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1

ANALYTIC FUEL FLOW ANALYSIS

In this analysis we identify functions for the fuel flow rate and Mach number for cruise at a fixed

flight level in Long Range Cruise (LRC) mode using only the aeroplane mass and static temperature

deviation as explanatory variables in solutions to two integrals determining the endurance and the

range potential.

4.1 Fuel Flow Rate

We adopt a linear model for the fuel flow rate using the aeroplane all up mass as the explanatory

variable. Let ��/�� be the fuel flow rate and � the aeroplane mass in the interval [160,230]

tonnes, then

���� = + �� (4.1.1)

This fuel flow rate linear differential equation accounts for aeroplane mass variation but fails to

account for non-standard ambient temperatures. The effect of non-standard temperature will be

modelled by increasing/decreasing the fuel flow rate by 3% for each 10 K increase/decrease in

standard total air temperature The total air temperature (TAT), more usually referred to as the

stagnation temperature, is the absolute temperature of a flow brought to rest isentropically from an

initial Mach number. Its value depends only on the initial Mach number, adiabatic index and static

temperature of the flowing fluid. Let � be the total temperature, �� the static temperature and �

the Mach number1,

� = ���1 + ����� (4.1.2) �� = ���� , a constant that depends only on the adiabatic index γ.

If the static temperature �� is written as the sum of the standard static temperature2 and a

temperature deviation term ∆��, viz., �� = ���� + ∆��, then the change in � caused by the presence

of ∆�� is ∆���1 + �����. We rewrite (4.1.1) as,

���� = � + ����1 + ��∆���1 + ������ (4.1.3)

where �� = ��

In a standard atmosphere ∆�� = 0 and (4.1.3) reduces to (4.1.1). The constant �� is a fuel flow rate

adjustment parameter for non-standard temperature conditions.

1 Shapiro, A. H. (1953). The Dynamics and Thermodynamics of Compressible Fluid Flow. Vol. I. New York,The

Ronald Press Company. 2 A static temperature at a given pressure altitude is considered to be standard if it is equal to the value

assigned to that pressure altitude by the International Standard Atmosphere (ISA) model. In the troposphere,

extending from the surface to 11 000 metres geopotential altitude, the temperature variation with altitude

under the ISA model is given by Touissaint’s law: if ���ℎ�is the standard static temperature at geopotential

altitude h, then ���ℎ� = �!�" − Γℎ, where �!�" is the standard mean sea level temperature of 288.15 K and Γ

is the fixed temperature lapse rate of 0.0065 K m��. The lower stratosphere, extending from 11 000 to 20 000

metres geopotential altitude, is considered isothermal in the ISA model. The U.S. Standard Atmosphere 1976,

which is the same as ISA up to 32 000 metres altitude, is available online at: [17 MB]

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770009539_1977009539.pdf

2

Separating the variables in (4.1.3) and integrating yields the time required for the aeroplane mass to

change between limits �� and � as fuel is consumed:

� = ( ����)*����)+,∆-.��)+/!,���/�0 (4.1.4)

If the end mass � is set to the zero fuel mass3 (ZFM) then the time in (4.1.4) will be called

endurance.

The Mach number in (4.1.4) is not independent of the aeroplane mass but we will assume it is

independent of temperature deviation. The variation of the square of the Mach number is modelled

as a quadratic in mass defined on the interval [160,230] tonnes:

�� = 1 + 2� + 3�� (4.1.5)

The fuel flow rate, after some manipulations, is re-written as,

���� = � + ����4 + 26� + 7��� (4.1.6)

4 = 1 + ��∆���1 + ��1�, 6 = �� ����∆��2 , 7 = ����∆��3

By using (4.1.6) in (4.1.4) the mass integral becomes,

� = ( ����)*���8)�9�):�,��/�0 (4.1.7)

Decomposing the integrand in (4.1.7) into partial fractions:

���)*���8)�9�):�,� ≡ < = ��)*� + >�)?8)�9�):�,@ (4.1.8)

< = *,�,:���*9)*,8 , A = − :*, B = �:��*9*,

now (4.1.7) becomes,

� = < ( ��)*��/�0 �� + < ( >�)?8)�9�):�, ���/�0 (4.1.9)

The first integral evaluates to,

< ( ��)*��/�0 �� = C* Dln| + ��|H�0�/ (4.1.10)

and the solution to the second integral depends on the sign of the discriminant of the quadratic in

the denominator4. We omit the limits of integration for clarity,

< ( >�)?8)�9�):�, �� (4.1.11a)

3 The zero fuel mass is the total aeroplane mass excluding usable fuel and is equal to the sum of the masses of

the aeroplane with its basic equipment, inconsumable fluids, crew, crew baggage, any removable equipment

required for a flight (catering equipment for example), passengers and their baggage, and freight including

non-revenue loads. 4 Gradshteyn, I.S. and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products. 7

th Ed. Section 2.103(5).

London: Elsevier.

3

= < = >�: ln|4 + 26� + 7��| + ?:�>9:√8:�9, tan�� :�)9√8:�9,@ for D47 > 6�H = < M >�: ln|4 + 26� + 7��| + ?:�>9�:√9,�8: ln N:�)9�√9,�8::�)9)√9,�8:NO for D47 < 6�H

Neither solution in (4.1.11a) is defined when 47 = 6�. In this case the differential equation in

(4.1.3) reduces to equation (4.1.1) and the required solution for endurance is, from (4.1.10), noting

that if ∆�� = 0 then < = 1,

� = �* Dln| + ��|H�0�/ for D∆�� = 0H (4.1.11b)

If ∆�� is defined on the interval [-50,50] kelvin and the non-standard temperature fuel flow rate

parameter �� on the interval (0, 0.02], then for coefficients C and E as identified above, requiring

additionally that �� > � , the three solutions for endurance are:

� = <� Dln| + ��|H�0�/ +QRRSRRT

0 < U A27 ln|4 + 26� + 7��| + B7 − A67√47 − 6� tan�� 7� + 6

√47 − 6�V�0

�/ < W A27 ln|4 + 26� + 7��| + B7 − A627√6� − 47 ln X7� + 6 − √6� − 477� + 6 + √6� − 47XY

�0

�/

if 47 = 62 if 47 > 62

if 47 < 62

(4.1.12)

In (4.1.12), if ∆�� = 0 then < = 1.

To introduce an additional uniform increase in the fuel flow rate it is only necessary to scale the

coefficients A and B by an appropriate factor.

Coefficients A and B for the fuel flow rate model, and coefficients C, D and E for the square of the

Mach number, are included in appendices B.1. and B.2.

4.2 Range Potential

In this subsection we identify the range potential as the distance the aeroplane can fly at a fixed

flight level in LRC cruise given a specified fuel quantity, an initial or final aeroplane mass and an

assumed uniform temperature deviation.

In an LRC cruise the Mach number depends on the aeroplane mass and is assumed to be

independent of temperature deviation. The speed of the aeroplane with respect to the air mass in

which it flies is related to the Mach number by the speed of sound which depends on the air mass

temperature ��. Let �\/�� be the aeroplane speed, � the Mach number and ]� the speed of sound:

���� = � ]� (4.2.1)

where ]� = ^_`�� , γ is the adiabatic index and R the gas constant for air.5

The variation of the Mach number with aeroplane mass is modelled as a quadratic defined on the

mass interval [160,230] tonnes:

5 The adiabatic index is constant below approximately Mach 3 in the troposphere and stratosphere, and the

gas constant remains constant at all altitudes reachable by transonic commercial aeroplanes.

4

� = 1� + 2�� + 3��� (4.2.2)

The aeroplane speed in (4.2.1) is a function of the aeroplane mass and ambient temperature,

���� = �1� + 2�� + 3����^_`�� (4.2.3)

Recalling the fuel flow rate ��/�� from (4.1.6),

���� = � + ����4 + 26� + 7��� (4.1.6)

and using the chain rule, the rate of change of path length with respect to mass is

���� ���� = ���� = �a,)b,�)c,�,�^�d-. ��)*���8)�9�):�,� (4.2.4)

separating the variables and integrating yields

\ = ^_`�� ( a,)b,�)c,�, ��)*���8)�9�):�,��/�0 �� (4.2.5)

Decomposing the integrand in (4.2.5) into partial fractions,

a,)b,�)c,�, ��)*���8)�9�):�,� ≡ <� = ��)*� + >,�)?,8)�9�):�,@ (4.2.6)

<� = �,c,��*b,)*,a,�,:���*9)*,8 , A� = �b,:���c,9�*a,:)*c,8�,c,��*b,)*,a, , B� = �a,:��c,8��*a,9)*b,8�,c,��*b,)*,a,

and re-writing the range potential integral (4.2.5),

\ = <�^_`�� ( ��)*� ���/�0 + <�^_`�� ( >,�)?,8)�9�):�, ���/�0 (4.2.7)

The first integral in (4.2.7) evaluates to,

<�^_`�� ( ��)*� ���/�0 = C,* ^_`��Dln| + ��|H�0�/ (4.2.8)

and the solution to the second integral depends on the sign of the discriminant of the quadratic in

the denominator6. We omit the limits of integration for clarity,

<�^_`�� ( >,�)?,8)�9�):�, �� (4.2.9)

= <�^_`�� =>,�: ln|4 + 26� + 7��| + ?,:�>,9:√8:�9, tan�� :�)9√8:�9,@ for D47 > 6�H = <�^_`�� M>,�: ln|4 + 26� + 7��| + ?,:�>,9�:√9,�8: ln N:�)9�√9,�8::�)9)√9,�8:NO for D47 < 6�H

Neither solution in (4.2.9) is defined when 47 = 6�, for which case we must re-write (4.2.4) using

the fuel flow rate given in (4.1.1) instead of (4.1.6). Recalling (4.1.1) for the fuel flow rate,

���� = + �� (4.1.1)

6 Ibid, §2.103(5).

5

then the rate of change of path length with mass becomes,

���� ���� = ���� = �a,)b,�)c,�,�^�d-. �)*� (4.2.10)

separating the variables and integrating yields

\ = ^_`�� ( a,)b,�)c,�, �)*� ���/�0 (4.2.11)

separating the improper rational function in the integrand into an integral part and remainder,

\ = ^�d-.*, =( ��3�� + �2� − 3�� ���/�0 + ( e,f,�egh,)g,i, �)*� ���/�0 @ (4.2.12) which after integrating leads to,

\ = ^�d-.�*, j��3��� + 2��2� − 3��� + �* �A�E� − ABD� + B�C�� ln| + ��|p�0�/ (4.2.13)

Defining ∆�� on the interval [-50,50] K and the non-standard temperature fuel flow rate parameter �� on (0, 0.02], additionally requiring �� > � , the three solutions for range potential s are:

\ =

QRRSRRT ^_`��2�� U��3��� + 2��2� − 3��� + 2� �A�E� − ABD� + B�C�� ln| + ��|V�0

�/ <�^_`�� U A�27 ln|4 + 26� + 7��| + B�7 − A�67√47 − 6� tan�� 7� + 6

√47 − 6�V�0

�/ <�^_`�� W A�27 ln|4 + 26� + 7��| + B�7 − A�627√6� − 47 ln X7� + 6 − √6� − 477� + 6 + √6� − 47XY

�0

�/

if 47 = 6� if 47 > 6�

if 47 < 6�

(4.2.14)

The condition 47 = 6� for the first solution in (4.2.14) is equivalent to ∆�� = 0.

The coefficients in the appendices that follow assume working in SI units.

Barry Martin

London, 25th

September 2014

Aqqa.org

6

APPENDIX A.1. – FUEL FLOW MODEL COEFFICIENTS

Defined on [160,230] tonnes.

FL A (const) B (for mass) Adjusted R2

400 -2.593056E-01 9.750000E-06 9.559116E-01

390 1.234127E-02 8.211640E-06 9.844439E-01

380 1.270833E-01 7.546296E-06 9.965852E-01

370 1.679167E-01 7.296296E-06 9.995254E-01

360 1.734722E-01 7.259259E-06 9.998555E-01

350 1.448413E-01 7.443122E-06 9.996202E-01

340 1.222024E-01 7.586640E-06 9.991936E-01

330 9.656746E-02 7.743386E-06 9.985727E-01

320 6.978175E-02 7.897487E-06 9.983050E-01

310 3.688492E-02 8.083995E-06 9.983037E-01

300 4.920635E-02 8.058201E-06 9.979309E-01

290 9.454365E-02 7.904762E-06 9.978443E-01

280 5.331349E-02 8.174603E-06 9.966678E-01

270 4.837302E-02 8.210979E-06 9.974304E-01

260 5.952381E-02 8.162698E-06 9.978099E-01

250 7.900794E-02 8.068122E-06 9.979861E-01

240 6.154762E-02 8.181878E-06 9.981208E-01

230 2.744048E-02 8.380291E-06 9.991767E-01

220 5.456349E-02 8.244048E-06 9.991549E-01

210 8.478175E-02 8.096561E-06 9.990629E-01

200 1.196230E-01 7.933201E-06 9.989627E-01

190 1.484921E-01 7.811508E-06 9.990911E-01

180 1.689484E-01 7.740079E-06 9.992798E-01

170 1.797619E-01 7.720238E-06 9.995502E-01

160 1.844841E-01 7.729497E-06 9.998090E-01

150 1.918254E-01 7.728175E-06 9.999463E-01

140 2.048016E-01 7.708995E-06 9.999573E-01

130 2.150794E-01 7.718254E-06 9.998284E-01

120 2.170635E-01 7.781085E-06 9.997187E-01

100 2.026389E-01 7.995370E-06 9.997927E-01

80 2.116468E-01 8.058862E-06 9.999806E-01

60 2.531548E-01 7.964947E-06 9.999876E-01

7

APPENDIX A.2. – COEFFICIENTS FOR THE SQUARE OF THE MACH NUMBER

Defined on [140,230] tonnes.

FL C (const) D (for mass) E (for mass^2) Adjusted R2

400 -6.338725E-02 7.556565E-06 -1.852864E-11 9.574705E-01

390 -1.736737E-01 8.452388E-06 -2.029970E-11 9.904565E-01

380 -2.510750E-01 8.917025E-06 -2.080977E-11 9.963347E-01

370 -3.045117E-01 9.085627E-06 -2.048360E-11 9.983702E-01

360 -3.125667E-01 8.716058E-06 -1.868966E-11 9.978348E-01

350 -3.264184E-01 8.435672E-06 -1.725057E-11 9.984950E-01

340 -3.475452E-01 8.279845E-06 -1.634170E-11 9.982636E-01

330 -3.852806E-01 8.331882E-06 -1.608364E-11 9.989316E-01

320 -3.672948E-01 7.813449E-06 -1.441890E-11 9.987128E-01

310 -2.717189E-01 6.459857E-06 -1.056402E-11 9.973457E-01

300 -1.647256E-01 5.007037E-06 -6.531402E-12 9.978654E-01

290 -4.158265E-02 3.432551E-06 -2.296439E-12 9.985144E-01

280 4.714455E-02 2.298811E-06 5.775758E-13 9.993730E-01

270 8.778279E-02 1.741698E-06 1.793636E-12 9.998368E-01

260 1.223446E-01 1.279644E-06 2.695379E-12 9.999372E-01

250 1.242464E-01 1.197379E-06 2.565530E-12 9.999518E-01

240 1.115891E-01 1.282003E-06 1.995227E-12 9.998870E-01

230 1.035701E-01 1.316914E-06 1.586591E-12 9.999655E-01

220 9.411223E-02 1.367166E-06 1.172121E-12 9.999379E-01

210 9.182657E-02 1.333061E-06 1.020492E-12 9.999294E-01

200 7.825889E-02 1.407630E-06 6.322727E-13 9.998996E-01

190 6.325375E-02 1.523002E-06 9.250000E-14 9.998097E-01

180 5.166243E-02 1.579349E-06 -2.055303E-13 9.998647E-01

170 3.807934E-02 1.662496E-06 -5.761364E-13 9.998368E-01

160 2.455800E-02 1.744623E-06 -9.376515E-13 9.997880E-01

150 1.736805E-02 1.765414E-06 -1.138333E-12 9.998578E-01

140 3.334485E-03 1.848065E-06 -1.451970E-12 9.999078E-01

130 2.127039E-03 1.784318E-06 -1.350985E-12 9.999539E-01

120 -2.481791E-03 1.765565E-06 -1.383258E-12 9.999038E-01

100 5.187879E-06 1.612999E-06 -1.147424E-12 9.999788E-01

80 6.216523E-03 1.420912E-06 -7.814015E-13 9.999127E-01

60 8.666985E-03 1.291416E-06 -6.017424E-13 9.998623E-01

8

APPENDIX A.3. – COEFFICIENTS FOR THE MACH NUMBER

Defined on [140,230] tonnes.

FL C2 (const) D2 (for mass) E2 (for mass^2) Adjusted R2

400 3.688970E-01 4.632424E-06 -1.136364E-11 9.568275E-01

390 2.966182E-01 5.230606E-06 -1.257576E-11 9.902816E-01

380 2.408152E-01 5.601364E-06 -1.310606E-11 9.967217E-01

370 1.956864E-01 5.830076E-06 -1.321970E-11 9.985497E-01

360 1.737561E-01 5.772955E-06 -1.253788E-11 9.972979E-01

350 1.468470E-01 5.779318E-06 -1.208333E-11 9.977177E-01

340 1.149803E-01 5.859621E-06 -1.193182E-11 9.976223E-01

330 7.384242E-02 6.050303E-06 -1.212121E-11 9.988682E-01

320 7.059242E-02 5.846894E-06 -1.132576E-11 9.991947E-01

310 1.204515E-01 5.067727E-06 -9.015152E-12 9.981204E-01

300 1.781470E-01 4.213712E-06 -6.553030E-12 9.984434E-01

290 2.522061E-01 3.215455E-06 -3.787879E-12 9.988326E-01

280 3.067212E-01 2.470303E-06 -1.818182E-12 9.994943E-01

270 3.307333E-01 2.091515E-06 -9.090909E-13 9.998664E-01

260 3.530364E-01 1.751212E-06 -1.515152E-13 9.999303E-01

250 3.542636E-01 1.658333E-06 -7.575758E-14 9.999534E-01

240 3.432394E-01 1.705606E-06 -3.787879E-13 9.998794E-01

230 3.372030E-01 1.700455E-06 -5.303030E-13 9.999679E-01

220 3.281030E-01 1.730000E-06 -7.575758E-13 9.999263E-01

210 3.243833E-01 1.696742E-06 -7.954545E-13 9.999199E-01

200 3.103576E-01 1.763333E-06 -1.060606E-12 9.998774E-01

190 2.958394E-01 1.856818E-06 -1.439394E-12 9.997681E-01

180 2.827212E-01 1.917273E-06 -1.666667E-12 9.998316E-01

170 2.674970E-01 2.004545E-06 -1.969697E-12 9.997833E-01

160 2.520485E-01 2.093030E-06 -2.272727E-12 9.997307E-01

150 2.427333E-01 2.121818E-06 -2.424242E-12 9.997953E-01

140 2.256182E-01 2.223636E-06 -2.727273E-12 9.998545E-01

130 2.202455E-01 2.190758E-06 -2.651515E-12 9.999466E-01

120 2.106970E-01 2.209697E-06 -2.727273E-12 9.998250E-01

100 2.059848E-01 2.100758E-06 -2.500000E-12 9.999830E-01

80 2.043470E-01 1.955833E-06 -2.159091E-12 9.999291E-01

60 2.002424E-01 1.857273E-06 -1.969697E-12 9.998875E-01

9

APPENDIX A.4. – Adjusted R2 for the linear fuel flow model (using coefficients A & B), and the

quadratic models for M2 (using coefficients C, D & E) and M (using coefficients C2, D2 & E2).

The linear fuel flow rate function fitted to the LRC cruise data for flight levels around FL280 shows a

small reduction in the coefficient of determination owing to a small jump in the source data. The

quadratic functions identified for the Mach number, and Mach number squared, cease providing a

satisfactory fit at flight levels in the stratosphere. At these high flight levels, from FL370 and above,

the LRC Mach number remains constant at Mach 0.838 over a wide portion of the aeroplane mass

interval, before eventually reducing with decreasing mass. This is in contrast to the lower flight levels

at which there is an immediate reduction in Mach number with decreasing mass. There is a second

reduction in the coefficient of determination for the fuel flow rate function after FL370 caused by

the presence of significant wave drag that requires additional thrust, and therefore an increased fuel

flow rate, to overcome.

9.960E-01

9.965E-01

9.970E-01

9.975E-01

9.980E-01

9.985E-01

9.990E-01

9.995E-01

1.000E+00

1.001E+00

50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390

Ad

just

ed

R2

FLIGHT LEVEL (geopotential standard altitude in hectofeet)

Adjusted R2 for the fuel flow linear model, and the quadratic models for M2 and M.

Fuel flow Adj R2

MachSqrd Adj R2

Mach Adj R2

B Martin, 2014. Aqqa.org

APPENDIX A.5. – Range potential for LRC cruise at FL350. With 3% performance degradation.

y = 1.897928E+00x2 - 5.578938E+02x + 1.242380E+04

R² = 9.999997E-01

0

500

1000

1500

2000

2500

3000

3500

16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0

Ra

ng

e p

ote

nti

al (

na

uti

cal a

ir m

iles)

Time UTC (hours)

FL350 LRC range potential (nautical air miles)

Boeing 777-200ER/GE90-94B. Uniform temperature deviation 10 K.

44 660 kg fuel on board (49.1 tonnes at brake-release, estimated 4 440 kg consumed in

climb); ZFM 174 tonnes. Includes 3% increased fuel flow rate

Barry Martin, 2014. Aqqa.org

APPENDIX A.6. – Mass profile. LRC cruise at FL350. With 3% performance degradation.

y = 8.478086E+01x2 - 9.632870E+03x + 3.578987E+05

R² = 9.999998E-01

170000

175000

180000

185000

190000

195000

200000

205000

210000

215000

220000

16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0

Ae

rop

lan

e a

ll u

p m

ass

(ki

log

ram

s)

Time UTC (hours)

FL350 LRC mass profile

Boeing 777-200ER/GE90-94B. Uniform temperature deviation 10 K.

44 660 kg fuel on board (49.1 tonnes at brake-release, estimated 4 440 kg consumed in

climb); ZFM 174 tonnes. Includes 3% increased fuel flow rate.

Barry Martin, 2014. Aqqa.org

APPENDIX A.7. – True airspeed profile. LRC cruise at FL350. With 3% performance degradation.

y = -2.156780E-01x2 + 5.122395E+00x + 4.662736E+02

R² = 9.999598E-01

460

465

470

475

480

485

490

495

16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0

Tru

e a

irsp

ee

d (

kno

ts)

Time UTC (hours)

FL350 LRC true airspeed profile (knots)

Boeing 777-200ER/GE90-94B. Uniform temperature deviation 10 K.

44 660 kg fuel on board (49.1 tonnes at brake-release, estimated 4 440 kg

consumed in climb); ZFM 174 tonnes. Includes 3% increased fuel flow rate

Barry Martin, 2014. Aqqa.org