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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