al.^{(Reference Howe23)} (TSFC-2) and Roux^{(Reference Roux24)} (TSFC-3) described in Equations (8)–(10), respectively.

Once the TSFC values have been obtained, thrust values are estimated using level-flight conditions at the given cruise altitude and true airspeed:

(14) \begin{equation}T = D = \frac{1}{2}\rho {V_{TAS}}^2S{C_D}.\end{equation}

In Equation (14), the air density (ρ) is calculated for the given pressure altitude from the QAR data and the aircraft wing planform area obtained from the aircraft manufacturer. In this step, a simple parabolic drag polar model in Equation (12) is assumed to estimate the thrust (T). After estimating the thrust values, in Step 3.1.2 the fuel flow rate values FF₁, FF₂ and FF₃ are calculated using Equation (7) for TSFC₁, TSFC₂ and TSFC₃, respectively. In Step 3.1.3, the MAPE is estimated for each fuel flow rate with reference to the real fuel flow rate (FF_QAR) from the QAR data:

(15) \begin{equation}MAP{E_i} = \frac{{100}}{N}\mathop \sum \limits_{j=1}^N \left| {\frac{{F{F_{QAR}} - F{F_{ij}}}}{{F{F_{QAR}}}}} \right|\left( {i=1,2,3} \right).\end{equation}

The TSFC resulting in the lowest MAPE in Equation (15)^{(Reference De Myttenaere, Golden, Le Grand and Rossi41)} is selected as the TSFC to be used in the generation of the drag polar model from the QAR data.

4.2 Estimation of the drag polar model

After choosing the appropriate TSFC model (TSFC-X), the next step is the generation of the aerodynamic performance model in Step 3.2. Using Equation (7), the thrust values are calculated for the TSF-X and the real fuel flow rate values for the QAR data. To perform the aerodynamic modelling for the aircraft, the drag polar equation must be formed, taking the effects of the Mach number and cambered wing into account for commercial aircraft. In Step 3.2.1, the drag coefficients (C _D) are estimated using the TSFC-X and QAR data, including the fuel flow rate, altitude and true airspeed for each cruise phase. As suggested in Young^{(Reference Young38)}, the most appropriate form for the drag polar is a sixth-order polynomial when considering commercial aircraft flying at or above Mach 0.6. Therefore, the drag polar model is formulated as follows:

(16) \begin{equation}{C_D}\left( {{C_L},M} \right) = {C_{{D_0}}}(M) + \left( {{K_1}(M)} \right){C_L}^2 + \left( {{K_2}(M)} \right){C_L}^6,\end{equation}

In Equation (16), C _D0, K ₁ and K ₂ are expressed as functions of the Mach number based on regression analysis of the QAR data:

(17) \begin{equation}{C_{D0}}(M) = {a_1} + {a_2}{\left( {M - {M_{Ref}}} \right)^2} + {a_3}{\left( {M - {M_{Ref}}} \right)^3},\end{equation}

(18) \begin{equation}{K_1}(M) = {b_1} + {b_2}{\left( {M - {M_{Ref}}} \right)^2}\; + {b_3}{\left( {M - {M_{Ref}}} \right)^3} + {b_4}{\left( {M - {M_{Ref}}} \right)^4}, \end{equation}

(19) \begin{equation}{K_2}(M) = {c_1}\left( {M - {M_{Ref}}} \right) + {c_2}{\left( {M - {M_{Ref}}} \right)^2}.\end{equation}

In Equation (17)–(19), the reference Mach number (M_Ref) is chosen as 0.6. The coefficients a ₁, a ₂, a ₃, b ₁, b ₂, b ₃, b ₄, c ₁ and c ₂ in the drag polar expression are found by using the nonlinear least-squares method, which is an optimisation model. To check the goodness of fit of the resulting regression model, R ² values were also calculated. R ² takes values between 0 and 1. An R ² value of 1 indicates that the regression model accounts for nearly all of the variability, where R ² = 0 denotes that the regression model cannot explain the variability^{(Reference Hagquist and Stenbeck42)}. Finally, the improved drag polar model is generated using regression analysis between C _D, as the dependent variable, and M and C _L, as the independent variables. After obtaining the parameters for the drag coefficients, the drag polar model is verified on the different aircraft and engine type groups.

4.3 Evaluation of cruise ranges

In the final step of this study, a range model is proposed to estimate more accurate range values. After the examination of different TSFC models, the TSFC with the minimum MAPE is implemented in the Breguet equation. Moreover, a new high-order polynomial drag polar model is generated and included in the range equation to include the effects of camber and compressibility using real flight data. This enhanced range model is expressed as R _E;

(20) \begin{equation}{R_E} = \frac{{{V_{TAS}}}}{{{\rm{TSFC}}\left( {M,h} \right)}}\frac{{{C_L}}}{{{C_D}\left( {{C_L},\;M} \right)}}\ln \frac{{{W_0}}}{{{W_1}}}.\end{equation}

To evaluate the accuracy of the new range model, two existing range equations are also estimated based on aerodynamic and propulsive data provided by Eurocontrol’s BADA, which is a database used to determine aircraft performance, including more than 300 different types of aircraft. Data in this model are derived from various aircraft documents provided by aircraft manufacturers or operators. It is necessary to point out, however, that the BADA aerodynamic model does not consider the effects of compressibility on the aerodynamic behaviour of the aircraft. Hence, BADA uses a simple parabolic drag model. In BADA, TAS is the only variable affecting the TSFC model⁽³⁶⁾. The first range equation is referred to as the basic Breguet range equation and is formulated using Equation (21), along with simple drag polar coefficients and the TSFC model from BADA:

(21) \begin{equation}{R_{BB}} = \frac{{{V_{TAS}}}}{{{\rm{TSFC}}\left( {{V_{TAS}}} \right)}}\frac{{{C_L}}}{{{C_D}\left( {{C_L}} \right)}}\ln \frac{{{W_0}}}{{{W_1}}}.\end{equation}

The second range equation is referred to as the basic Hale range equation, based on linearised constant-altitude and constant-airspeed cruise, in Equation (6):

(22) \begin{equation}{R_{BH}} \cong 2\frac{{{V_{TAS}}}}{{{\rm{TSFC}}\left( {{V_{TAS}}} \right)}}\frac{1}{{2\sqrt {{C_{D0}}K} }}\left[ {\frac{{{C_{D0}}}}{K}\frac{{\rho {V_{TAS}}^2S}}{{2{W_0}}}\frac{{1 - \frac{{{W_1}}}{{{W_0}}}}}{{\frac{{{C_{D0}}}}{K}{{\left( {\frac{{\rho {V_{TAS}}^2S}}{{2{W_0}}}} \right)}^2} + \frac{{{W_1}}}{{{W_0}}}}}} \right].\end{equation}

In Equation (22), the values of the coefficients C _D0 and K of the simple drag model and the TSFC model are also taken from BADA for the considered aircraft and engine type groups.