Nomenclature

A,B

coefficients in Equation (16)

A _e

sum of the engine core and bypass jet exit cross-sectional areas, multiplied by the number of engines

A _pax

reference area for a typical passenger in a single class cabin=0.65 m²

AR

wing aspect ratio

BAM

basic aircraft mass (operational empty mass – mass of operational items)

C _d

airframe drag coefficient=D/(0.5γp _∞ (M _∞) ² S _ref)

Cd _o

zero-lift drag coefficient

C _L

lift coefficient=mg/(0.5γp _∞ (M _∞) ² S _ref)

C _t

thrust coefficient=F _n /(0.5γp _∞ (M _∞) ² A _e)

D

drag force

e

aircraft Oswald efficiency factor

ETRW

ratio of energy used for a trip to revenue work done

FL

flight level

FM

fuel mass

F _g

gross thrust, summed over all engines

F _n

net thrust, summed over all engines

G

(C _L )_ηLDm $M_{\infty}^{2} $

g

acceleration due to gravity (9.81 m/s² at sea level)

h

true altitude relative to mean sea level

IA

indicated, or ‘pressure’, altitude

IFL

ideal flight level

L

lift force

LCV

lower calorific value of fuel (≈ 43.10⁶ J/kg for kerosene)

L/D

aircraft lift-to-drag ratio

LM

landing mass

m

instantaneous total aircraft mass

$\dot{m}_{{{\rm air}}} $

total mass flow rate of air entering engine intakes

$\dot{m}_{f} $

total instantaneous fuel flow rate to the engines

M _ref

reference mass for a ‘typical’ passenger plus normal baggage allowance (= 95 kg)

M

Mach number

MEM

manufacturer’s empty mass (=BAM)

MFM

maximum total fuel mass

MLM

maximum permitted landing mass

MPM

maximum payload (passengers+cargo) mass (=MZFM−OEM)

MTOM

maximum permitted take-off mass

MZFM

maximum zero fuel mass (maximum permitted aircraft mass without fuel)

n

ratio of (η _o L/D)_avg to (η _o L/D)_opt

OEM

aircraft operational empty mass (mass of aircraft without payload and without fuel)

OIM

mass of the operational items

PM

payload mass (passengers+cargo)

p _∞

ambient static pressure

p _o

total pressure – see Equation (A.1)

p _SL

static pressure at sea level (1,013.25 hPa in the ISA)

R

distance measured along the great circle connecting the departure and destination points, with the origin at the departure point

ℜ

gas constant for air (287 J/kg/K)

S

distance measured along the ground track connecting the departure and destination points, with the origin at the departure point

S _ref

aircraft reference wing area (gross wing plan area)

T _∞

ambient static temperature

T _o

total temperature – see Equation (A.2)

TFM

mass of the trip fuel (fuel burned between ‘brakes off’ at take-off and ‘brakes off’ at the end of the landing run)

TOM

total aircraft mass at the start of the take-off run

V _∞

true air speed

V _cw

component of wind speed normal to the ground track (crosswind)

V _gt

speed along the ground track

V _hw

component of wind speed parallel to the ground track (headwind)

W

headwind speed normalised with true air speed (V _hw/V _∞)

X

non-dimensional great circle distance (R.g/(LCV.(η _o L/D)_opt))

ZFM

zero fuel mass (mass of aircraft, including payload ,but without any fuel)

α

trip fuel mass/aircraft take-off mass

β

mass of fuel carried, but not to be consumed during flight (= reserve, contingency and tankered)/aircraft take-off mass

Γ

extra distance travelled relative to great circle track per unit great circle distance travelled (= ΔR/R)

γ

ratio of specific heats for air (=1.4)

γ_c

average climb gradient

γ_d

average descent gradient

ΔR

difference between ground track length and great circle length between two points

ΔX

non-dimensional ΔR

ε_cl

climb ‘lost’ fuel index – see Equation (D.3).

ε_dl

descent ‘recovered’ fuel index – see Equation (D.6)

ε _t

overall ‘lost’ fuel index – Equation (D.11)

η _o

propulsion system overall efficiency

2.0 BACKGROUND

In standard aircraft design and performance texts, e.g. Refs 12–14, the material is usually presented in terms of fundamental quantities such as Mach number, dynamic pressure, engine overall efficiency, lift coefficient and drag coefficient. However, some of the data used in this analysis come from sources intended for use by flight crew, e.g. flight crew operating manuals (FCOMs), and these contain terminology and vocabulary that may be unfamiliar. Aircrew work with direct measurements of total pressure, total temperature and static pressure and they merge these with weather information and air traffic directives to manage the ground track, time to destination, fuel usage and separation from other traffic. They describe the performance of the aircraft in terms of Mach number, indicated air speed (IAS), calibrated air speed (CAS), true air speed (TAS) and conceptual parameters such as indicated altitude (IA) and flight level (FL). Therefore, data from aircraft manuals usually require translation into the more familiar, fundamental parameters. Consequently, the definitions of the various terms and the relationships linking them are discussed in detail in Appendix A.

3.0 FUNDAMENTAL CONSIDERATIONS

The overall propulsion efficiency is defined as

(1) $${\rm \reta }_{o} {\equals}{{F_{n} .V_{\infty} } \over {\dot{m}_{{\rm f}} .{\rm LCV}}},$$

where F _n is the total delivered, or net, thrust from the engines, V _∞ is the true airspeed, $\dot{m}_{f} $ is the total instantaneous fuel flow rate and LCV is the lower calorific value of the fuel (43.10⁶ J/kg for kerosene). Hence, in straight and level flight at constant speed, the instantaneous fuel flow rate is

(2) $$\dot{m}_{f} {\equals}{\minus}{{{\rm d}m} \over {{\rm d}t}}{\equals}{{D.V_{\infty} } \over {{\rm \reta }_{o} .{\rm LCV}}}{\equals}{{mg} \over {\left( {{\rm \reta }_{o} L\,/\,D} \right)}}{{V_{\infty} } \over {{\rm LCV}}},$$

where L is the lift, D is the drag, m is the instantaneous total mass of the aircraft and g is the acceleration due to gravity. When the aircraft is following a prescribed ground track, whilst being subjected to an opposing wind whose parallel to track (headwind) and normal to track components (crosswind) are V _hw and V _cw respectively, the resultant speed along the ground track, V _gt, is

(3) $$V_{{{\rm gt}}} {\equals}{{{\rm d}S} \over {{\rm d}t}}{\equals}V_{\infty} \left( {1{\minus}\left( {{{V_{{{\rm cw}}} } \over {V_{\infty} }}} \right)^{2} } \right)^{{{1 \over 2}}} {\minus}V_{{{\rm hw}}} ,$$

where S is the distance travelled. However, provided that V _cw is less than 15% of the aircraft’s true airspeed (roughly 70 kn in practice), the crosswind component may be ignored and, with no restriction placed upon the magnitude of the headwind, the fuel consumption per unit distance travelled along the ground track is

(4) $${{{\rm d}m_{f} } \over {{\rm d}S}}{\equals}{\minus}{{{\rm d}m} \over {{\rm d}S}}\,\approx\,{{mg} \over {\left( {{\rm \reta }_{o} L\,/\,D} \right){\rm LCV}}}\left( {1{\minus}W} \right)^{{{\minus}1}} ,$$

where W is the ratio of headwind speed to the aircraft’s true airspeed. Clearly, for a given aircraft, the required fuel consumption per unit distance travelled is smallest when (η _o L/D) is as large as possible.

In straight and level flight at a constant speed, the total thrust, F _n , is equal to the aircraft drag. Therefore, thrust and, consequently, overall efficiency depend on those parameters that govern the drag. Hence, for a given aircraft and engine combination, the product of the engines’ overall efficiency and the airframe’s lift-to-drag ratio (η _o L/D), is, primarily, a function of the flight Mach number, M _∞, and the lift coefficient, C _L.Footnote ² This is illustrated in Fig. 1, which shows these variations for the Douglas DC-10-10. The curves are cross plots of data taken from the aircraft’s flight crew operating manual (FCOM), reproduced by Shevell (Ref. 12, Fig. 15.17). The data show that curves of (η _o L/D) versus lift coefficient at fixed Mach number and versus Mach number at fixed lift coefficient both exhibit maxima. Hence, each aircraft has an absolute maximum, or optimum, value of (η _o L/D) occurring at a particular combination of flight Mach number, (M _∞)_opt, and lift coefficient, (C _L )_opt. In the case of the DC-10-10, (η _o L/D)_opt is 4.86 when M _∞ is 0.80 and C _L is 0.48. Flight at this condition requires the absolute minimum amount of fuel to be consumed per unit distance travelled over the ground. Conversely, for a given quantity of fuel, the aircraft covers the largest possible distance. Hence, this particular Mach number and lift coefficient pair are termed the ‘maximum range cruise’, or MRC, values.

Figure 1 The variation of (η _o L/D) with lift coefficient at constant Mach number for the Douglas DC-10-10 aircraft, after Shevell (Ref. 12).

Differentiating Equation (4) gives the sensitivity of fuel consumption per unit distance flown to changes in headwind, i.e.

(5) $${{{\rm d}\left( {{\rm d}m_{{\rm f}} \,/\,{\rm d}S} \right)} \over {{\rm d}m_{{\rm f}} \,/\,{\rm d}S}}\,\approx\,{W \over {\left( {1{\minus}W} \right)}}{{{\rm d}W} \over W}.$$

The sensitivity decreases as the true airspeed increases and, since weather conditions may change en route, it is prudent to plan the fuel requirement on the basis of a cruising speed that is somewhat higher than the MRC value, i.e. some fuel is sacrificed for higher speed. This is the basis for the use of ‘long-range cruise’, or LRC, conditions frequently encountered in operations literature. The LRC Mach number is defined as the largest Mach number at which (η _o L/D) is 99% of (η _o L/D)_opt. Referring again to Fig. 1, this definition delivers a unique Mach number and lift coefficient pair, which for the DC-10-10 are, 0.83 and 0.45, respectively. Hence, a 1% fuel sacrifice delivers about a 3.5% increase in cruise Mach number. Alternatively, if viewed as a potential time saving, this would give more than 20 min on a 10 h flight. Shorter flight times reduce time-based maintenance costs and may help with flight scheduling.

The maximum cruise Mach number is reached when either the engines are operating at the maximum cruise rating, or the achieved Mach number reaches the regulated maximum permitted value. The relevant regulated speed is the maximum operating Mach number, M _MO. According to Schaufele (Ref. 13), M _MO is usually equal to, or slightly above (by 0.01 or 0.02), the aircraft’s design cruise Mach number, the latter being determined by either maximum manoeuvre loads, maximum gust loads or buffet onset. In the case of the DC-10-10, M _MO is 0.88, which gives a typical margin of 0.08 over M _MRC and 0.05 over M _LRC. In routine operations, the fastest condition is the ‘catch up’, or high-speed cruise, Mach number, M _HSC, which is about 0.02 above M _LRC.

Referring once again to Fig. 1, it can also be seen that, as the flight Mach number increases, the lift coefficient for maximum (η _o L/D), (C _L )_ηLDm, decreases. Moreover, to a good approximation, the expression

(6) $$\left( {C_{{\rm L}} } \right)_{{{\rm \reta LDm}}} M_{\infty}^{2} \,\approx\,0.305$$

captures the relevant C _L values for Mach numbers in the range 0.80–0.86, i.e. the normal operating range for the aircraft. Since the variation is driven, primarily, by changes in wave drag and engine overall efficiency, this simple relation cannot be exactly true. Nevertheless, it is very useful simplification when analysing cruising performance.

4.0 GENERAL RELATIONSHIPS

For the past 50 years, market demands and a strictly controlled operating environment have constrained aircraft configuration development. Consequently, design changes have been, and continue to be, largely incremental. The implication is that the overall performance of aircraft, both old and new, can be captured by the same key parameters and that any deviations from the characteristics governed by these parameters ought to be small. This hypothesis can be tested by using the DC-10-10 as the baseline.

In compressible flow, but in the absence of wave drag, an aircraft’s cruise drag polar may be approximated by

(7) $$Cd{\equals}{D \over {\left( {{\rm \rgamma }\,/\,2} \right)p_{\infty} M_{\infty}^{2} S_{{{\rm ref}}} }}\,\approx\,Cd_{o} {\plus}\left( {{1 \over {{\rm \rpi }.AR.e}}} \right)C_{{\rm L}}^{2} ,$$

where S _ref is the aircraft’s gross wing area, Cd _o is the zero-lift drag coefficient, AR is the wing aspect ratio and e is the Oswald efficiency factor. All these quantities are dependent on the aircraft geometry. In addition, the Oswald efficiency factor has a very weak dependency upon Mach number – see for example Ref. 15.Footnote ³ Noting that L/D (= C _L/C _D ) has a maximum when the lift coefficient, (C _L)_LDm, is equal to (π.AR.e.Cd _o )^0.5, it can be shown that

(8) $${{\left( {L\,/\,D} \right)} \over {\left( {L\,/\,D} \right)_{{{\rm max}}} }}{\equals}{{2\left( {C_{{\rm L}} \,/\,\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} } \right)} \over {1{\plus}\left( {C_{{\rm L}} \,/\,\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} } \right)^{2} }}$$

and

(9) $${{Cd} \over {\left( {Cd} \right)_{{{\rm LDm}}} }}{\equals}{{C_{t} } \over {\left( {C_{t} } \right)_{{{\rm LDm}}} }}{\equals}{{1{\plus}\left( {C_{{\rm L}} \,/\,\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} } \right)^{2} } \over 2}.$$

Therefore, normalising the lift coefficient with the value for maximum lift-to-drag ratio removes all the aircraft specific, geometric parameters. Furthermore, when C _L is close to (C _L)_LDm, it can be shown that

(10) $${{\left( {L\,/\,D} \right)} \over {\left( {L\,/\,D} \right)_{{{\rm max}}} }}\,\approx\,1{\minus}0.5\left( {{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} }}{\minus}1} \right)^{2} {\plus}0.5\left( {{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} }}{\minus}1} \right)^{3} $$

and

(11) $${{Cd} \over {\left( {Cd} \right)_{{{\rm LDm}}} }}\,\approx\,{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} }}.$$

As demonstrated in Appendix B, for a gas turbine powered aircraft, the overall propulsion efficiency depends only on the flight Mach number and the total delivered thrust coefficient,

(12) $$C_{t} {\equals}{{F_{n} } \over {\left( {{\rm \rgamma }\,/\,2} \right)p_{\infty} M_{\infty}^{2} A_{{\rm e}} }},$$

where A _e is the sum of the core and bypass jet exit cross-sectional areas multiplied by the number of engines. In general, at a given Mach number, an engine will have a maximum overall efficiency, (η _o )_max, at a particular value of the thrust coefficient, (C _t )_ηm. An example of the variation of η _o with C _t for an existing engine is given in Fig. B1 of Appendix B. In general, the maximum value of η _o is observed to be, primarily, a function of the Mach number. The relationship has a near ‘power law’ form, with the exponent being a function of the bypass ratio and, typically, having a value in the range 0.6–0.7, see Ref. 16. For the example given in Appendix B, the exponent is about 0.65. This being the case, normalising η _o and C _t with their maximum values will remove much of the Mach number dependence. Figure 2 shows the effect for the engine used in Appendix B.

Figure 2 The variation of normalised engine overall efficiency with normalised thrust coefficient and Mach number for an existing turbofan engine, derived from Fig. 8.2 of Ref. 28.

In this variation, there is a clear speed effect that is significant for Mach numbers above 0.6, with the engine efficiency increasing as Mach number increases. This may be due to compressibility, or to variations in the inlet and nozzle operating efficiencies. However, the improvement is always less than 5% for normalised thrust coefficients between 0.6 and 1.2. Therefore, to a good approximation, normalising the efficiency and the corresponding thrust coefficient with (η _o )_max and (C _t )_ηm, respectively, produces a near single curve for Mach numbers below 0.6, i.e.

(13) $$\matrix{\displaystyle {{{{\rm \reta }_{o} } \over {\left( {{\rm \reta }_{o} } \right)_{{{\rm max}}} }}} \hfill & \,\approx\, \hfill & {{\rm function}\left( \displaystyle{{{C_{t} } \over {\left( {C_{t} } \right)_{{{\rm \reta m}}} }}} \right)} \hfill \cr {} \hfill & \,\approx\, \hfill & {1{\minus}0.50\left( \displaystyle{{{C_{t} } \over {\left( {C_{t} } \right)_{{{\rm \reta m}}} }}{\minus}1} \right)^{2} {\plus}0.10\left( {{{C_{t} } \over {\left( {C_{t} } \right)_{{{\rm \reta m}}} }}{\minus}1} \right)^{3} .} \hfill \cr } $$

where the approximating function is a truncated Taylor expansion about the point (1,1) and the coefficients are those for the lowest Mach number (0.5).

From Equation (9), it can be seen that since thrust and drag are equal in straight and level flight at constant speed, when C _L/(C _L)_LDm is in the range 0.6–1.2,

(14) $${{C_{t} } \over {\left( {C_{t} } \right)_{{{\rm \reta m}}} }}\,\approx\,\left( {{{\left( {C_{t} } \right)_{{{\rm LDm}}} } \over {\left( {C_{t} } \right)_{{{\rm \reta m}}} }}} \right)\left( {{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} }}} \right).$$

Engines are usually sized so that, in the cruise, L/D and η _o are both close to their respective maximum values, i.e. (C _t )_LDm and (C _t )_ηm are approximately equal. Hence, combining Equations (10), (13) and (14) gives

(15) $$\matrix{ \displaystyle {{{\left( {{\rm \reta }_{o} L\,/\,D} \right)} \over {\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm max}}} }}} \hfill & \,\approx\, \hfill & {{\rm function}\left(\displaystyle {{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} }}} \right)} \hfill \cr {} \hfill & \,\approx\, \hfill & {1{\minus}1.00\left(\displaystyle {{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} }}{\minus}1} \right)^{2} {\plus}0.60\left( {{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm LDm}}} }}{\minus}1} \right)^{3} .} \hfill \cr } $$

When wave drag is taken into account, as the Mach number increases, the drag rises above the level given by Equation (7), leading to a corresponding reduction in the lift-to-drag ratio. This effect opposes the growth of η _o and, hence, reduces the rate of rise of (η _o L/D) with Mach number. Eventually, wave drag becomes the dominant effect. Consequently, as illustrated in Fig. 1, rather than continuing to increase monotonically, (η _o L/D)_max passes through a maximum as Mach number increases, i.e. (η _o L/D) has an optimum value. Wave drag is a complex function of both Mach number and lift coefficient. However, in normal aircraft operations, whilst the effects of wave drag are important, its magnitude is quite small; typically less than 10 drag counts (1 drag count=ΔCd of 0.001), which is usually less than 5% of the total.Footnote ⁴ Therefore, given that wave drag is relatively small, the relation given in Equation (15) should still be valid, at least approximately, if the values of the various normalising parameters are taken at the condition for maximum (η _o L/D), rather than those for maximum (L/D).

Taking the data from Fig. 1, for each combination of (η _o L/D), M _∞ and C _L, (η _o L/D) is normalised with the maximum value for that particular Mach number, each C _L is normalised with the value giving the maximum value of (η _o L/D) for that Mach number and each Mach number is normalised with the value that gives the optimum (= absolute maximum) (η _o L/D). The results are given in Fig. 3 where it can be seen that the curves for constant (M _∞/M _opt) form a nested set, with those for progressively higher Mach numbers exhibiting greater dependence on the normalised lift coefficient. This reflects the increasing importance of wave drag, offset slightly by increasing engine overall efficiency, as the Mach number increases. Furthermore, these curves are not symmetrical about the line C _L/(C _L)_ηLDm equal to unity, reflecting the fact that increasing the lift coefficient, at fixed Mach number, always increases the wave drag. The figure confirms the expectation that the principal variable controlling the normalised (η _o L/D) is the normalised lift coefficient.

Figure 3 The variation of normalised (η _o L/D) with normalised C _L and M _∞ for the DC-10-10.

Equations (8) and (15) are also included in the figure. As expected, Equation (15) is good approximation to the low Mach number (incompressible) limiting curve. Comparing Equations (10) and (13) shows that the contributions of induced drag and engine overall efficiency to normalised (η _o L/D) in response to changes in the normalised lift coefficient are large and approximately equal, whilst the effects of wave drag are somewhat smaller.

To aid interpolation, the curves may be represented by a truncated Taylor expansion about the point (1,1),

(16) $${{\left( {{\rm \reta }_{o} L\,/\,D} \right)} \over {\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm max}}} }}\,\approx\,1{\plus}{A \over 2}\left( {{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm ηLDm}}} }}{\minus}1} \right)^{2} {\plus}{B \over 6}\left( {{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm ηLDm}}} }}{\minus}1} \right)^{3} $$

Therefore, the coefficients A and B are estimates of the normalised second and third derivatives of (η _o L/D) with respect to C _L at (1,1). Recalling that the engine maximum efficiency exhibits a near power law variation with Mach number, A and B can be approximated by simple functions of the normalised Mach number, i.e. for (M _∞/M _opt)<0.975,

(17) $$A{\equals}B\,\approx\,{\minus}2.6,$$

otherwise

(18) $$A\,\approx\,{\minus}\left( {2.6{\plus}120\left( {{{M_{\infty} } \over {M_{{{\rm opt}}} }}{\minus}0.975} \right)^{2} } \right)$$

and

(19) $$B\,\approx\,{\minus}\left( {2.6{\plus}270\left( {{{M_{\infty} } \over {M_{{{\rm opt}}} }}{\minus}0.975} \right)^{2} } \right).$$

Equations (16)–(19) fit the DC-10-10 data to better than 1% for C _L/(C _L)_ηLDm in the range 0.6–1.2 and for all values of M _∞/M _opt up to 1.1.

Figures 4–7 show the variation of normalised (η _o L/D) with normalised lift coefficient at fixed normalised Mach number for four aircraft that vary, substantially, in terms of size, design range and age. The data are derived from tables in the flight crew operations manual and, whilst not being generally available, a number of these documents, or relevant extracts from them, can be easily found on the internet, see also, for example, Refs 17–19. These data do exhibit some scatter, as illustrated in Fig. 4, but this is limited to a maximum of about 1%. In Figs. 5, 6 and 7, the raw data have been smoothed by using a function of the form given in Equation (16), with the coefficients being determined by the least squares error criterion. The estimates generated by applying Equations (16) to (19) are also shown. In all cases, these estimates, based solely on the DC-10-10 characteristics, lie within 2% of the FCOM data and it is quite impossible to identify any particular aircraft from such normalised plots. Therefore, the general conclusion is that, to a very good approximation, these equations are valid for all aircraft.Footnote ⁵

Figure 4 The variation of normalised (η _o L/D) with normalised C _L for Aircraft 1 for a range of Mach numbers below M _opt. Open symbols are data and the solid line is Equation (16).

Figure 5 The variation of normalised (η _o L/D)with normalised C _L for M _∞/M _MRC for Aircraft 2. Solid lines are based on smoothed data and dashed lines are estimates from Equation (16).

Figure 6 The variation of normalised (η _o L/D) with normalised C _L for M _∞/M _MRC for Aircraft 3. Solid lines are based on smoothed data and dashed lines are estimates from Equation (16).

Figure 7 The variation of normalised (η _o L/D) with normalised C _L for M _∞/M _MRC for Aircraft 4. Solid lines are based on smoothed data and dashed lines are estimates from Equation (16).

Having demonstrated that the normalised distributions of (η _o L/D) with lift coefficient and Mach number are essentially ‘universal’, the next step is to examine the variation of (C _L)_ηLDm and (η _o L/D) _max , with Mach number. These results are presented in Figs. 8 and 9.

Figure 8 The variation of normalised (C _L)_ηLDm with normalised Mach number for a number of different aircraft.

Figure 9 The variation of normalised (η _o L/D)_max with normalised Mach number for a number of different aircraft.

Figure 8 shows the variation of (C _L)_ηLDm/(C _L )_opt with M _∞/M _opt for the DC-10-10, Aircraft 1 and Aircraft 3. Once again, all the data are well represented by a single curve that can be approximated by

(20) $${{\left( {C_{{\rm L}} } \right)_{{{\rm \reta LDm}}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm opt}}} }}\,\approx\,1.05,$$

for M _∞/M _opt <0.8, whilst for 0.8<M _∞/M _opt<1.1,

(21) $${{\left( {C_{{\rm L}} } \right)_{{{\rm \reta LDm}}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm opt}}} }}\,\approx\,1.05{\plus}5.0\left( {{{M_{\infty} } \over {M_{{{\rm opt}}} }}{\minus}0.80} \right)^{3} {\minus}55\left( {{{M_{\infty} } \over {M_{{{\rm opt}}} }}{\minus}0.80} \right)^{4} .$$

The shape of the curve is determined by the wave drag. At the lower Mach numbers, where wave drag is very small, the lift coefficient for maximum lift-to-drag ratio is constant. As the flight Mach number increases, (C _L)_ηLDm decreases monotonically. Initially, the decline is gentle, with a 5% reduction from the incompressible value being felt at M _opt. At higher Mach numbers, the aircraft enters the ‘drag rise’ region and, consequently, the drop off becomes rapid. Also shown is the parameter, G, where

(22) $$G{\equals}\left( {C_{{\rm L}} } \right)_{{{\rm \reta LDm}}} M_{\infty}^{2} \,\approx\,1.01\left( {C_{{\rm L}} } \right)_{{{\rm opt}}} M_{{{\rm opt}}}^{2} .$$

This approximate relation provides an estimate of the variation of (C _L)_ηLDm with Mach number for Mach numbers in the range M _opt to 1.06M _opt, i.e. the practical operating range for aircraft in the cruise. The constant of proportionality has been chosen to give the best fit (±1%) over this limited range.

Finally, the variation of normalised (η _o L/D)_max with normalised Mach number is presented in Fig. 9 and, once again, all data fall upon a near ‘universal’ curve. For 0.80<M _∞/M _opt<1.0, the curve may be represented by the approximate relation

(23) $${{\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm max}}} } \over {\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm opt}}} }}\,\approx\,1{\minus}6.00\left( {{{M_{\infty} } \over {M_{{{\rm opt}}} }}{\minus}1} \right)^{2} {\minus}15.0\left( {{{M_{\infty} } \over {M_{{{\rm opt}}} }}{\minus}1} \right)^{3} $$

and, for 1.0≤M _∞/M _opt <1.08,

(24) $${{\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm max}}} } \over {\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm opt}}} }}\,\approx\,1{\minus}233\left( {{{M_{\infty} } \over {M_{{{\rm opt}}} }}{\minus}1} \right)^{3} .$$

At the lower Mach numbers, (η _o L/D)_max increases with increasing Mach number due to the effect of Mach number on the overall propulsion efficiency. However, this benefit is progressively reduced by wave drag development at the higher Mach numbers. Eventually, wave drag becomes the dominant effect and (η _o L/D)_max peaks before dropping rapidly as the aircraft enters the ‘drag rise’ regime.

Given the near universal nature of the normalised curves, good estimates for the performance of any particular aircraft can be determined once the values of (η _o L/D), C _L and M _∞ are known at the optimum (maximum range cruise) condition. Importantly, these parameters may be estimated using information that is in the public domain, see for example Refs 20 and 21, and standard theoretical, or empirical, methods. Hence, the method can be used without the need for aircraft type specific, potentially confidential, FCOM data.

By way of example, approximate values of these quantities for a number of aircraft are listed in Table 1.

Table 1 Approximate values for the principal characteristics of a number of aircraft types

5.0 OPTIMUM CRUISE ALTITUDE

Atmospheric pressure, p _∞, and geometric altitude above sea level, h, are linked by the hydrostatic equation,

(25) $${{p_{\infty} } \over {p_{{{\rm SL}}} }}{\equals}1{\minus}\mathop{\int}\limits_0^h {\rho _{\infty} g{\rm d}h} ,$$

where ρ _∞ is the air density, and a solution requires knowledge of the complete variation of density, or temperature, with altitude. On any given day, the dependency of air temperature upon height is not known. However, in aircraft operations, the variation of pressure with height in the ‘International Standard Atmosphere’ (ISA – see Ref. 22) is universally adopted for the determination of ‘altitude’ from pressure – see Appendix A. The exact relations are cumbersome. However, since aircraft usually cruise at indicated altitudes (IA) between 30,000 and 40,000 ft, it is convenient to specify ‘altitude’ in terms of the non-dimension flight level, FL, and this can be linked to the local static pressure by a simple power-law without incurring a significant loss of accuracy. Hence, for this limited range of altitudes, the flight level is given by

(26) $${\rm FL}\,\approx\,144\left( {{{p_{\infty} } \over {p_{{{\rm SL}}} }}} \right)^{{{\minus}0.61}} .$$

This relation may be differentiated to give the effect of small fractional changes, i.e.

(27) $${{{\rm d}\left( {{\rm FL}} \right)} \over {{\rm FL}}}\,\approx\,{{{\rm d}\left( {{\rm IA}} \right)} \over {{\rm IA}}}\,\approx\,{\minus}0.61{{{\rm d}p_{\infty} } \over {p_{\infty} }}.$$

In straight and level flight, lift is equal to aircraft weight and, from the definition of lift coefficient,

(28) $${{p_{\infty} } \over {p_{{{\rm SL}}} }}{\equals}\left( {{2 \over {{\rm \rgamma }C_{{\rm L}} M_{\infty}^{2} }}} \right)\left( {{{mg} \over {p_{{{\rm SL}}} S_{{{\rm ref}}} }}} \right).$$

As previously noted, for aircraft operating in the normal Mach number range, (C _L)_ηLDm(M_∞)² (=G) is almost constant. Consequently, Equations (22), (26) and (28) reveal that each aircraft type has a single ‘ideal’ cruising flight level, IFL, at which (η _o L/D) is approximately a maximum for all Mach numbers in the normal operating range. This ideal altitude depends on the aircraft’s instantaneous weight and so it gradually increases as the flight progresses. An aircraft flying at constant Mach number may be kept at the ideal altitude continuously by climbing at exactly the rate required to keep the lift coefficient constant, i.e. following the so-called ‘cruise-climb’ trajectory.

All aircraft have a number of ‘certified’ parameters that are approved by the Regulating Authority and which are set out in the aircraft’s Type Certificate Data Sheet, e.g. Ref. 23. The list includes a maximum permitted take-off mass, MTOM, a maximum permitted landing mass, MLM, a maximum permitted mass before any fuel is loaded, known as the maximum permitted ‘zero fuel’ mass, MZFM, and a maximum permitted operating altitude, FL_max. In addition, the internal volume of the fuel tanks determines the maximum fuel mass, MFM. These parameters are fundamental characteristics that define the aircraft’s legal operating boundaries.

In Ref. 24, it is shown that all current, turbo-fan powered, civil transport aircraft with more than 100 seats have approximately the same wing loading at their certified maximum landing masses, i.e.

(29) $${{g{\rm MLM}} \over {p_{{{\rm SL}}} S_{{{\rm ref}}} }}\,\approx\,{\rm constant}{\equals}3.50\left( {{{gM_{{{\rm ref}}} } \over {p_{{{\rm SL}}} A_{{{\rm pax}}} }}} \right),$$

where M _ref is 95 kg and A _pax is 0.65 m². For this class of aircraft, the wing loading is determined, primarily, by the need to minimise the fuel requirement, although it does have other implications, e.g. take-off and landing distance, – see, for example, Ref. 14. This being the case, Equation (29) may not be valid for types where the wing loading is determined by other criteria, e.g. business jets. Nevertheless, at any point in the cruise, the ideal flight level (IFL) for a large, turbo-fan aircraft is given by

(30) $${\rm IFL}\,\approx\,725\left( {G{{{\rm MLM}} \over m}} \right)^{{0.61}} .$$

Clearly, if an aircraft is to land without first dumping fuel, its mass at the end of the cruise phase, m _fc, must not exceed MLM by more than the mass of the fuel to be used during descent, approach and landing, MF_dl. The determination of MF_dl requires a detailed knowledge of the descent approach and landing profiles and the appropriate relations are developed in Appendix D. However, MF_dl is typically about 1% of the landing mass (see for example Ref. 20). Moreover, as shown in Table 1, G shows little variation between aircraft types, with 0.335 being a good average value. Hence, the minimum value for the ideal altitude at the final cruise location is approximately the same for all aircraft, being

(31) $$\left( {{\rm IFL}_{{{\rm fc}}} } \right)_{{{\rm min}}} \,\approx\,372\left( {1{\minus}\left( {{{{\rm MF}_{{{\rm dl}}} } \over {m_{{{\rm fc}}} }}} \right)} \right)^{{0.61}} \,\approx\,370.$$

If an aircraft is to land with a mass lower than the certified maximum value, the ideal altitude will be greater than FL 370, with the upper limit being fixed by the certified FL_max. This maximum altitude is determined by a combination of design decisions and, whilst it should exceed the altitude for minimum fuel burn, there are at least three reasons why the excess should be small. First, the engine size (weight) increases with increasing FL_max. Second, the weight of the fuselage structure increases as Fl_max increases and, third, there is a passenger physiological limit imposed by the impact of hypoxia in the event of a sudden loss of cabin pressure. However, since these issues are common to all large passenger transport aircraft, the maximum value is almost the same for all current types,Footnote ⁶ being about FL 410 ± 20. Therefore, if an aircraft is to fly at the ideal altitude for the whole trip, without dumping fuel before landing,

(32) $$370\leq {\rm IFL}_{{fc}} \leq 410$$

or

(33) $$185\:\:{\rm hPa}\leq \left( {p_{\infty} } \right)_{{fc}} \leq 215\:{\rm hPa}$$

and, hence, the mass at landing must lie in the range

(34) $$0.845\leq {{{\rm LM}} \over {{\rm MLM}}}\leq 1.0.$$

When the aircraft lands, it will have payload and unused fuel on board. The combination of the payload and residual fuel masses, including reserve, contingency, tankered and taxi-in fuel, is known as the ‘disposable’ mass, DM. This can be any permitted combination of payload and fuel and is given by

(35) $${\rm DM}{\equals}{\rm PM}{\plus}{\rm FM}_{{nc}} {\equals}{\rm LM}{\minus}{\rm OEM}{\equals}{\rm LM}{\minus}\left( {{\rm ZFM}{\minus}{\rm PM}} \right).$$

where PM is the payload mass, FM_nc is the mass of the fuel that is not consumed during the flight and OEM is the operational empty mass. The OEM is the mass of the aircraft before any payload and fuel are loaded. Strictly speaking, OEM should be further subdivided into the manufacturers empty mass, MEM, (sometimes known as the basic aircraft mass, BAM) and the operational items mass, OIM, which, in general, depends upon the payload and the route. However, in this analysis, no accuracy is lost by treating OEM as a fixed quantity. In addition, the non-consumed fuel, some of which may be used on subsequent flights, must be greater than, or equal to, the minimum reserve fuel required by the regulatory authority, FM_res, plus any contingency fuel, FM_cont, specified by the operator or the crew. The fuel needed for the flight is known as the trip fuel and the trip fuel mass, TFM, is obtained by integrating Equation (4) all the way from departure to destination. This process is set out in Appendix D and an approximate, though accurate, solution is given by

(36) $${{{\rm TFM}} \over {{\rm TOM}}}{\equals}{\rm \ralpha }_{t} \,\approx\,1{\minus}{\rm EXP}\left( {{\minus}\left( {X_{t} {\plus}{\rm {\rvarepsilon}}_{t} } \right)} \right){\equals}1{\minus}{\rm EXP}\left( {{\minus}\bar{X}_{t} } \right),$$

where X _t is the non-dimensional trip distance, given by

(37) $$X_{t} {\equals}{{gR_{t} } \over {\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm opt}}} {\rm LCV}}},$$

with R _t being the great circle distance between the departure and destination points, ε _t is the total ‘lost fuel’ index and LCV is the lower calorific value of the fuel. The total lost fuel index captures the additional fuel used in the climb and descent phases, fuel wasted by not cruising at the optimum speed and height, fuel used to fly extra distance due to route deviations and fuel lost, or saved, because of the wind. Using the general definition for ‘lost fuel’ given in Appendix D,

(38) $${\rm {\rvarepsilon}}_{t} \,\approx\,{\rm {\rvarepsilon}}_{{{\rm cd}}} {\plus}{{{\rm \Delta }X} \over {n\left( {1{\minus}W_{{{\rm avg}}} } \right)}}{\plus}\left( {{{\left( {1{\minus}n\left( {1{\minus}W_{{{\rm avg}}} } \right)} \right)} \over {n\left( {1{\minus}W_{{{\rm avg}}} } \right)}}} \right)X_{t} ,$$

where ε_cd is the sum of the indices for the climb, ε_cl, and the descent, ε_dl, ΔX is the normalised total deviation from the great circle track, W _avg is the normalised headwind averaged over the whole route and n is

(39) $$n{\equals}{{\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm avg}}} } \over {\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm opt}}} }}$$

see Appendix D (Equation (D.2)). It should be noted that ε _t may be interpreted as an ‘additional non-dimensional, distance flown at optimum cruise conditions’ and that it is not necessarily small compared to the actual non-dimensional distance covered, X _t , particularly when there is a strong headwind.

At this point, it is useful to introduce the concept of ‘design’ range, R _des, being defined here as the absolute maximum distance that can be flown in still air conditions when carrying the largest permissible payload mass and the minimum permissible reserve fuel. This means that payload is added until the aircraft mass reaches the MZFM and then fuel is added until MTOM is reached and, for this reason, R _des is sometimes called the ‘harmonic’ range. The net lost fuel index for climb and descent must also have its minimum value, (ε_cd)_min, implying that optimum climb, cruise and descent trajectories are flown, with no wind and no air traffic imposed deviations from the great circle route. In which case,

(40) $${{{\rm MZFM}} \over {{\rm MTOM}}}{\equals}{\rm EXP}\left( {{\minus}\left( {X_{{{\rm des}}} {\plus}\left( {{\rm {\rvarepsilon}}_{{{\rm cd}}} } \right)_{{{\rm min}}} } \right)} \right){\minus}{\rm \rbeta }_{{{\rm min}}} ,$$

where X _des is the non-dimensional design range and

(41) $${{{\rm FM}_{{{\rm res}}} } \over {{\rm TOM}}}{\equals}{{\left( {{\rm FM}_{{{\rm nc}}} } \right)_{{{\rm min}}} } \over {{\rm TOM}}}{\rm {\equals}\rbeta }_{{{\rm min}}} .$$

The design range, (η _o L/D)_opt, (ε_cd)_min and β_min are all fundamental characteristics of the aircraft and, as such, can be established to any desired level of accuracy.

The take-off mass, TOM, is the sum of the landing and the trip fuel masses, i.e.

(42) $${\rm TOM}{\equals}{\rm LM}{\plus}{\rm TFM}{\equals}{{{\rm LM}} \over {\left( {1{\minus}{\rm \ralpha }_{t} } \right)}}{\equals}{{{\rm LM}} \over {{\rm EXP}\left( {{\minus}\bar{X}_{t} } \right)}}\leq {\rm MTOM}.$$

Hence, for a given landing mass, as the trip length increases more trip fuel is needed and, eventually, either the take-off mass reaches the maximum permitted value, MTOM, or the fuel mass reaches its maximum value, MFM, i.e. the tanks are full.

Combining Equations (40) and (42) reveals that, for a given route, i.e. known values for X _t and (ε_cd)_min, the aircraft take-off mass fraction, TOM/MTOM, landing mass fraction, LM/MLM, and the non-dimensional design range are linked by the relation

(43) $$X_{{{\rm des}}} {\equals}{\minus}{\rm LN}\left( {{\rm EXP}\left( {\left( {{\rm {\rvarepsilon}}_{{{\rm cd}}} } \right)_{{{\rm min}}} } \right)\left( {{{\left( {{\rm TOM\,/\,MTOM}} \right){\rm EXP}\left( {{\minus}\bar{X}_{t} } \right)} \over {\left( {{\rm LM\,/\,MLM}} \right)\left( {{\rm MLM\,/\,MZFM}} \right)}}{\plus}{\rm \rbeta }_{{{\rm min}}} } \right)} \right).$$

As will be discussed later, according to Randle et al.⁽ Reference Randle, Hall and Vera-Morales ^{20 )}, (ε_cd)_min is about 0.0067, whilst, from Ref. 24, β_min is about 0.045 and, for the aircraft currently in service,

(44) $${{{\rm MLM}} \over {{\rm MZFM}}}\,\approx\,1.075.$$

This being the case, with a little rearrangement and approximation, for a given route, the aircraft that can take off and land at the maximum permitted masses, has a design range given by

(45) $$X_{{{\rm des}}} \,\approx\,0.018{\plus}0.952\bar{X}_{t} ,$$

Moreover, if, as is usually the case,

(46) $${\rm {\rvarepsilon}}_{t} \geq 0.051X_{t} {\minus}0.0194,$$

X _des will be greater than X _t . Consequently, this aircraft can carry the maximum possible disposable load, can carry the maximum permitted payload, has the lowest MTOM for a given payload, i.e. it is the smallest aircraft needed to serve the route, and has an energy to revenue work ratio, ETRW, that is at, or close to, the optimum value. Therefore, from an operational, a commercial and an environmental perspective, this is the ‘perfect’ aircraft for the route.

The mass of the aircraft at the beginning of the cruise, m _ic, is equal to the take-off mass, TOM, less the mass of the fuel, MF_cl, used to accelerate and climb to the initial cruise height and speed. Hence, the ideal initial cruise altitude is

(47) $${\rm IFL}_{{{\rm ic}}} \,\approx\,372.3\left( {{{{\rm MLM}} \over {{\rm LM}}}} \right)^{{0.61}} \left( {{\rm EXP}\left( {{\minus}0.61\left( {\bar{X}_{t} } \right)} \right)} \right)\left( {1{\minus}{{{\rm MF}_{{{\rm cl}}} } \over {{\rm TOM}}}} \right)^{{{\minus}0.61}} .$$

Here too, the determination of climb fuel requires a detailed knowledge of the take-off and climb profiles and, again, the appropriate relations are developed in Appendix D. However, since MF_cl is typically about 2.5% of the take-off mass, see Ref. 20, it follows from Equation (34) that the initial IFL must lie in the range

(48) $$376\leq {{{\rm IFL}_{{{\rm ic}}} } \over {{\rm EXP}\left( {{\minus}0.61\left( {\bar{X}_{t} } \right)} \right)}}\leq 419.$$

On a given route, the ‘perfect’ aircraft carrying its maximum disposable mass will have the lowest IFL at all points in the flight. When this aircraft is landing with a mass below the maximum permitted, the IFL will be somewhere between the minimum and the maximum value. Clearly, it is also possible to carry the same payload in aircraft with design ranges both greater than and smaller than the ‘perfect’ value. However, in all cases, the final and initial ideal cruise altitudes will fall within the range given by Équations (32) and (48). Taking all possible scenarios into account,

(49) $${\rm IFL}_{{{\rm ic}}} \,\approx\,397{\rm EXP}\left( {{\minus}0.61\left( {\bar{X}_{t} } \right)} \right),\quad \left( {p_{\infty} } \right)_{{{\rm ic}}} \,\approx\,192.5 {\rm EXP}\left( {\bar{X}_{t} } \right) {\rm hPa}$$

and

(50) $${\rm IFL}_{{{\rm fc}}} \,\approx\,390,\left( {p_{\infty} } \right)_{{{\rm fc}}} \,\approx\,200 {\rm hPa},$$

with an error range of ± 5%, or FL ± 20, whilst the change in the IFL over the course of the trip is

(51) $${\rm IFL}_{{{\rm fc}}} {\minus}{\rm IFL}_{{{\rm ic}}} \,\approx\,242\left( {\bar{X}_{t} {\minus}0.03} \right)\,{\rm or}\,{\rm \Delta }p_{\infty} \,\approx\,192.5\left( {0.04{\minus}\bar{X}_{t} } \right) {\rm hPa}$$

Therefore, to a good approximation, the final ideal cruise flight level is the same for all aircraft under all operating conditions and is close to the 200 hPa isobar. This simple, rather surprising, result is a direct consequence of designing aircraft for (near) minimum fuel burn, i.e. the final optimum cruise altitude is an output of the design process and not an input to it. On the other hand, the initial ideal cruise altitude is determined by the trip fuel requirement and this depends upon the route length, the headwind, the aircraft’s optimum (η _o L/D) and the value of the lost fuel index. However, importantly, it is almost independent of the size of the aircraft, its take-off mass and its design range.

All other things being equal, the more fuel efficient the aircraft type, the greater the initial ideal cruise altitude, with a 10% increase in (η _o L/D) increasing the initial altitude for the longest journeys flown by about 1,000 ft. Conversely, flights with air traffic imposed inefficiencies require more fuel and so the initial altitude for minimum fuel cruise will decrease. Therefore, the trend towards more fuel-efficient aircraft and more efficient ATM systems means that the initial ideal cruise altitude will increase and, for a given route, the difference between the initial and final ideal cruise altitudes will decrease, reducing the average gradient of the cruise-climb trajectory.

The mass of the aircraft at any intermediate point in the cruise, X, is given by

(52) $${m \over {{\rm TOM}}}{\equals}{\rm EXP}\left( {{\minus}\left( {X{\plus}{\rm {\rvarepsilon}}_{{{\rm cr}}} } \right)} \right){\equals}{\rm EXP}\left( {{\minus}\bar{X}_{c} } \right),$$

where,

(53) $$ \bar{X}_{c} \,\approx\,{\rm {\rvarepsilon}}_{{{\rm cl}}} {\plus}\left( {{{X{\plus}{\rm \Delta }X_{{{\rm cr}}} {\plus}{\rm \Delta X}_{{{\rm da}}} } \over {n\left( {1{\minus}W_{{{\rm avg}}} } \right)}}} \right).$$

Hence, for ${\rm }X_{{{\rm cl}}} \leq X\leq \left( {X_{t} {\minus}X_{{{\rm dl}}} } \right),$

(54) $${\rm IFL}\,\approx\,391\left( {{\rm EXP}\left( {{\minus}0.61\left( {\bar{X}_{t} {\minus}\bar{X}_{c} } \right)} \right)} \right)$$

and the non-dimensional climb gradient at any location is

(55) $${{{\rm d}\left( {{\rm IFL}} \right)} \over {{\rm d}X}}\,\approx\,0.61({\rm IFL}).$$

The ideal cruise-climb trajectory gives the lowest fuel burn for all Mach numbers in the normal operating range. However, currently and for safety reasons, aircraft are not permitted to cruise climb, nor are they guaranteed to be able to use the ideal initial cruise flight level. Cruise begins at the ATM specified height and must continue at that value. However, occasional step climbs of ΔFL +20 (≈2,000 ft) may be requested and, depending on the circumstances, may be permitted. Therefore, in current operations, the closest approximation to the ideal cruise climb trajectory begins with the aircraft at the ideal initial cruise altitude. This is maintained until the aircraft weight is such that the ideal cruise altitude is 1,000 ft higher (ΔFL+10). At this point, the aircraft performs a ‘step climb’ of 2,000 ft to an altitude of 1,000 ft above the current ideal cruise value. The flight continues at this new FL until the aircraft is once again 1,000 ft below the ideal level when the step climb of 2,000 ft is repeated. This process continues until the aircraft reaches the end of the cruise. By following this procedure, the aircraft is never more than 1,000 ft from the ideal value. However, whilst this is a good approximation to the ideal climb profile, there is still a fuel penalty.

6.0 FUEL REQUIREMENTS

If an aircraft is kept at the IFL as given by Equation (54), the trip fuel requirement is that given by Equation (36). However, in general, whilst this will be a low fuel journey, it will not be the minimum fuel journey. In order to determine the minimum fuel journey, it is convenient to define a reference trip fuel, (TFM)_ref. Here the aircraft flies at the maximum range cruise Mach number, M _MRC, there are no air traffic imposed deviations from the great circle route and the lost fuel during the climb and descent phases is minimised by flying optimum speed versus height profiles. This being the case, if an aircraft, operating a given route, is to carry a specified disposable mass, i.e. X _t and the landing mass are fixed, then, from Equations (36), (38) and (42),

(56) $${{\left( {{\rm TFM}} \right)_{{{\rm ref}}} } \over {{\rm LM}}}\,\approx\,{{\left( {1{\minus}{\rm EXP}\left( {{\minus}\left( {\bar{X}_{t} } \right)_{{{\rm ref}}} } \right)} \right)} \over {{\rm EXP}\left( {{\minus}\left( {\bar{X}_{t} } \right)_{{{\rm ref}}} } \right)}}.$$

with

(57) $$\left( {\bar{X}_{t} } \right)_{{{\rm ref}}} {\rm {\equals}}\left( {{\rm {\rvarepsilon}}_{{{\rm cd}}} } \right)_{{{\rm min}}} {\plus}{{X_{t} } \over {\left( {1{\minus}W_{{{\rm avg}}} } \right)}}.$$

where W _avg is the headwind at the IFL averaged along the great circle track. This is the baseline against which the trip fuel requirement for all other trajectories should be judged and so, for a given value of X _t ,

(58) $${{{\rm TFM}{\minus}\left( {{\rm TFM}} \right)_{{{\rm ref}}} } \over {\left( {{\rm TFM}} \right)_{{{\rm ref}}} }}\,\approx\,{{{\rm EXP}\left( {\bar{X}_{t} {\minus}\left( {\bar{X}_{t} } \right)_{{{\rm ref}}} } \right){\minus}1} \over {1{\minus}{\rm EXP}\left( {{\minus}\left( {\bar{X}_{t} } \right)_{{{\rm ref}}} } \right)}}.$$

As a first step in determining the minimum fuel requirement, it is useful to determine the sensitivity of fuel requirement to variations in the average wind and to those quantities that are controlled by ATM. This is found by differentiating Equation (56), which gives

(59) $${{{\rm d}\left( {{\rm TFM}} \right)} \over {\left( {{\rm TFM}} \right)_{{{\rm ref}}} }}{\equals}f_{1} {{{\rm d}\left( {{\rm {\rvarepsilon}}_{{{\rm cd}}} } \right)} \over {\left( {{\rm {\rvarepsilon}}_{{{\rm cd}}} } \right)_{{{\rm min}}} }}{\plus}f_{2} \left( {{{{\rm d}\left( R \right)} \over {R_{t} }}{\minus}{{{\rm d}\left( n \right)} \over {\left( n \right)_{{{\rm max}}} }}{\plus}{{W_{{{\rm avg}}} } \over {\left( {1{\minus}W_{{{\rm avg}}} } \right)}}{{{\rm d}\left( {W_{{{\rm avg}}} } \right)} \over {W_{{{\rm avg}}} }}} \right),$$

where, with some manipulation and approximation,

(60) $$f_{1} \,\approx\,{{\left( {{\rm {\rvarepsilon}}_{{{\rm cd}}} } \right)_{{{\rm min}}} } \over 2}\left( {1{\plus}{{2\left( {1{\minus}W_{{{\rm avg}}} } \right)} \over {\left( {{\rm {\rvarepsilon}}_{{{\rm cd}}} } \right)_{{{\rm min}}} \left( {1{\minus}W_{{{\rm avg}}} } \right){\plus}X_{t} }}} \right),$$

and

(61) $$f_{2} \,\approx\,{{X_{t} } \over {2\left( {1{\minus}W_{{{\rm avg}}} } \right)}}\left( {1{\plus}{{2\left( {1{\minus}W_{{{\rm avg}}} } \right)} \over {\left( {{\rm {\rvarepsilon}}_{{{\rm cd}}} } \right)_{{{\rm min}}} \left( {1{\minus}W_{{{\rm avg}}} } \right){\plus}X_{t} }}} \right).$$

The cruise fuel consumed is obtained by integrating Equation (4) from the initial to the final cruise positions and, as explained in Appendix C, if the normalised headwind and the flight Mach number are constant, n is given by

(62) $$n{\equals}{{\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm avg}}} } \over {\left( {{\rm \reta }_{o} L\,/\,D} \right)_{{{\rm opt}}} }}{\equals}{{\left( {X_{{{\rm fc}}} {\minus}X_{{{\rm ic}}} } \right)} \over {\left( {{\rm \reta }_{o}L\,/\,D} \right)_{{{\rm opt}}} {\int}_{X_{{{\rm ic}}} }^{X_{{{\rm fc}}} } {\left( {{\rm \reta }_{o}L\,/\,D} \right)^{{{\minus}1}} {\rm d}X} }}.$$

When the aircraft cruise-climbs at the IFL, C _L is always equal to (C _L)_ηLDm and, from Equation (16), (η _o L/D) is constant and equal to the maximum value for the chosen cruise speed. For Mach numbers between the maximum range and the high-speed cruise values, n is given by Equation (24). Therefore, if the flight Mach number exceeds M _opt by ΔM, the resulting reduction in n relative to its maximum value (= unity) is approximately

(63) $${{{\rm d}\left( n \right)} \over {\left( n \right)_{{{\rm max}}} }}{\equals}{\rm d}\left( n \right)\,\approx\,{\minus}233\left( {{{{\rm \Delta }M_{\infty} } \over {M_{{{\rm opt}}} }}} \right)^{3} .$$

Since typical operating speeds lie somewhere between the maximum range and the long-range cruise values, n normally lies between 1 and 0.99.

Now consider an aircraft at the beginning of the cruise and flying at M _opt. If the flight level is the ideal value, (η _o L/D) will have the optimum value. However, if the aircraft begins the cruise at a higher flight level, differing from the ideal value by an amount ∆FL_ic, then

(64) $$\left( {{{{\rm FL}} \over {{\rm IFL}}}} \right)_{{{\rm ic}}} {\equals}1{\plus}\left( {{{{\rm \Delta FL}} \over {{\rm IFL}}}} \right)_{{{\rm ic}}} .$$

It follows from Equations (26) and (28) that

(65) $${{\left( {C_{{\rm L}} } \right)_{{{\rm ic}}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm opt}}} }}\,\approx\,\left( {{{{\rm FL}} \over {{\rm IFL}}}} \right)_{{{\rm ic}}}^{{{1 \over {0.61}}}} {\equals}1{\plus}\left( {{1 \over {0.61}}} \right)\left( {{{{\rm \Delta FL}} \over {{\rm IFL}}}} \right)_{{{\rm ic}}} {\plus}\left( {{{0.195} \over {0.61^{2} }}} \right)\left( {{{{\rm \Delta FL}} \over {{\rm IFL}}}} \right)_{{{\rm ic}}}^{2} {\plus} \cdots $$

If the aircraft subsequently cruise-climbs with the lift coefficient held constant at (C _L)_ic, the reduction in n resulting from beginning the cruise at a non-IFL is obtained from Equation (16) and may be expressed approximately as

(66) $${\rm d}\left( n \right)\,\approx\,{\minus}\left( {3.60\left( {{{{\rm \Delta FL}} \over {{\rm IFL}}}} \right)_{{{\rm ic}}}^{2} {\plus}4.3\left( {{{{\rm \Delta FL}} \over {{\rm IFL}}}} \right)_{{{\rm ic}}}^{3} } \right).$$

Ignoring products of small quantities, the change in n due to a combination of non-optimum Mach number and non-ideal initial altitude are given by the sum of Equations (63) and (66). Hence, for the cruise-climb, Equations (59), (61), (63) and (66) can be used to assess the impact of route deviations and changes in speed and altitude on the required trip fuel. However, since the current ATM environment requires aircraft to cruise at a constant flight level, with the possibility of an occasional step climb to a higher altitude, it is important to examine fuel usage in this situation.

If an aircraft maintains a constant speed and a constant flight level then, as it gets lighter, the lift coefficient decreases such that

(67) $${{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm \reta LDm}}} }}{\equals}{{C_{{\rm L}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm ic}}} }}{{\left( {C_{{\rm L}} } \right)_{{{\rm ic}}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm \reta LDm}}} }}{\equals}{{\left( {C_{{\rm L}} } \right)_{{{\rm ic}}} } \over {\left( {C_{{\rm L}} } \right)_{{{\rm \reta LDm}}} }}\left( {{m \over {m_{{{\rm ic}}} }}} \right).$$

This being the case, the determination of n (Equation (62)) involves the numerical solution of an implicit, integral equation. However, as demonstrated in Appendix C, an approximate explicit solution can be developed. The full, analytic, version is rather complex, but, for short cruise distances and small deviations from the optimum values for speed and altitude, the power series form given in Equation (C.21) of Appendix C can be used. Hence, in general, the additional change to n that occurs when the aircraft cruises at a fixed, non-IFL is given by

(68) $$\matrix{ {{\rm d}\left( n \right)} \hfill &#x0026; \,\approx\, \hfill &#x0026; {{\minus}\left( {3.60\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}}^{2} {\plus}4.3\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}}^{3} } \right)} \hfill \cr {} \hfill &#x0026; {} \hfill &#x0026; {{\plus}2.19\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}} \left( {1{\plus}2.81\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}} {\plus}10.13\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}}^{2} {\plus} \cdots } \right)\left( {{\textstyle{{X{\minus}X_{{{\rm ic}}} } \over {1{\minus}W_{{{\rm avg}}} }}}} \right)} \hfill \cr {} \hfill &#x0026; {} \hfill &#x0026; {{\minus}0.446\left( {1{\plus}6.615\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}} {\plus}39.60\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}}^{2} {\plus} \cdots } \right)\left( {{\textstyle{{X{\minus}X_{{{\rm ic}}} } \over {1{\minus}W_{{{\rm avg}}} }}}} \right)^{2} } \hfill \cr {} \hfill &#x0026; {} \hfill &#x0026; {{\plus}0.450\left( {1{\plus}11.89\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}} {\plus}68.31\left( {{\textstyle{{{\rm \Delta FL}} \over {{\rm IFL}}}}} \right)_{{{\rm ic}}}^{2} {\plus} \cdots } \right)\left( {{\textstyle{{X{\minus}X_{{{\rm ic}}} } \over {1{\minus}W_{{{\rm avg}}} }}}} \right)^{3} {\plus}{\rm } \cdots ,} \hfill \cr } $$

where the coefficients A and B have been evaluated at M _opt and products of all small quantities have been neglected.

This relation reveals the complex nature of the constant altitude cruise compared to the simple cruise climb. When the altitude is held constant, n not only depends on Mach number and any initial deviation from the IFL but also on the distance travelled and the headwind. If an aircraft is to maintain a flight level at, or below, the initial IFL, the fuel burn rate will increase steadily relative to the initial value as the flight progresses. However, if the flight level is above the initial IFL, the fuel burn rate initially decreases as distance flown increases, eventually passing through a minimum. The total change in n due to Mach number variations and a fixed altitude cruise is given by the sum of Equations (63) and (68).

The functions f ₁ and f ₂ are always positive. Hence, any increase in ε_cd will increase the fuel requirement above the reference value. However, if

(69) $$\left( {{{{\rm d}\left( R \right)} \over {R_{t} }}{\minus}{{{\rm d}\left( n \right)} \over {\left( n \right)_{{{\rm max}}} }}{\plus}{{W_{{{\rm avg}}} } \over {\left( {1{\minus}W_{{{\rm avg}}} } \right)}}{{{\rm d}\left( {W_{{{\rm avg}}} } \right)} \over {W_{{{\rm avg}}} }}} \right)\,\lt\,0,$$

the fuel requirement will be less than the reference value. This may be the case if the wind speed variations in either the vertical, or horizontal, planes are sufficiently large that the advantage of a moving to a position where the benefits of a more favourable wind outweigh the penalties of flying higher, or lower, or expending the length of the ground track. Therefore, the minimum fuel flight profile will be the one that minimises Equation (69)