The turbo fan aircraft minimum
cost climb technique
P. Mirosavljević, S. Gvozdenović and O. Čokorilo
The Faculty of Transport and Traffic Engineering, University of Belgrade, Belgrade, Serbia
Abstract
Purpose – The purpose of this paper is to define minimum cost technique of turbo fan transport aircraft in the presence of dynamic change of aircraft
performance. Results can be practical applicable in airlines for achieving minimal operation costs.
Design/methodology/approach – Logarithmic differential is applied for defining conditions in order to achieve optimal Mach number for minimal
climb cost. This condition is solved numerically by using Newton-Ramphson method, to obtain optimal Mach number distribution with altitude.
Conclusion about optimal top of climb (TOC) is defined after analyses for different aircraft mass and cost indexes.
Findings – Proposed method of minimum cost climb resulting in potential savings up to 5 per cent compared to Aircraft Flight Manual climb law.
Proposed method also made correction of climb law and optimal TOC under existence of aircraft performance degradation.
Practical implications – Use of defined climb law and optimal TOC will minimize cost of en route flight profile.
Originality/value – At present, there is no definition of climb technique for minimum cost of en route flight profile, under dynamic degradation of
aircraft performance. Final results are standardized to become applicable and easy to use with modern and old type of flight management system.
Keywords Air transport, Aeroplanes, Operating costs, Aerodynamics
Paper type Research paper
subject in many papers and research studies as a phase which
represent connection between initial climb and cruise with
relatively small investigation of costs generate during that
phase. In the operative flight preparation, planners can use
climb law which is published in Aircraft Flight Manual (AFM)
or climb law defined by flight management system (FMS)
according to set cost index (CI). FMS performance data base
is not aware of dynamic change of aircraft performance and
should be updated if is that possible. Dynamic changing in
airline environment for many air companies are often
overlooked and become source of hidden costs.
Optimal vertical short-haul flight profile was the topic for
many authors. In order to achieve optimal flight path, other
authors used calculus of variation which is apply on total
energy aircraft model. Such an example is work of Schultz and
Zagalsky (1972) who implemented calculus of variation to
develop optimal flight path. Barman and Erzebrger (1976)
developed optimal short-haul flight which includes minimum
costs trajectory. Optimal flight profile was developed
according to aircraft model based on approximation of
energetic condition of the aircraft. The calculus of variation
was again used to achieve optimal result. Simons and
Jenkinson (1985) developed sub-optimal, short-haul
minimum fuel flight profile, for propeller aircrafts by using
multivariable optimizations to develop flight profile with
minimum usage of fuel. Many authors analyzed flight only to
find result for optimizing climbing, descending or cruising
resulting minimum usage of time (Ojha, 1993) and fuel
(Calise, 1977).
In the paper (Ojha, 1993), authors achieved minimum time
climb with included presumptions. Authors like Maido
(2006) stress out the need of more detailed variation of data
and multiple approach in order to achieve quality results,
which would be lost if we would include presumptions. One
can notice that conditions for minimum costs climb are not
enough researched and defined.
Nomenclature
AF
¼ Acceleration factor
CASopt ¼ optimal CAS
CF
¼ Fuel price (USD/kg)
CT
¼ flight hour price (USD/Fhr)
CTOC ¼ total en route flight cost (USD)
¼ aerodynamic drag coefficient
Cx
Cz
¼ aerodynamic lift coefficient
F
¼ Fuel flow(kg/s)
K
¼ 1 þ AF
mcl1
¼ aircraft mass at the beginning of climb
RCC
¼ Rate of climb cost (USD/m)
ROC ¼ Rate of climb (m/s)
Rx
¼ aerodynamic drag force (N)
Tn
¼ turbo fan engine net thrust (N)
TOCopt ¼ Optimal top of climb
TOD ¼ Top of descent
VT
¼ True air speed [m/s]
X
¼ range [m]
r
¼ air density in ISA conditions (kg/m3)
Introduction
Recently, there is rising competition between air carriers
which require costs determination and reduction. The process
of cost reduction demand determination of all sources of costs
and possibilities for their reduction. The climb phase was
The current issue and full text archive of this journal is available at
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Aircraft Engineering and Aerospace Technology: An International Journal
81/4 (2009) 334– 342
q Emerald Group Publishing Limited [ISSN 1748-8842]
[DOI 10.1108/00022660910967327]
334
The turbo fan aircraft minimum cost climb technique
Aircraft Engineering and Aerospace Technology: An International Journal
P. Mirosavljević, S.Gvozdenović and O.Čokorilo
Volume 81 · Number 4 · 2009 · 334 –342
The present models of minimum time climb and minimum
fuel climb flight path model are optimized on the basis of
flight efficiency without concerning operative efficiency of
such flight. The paper which represent need for connection
between operational and flight efficiency is McLean (2006),
also highlighted need for accomplishment of both goals.
It was noticed that solution for minimum cost climb
technique which cover both operational and flight efficiency,
under of presence of dynamic disturbance, is not identified.
In this paper, is shown method for solution of that
optimization problem. The optimization problem solution
must be standardized in form of climb law and top of climb
(TOC), for application in aircraft operations. The aim of this
paper is defining optimal climb strategy on short haul flights
with satisfied criterion of minimal operations cost under
presence of dynamic disturbance, with results standardization
for operative application.
During transport aircraft exploitation is evident degradation
of aircraft performances, as a result of non-adequate aircraft
maintenance. The degradation range is from 1 per cent on the
first 3,000 Fhr to 10 per cent (IATA, 2004). Cook et al.
(2007) had defined importance of dynamic disturbance,
which arise before flight and can influence on total operation
costs. The problem which was analyzed is practical and is
present in everyday aircraft operations.
It should take optimal result get as close as possible to
operative use, which means air traffic control restrictions must
be obeyed. The transport aircraft in flight operations use
climb law in form of IAS/M (McLean, 2006). In the paper,
we also show technique which gives us sub-optimal law of
climbing based on optimization results. Sub-optimal law of
climbing has minimum exception margin from the optimal
speed distribution in climbing. Sub-optimal climb law
consists of calibrated air speed (CAS) speed and M
number, which makes it useful in operative exploitation
obeying system limitations.
In cases from works Schultz and Zagalsky (1972) and
Barman and Erzebrger (1976) in which authors researched
optimization of all flight phases, were not analyzed
acceleration phase on TOC, from speed in the end of
climbing to speed in the beginning of cruising. In this paper,
the influence of acceleration phase, on total operating cost, is
also calculate.
The aircraft climb optimization is possible on the base of
aircraft documents and/or FMS. The basic aircraft flight
planning documents are AFM, Flight Planning and
Performance Manual (FPPM) (The Boeing Company, 2003a,
b) and Performance Engineers Manual (PEM) (The Boeing
Company, 1985). In the content of AFM is usually only
one climb law available without options of optimization. The
FMS memory where is entered performance data base
(PDB), is not enough to contain all different aircraft
performance model. PDB depends on aircraft engine
combination. It is possible that FMS’s PDB not match to
aircraft on which is installed, exactly. In that case, air carrier
must conduct corrections of PDB to achieve total match
between aircraft model. During aircraft exploitation, PDB is
tune, when aircraft drag and engine characteristic begin to
deviate, form book level. Recently, there exists methods for
prediction of aircraft performance deviation, on which base is
determined PERF FACTOR (Airbus Industry, 2002).
Some aircraft like ERJ145, Fokker100, and B737300
do not have option of update PDB by PERF FACTOR.
For such cases, flight preparation must conduct on actual
aircraft performance data in pre flight preparation.
For aircraft equipped with older types of FMS without
option of update PDB flight must be prepare on the basis of
add-on software which is offered by aircraft manufacturer
(Airbus IFP) or as service offered from other companies
(Lido OC).
The trend of update of PERF FACTOR and also
possibilities of PDB update is possible on the basis of data
obtain by aircraft performance monitoring (APM), such as
flight operational quality assurance (FOQA). Form FOQA is
possible to gather data on the daily basis for every aircraft in
fleet. This opens possibility of daily aircraft performance
optimization based on real data, before each flight for each
aircraft in air carrier’s fleet.
Generally, this paper is organized in three parts. In the first
part of the paper proposes methodology for determination of
new minimum cost-climb technique, under aircraft drag and
engine performance degradation. The second part analyzes
influence of aircraft mass and CI on optimal TOC choice,
which minimize en route flight profile under consideration of
dynamic aircraft performance change. The third part of the
paper proposes method which prepare obtained solutions
in the first part of paper, for application in passenger
aircraft FMS.
In the case when aircraft is equipped with older type of
FMS without option of PDB update (IATA, 2004) by
proposed method final results are applicable during
pre-flight FMS preparation. In the case when is aircraft
without FMS by proposed method, final results are
applicable for flight on the base of air speed indicator,
Mach meter and altimeter.
The purposed method is alternative to flight preparation
based on AFM and FPPM data.
Assumptions
In the paper, B737-300 type of aircraft is accepted as
reference aircraft, equipped with older generation of FMS
without PDB update option.
In order to research climb impact, we analyzed simplified
flight profile which includes only en route part of flight.
In order to use comparison, in the paper, we use tabular
calculation method (Jenkinson et al., 1999) for fuel, time and
distance calculation. This model is based on “step by step”
calculation of each phase of chosen intervals heights
in climbing and descending, and for chosen parts of range
in cruising.
Assumptions that are accepted are as follows: small angle of
climb (Houghton and Brock, 1970), e.g. g , 138 and cos
g < 1, sin g < g, CAS ¼ IAS, aircraft’s weight is equal to lift
force and drag force is equal to cruising pressure power.
Besides these assumptions, we also used assumptions about
ISA flight conditions. The range that aircraft performs during
cruise is calculated as difference between total range R (km)
and range which aircraft makes during descending, climbing
and speeding:
Rcr ¼ R 2 Rdes 2 Rcl 2 Racc
ð1Þ
Speed limits in all flight phases are in form of maximum speed
CASMO and maximum MMO. In order to research climb
impact, we analyzed simplified flight profile which includes
only en route part of flight. En route phase starts with climbing
335
The turbo fan aircraft minimum cost climb technique
Aircraft Engineering and Aerospace Technology: An International Journal
P. Mirosavljević, S.Gvozdenović and O.Čokorilo
Volume 81 · Number 4 · 2009 · 334 –342
to 1,500 ft QNH to TOC based on optimal climb law,
acceleration phase on cruising height to cruising speed,
cruising with constant height from TOD, with constant
descent law 074/250 kt to 1,500 ft QNH. This cruise and
descent regime was taken from FPPM (The Boeing
Company, 2003b). Such cruising and descending phase
model was chosen in order to display the usefulness of
minimum costs climb technique. For climbing, there is used
maximum climb thrust setting, whereas in cruising, program
of continual thrust setting decrease is used as cruise
progresses, because the cruise is done under const h and
const Mcr. In descending, regime of minimum power is used
(low idle thrust).
In order to define optimal Mach number distribution for
minimum climb cost, it is necessary to present aerodynamic
data and aircraft engine data in functional form, with
influence of dynamic degradation.
Entry data for flight’s parameters definition are taken from
PEM. The aircraft high-speed drag polar is presented
through coefficient of aerodynamic drag Cx which is
function of two constants Cz and M. The program APM is
used for determination of DC drag correction, as a percent
of real data change compared to PEM data. Since function
which describes coefficient of aerodynamic drag, equation
(2) also includes coefficient Prandtl, we achieve the average
square of coefficient of correlation from 0.95, compared to
data of aircraft’s manufacturer (The Boeing Company,
1989):
k5 M
DC
ð2Þ
Cx ¼ k3 þ k4 M þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ
100
1 2 M2
on SL. Fuel flow for climb and cruise phases is presented by
equation (10):
2 !
pffiffiffiffi
Tn
Tn
F ¼ dt ut ffst þ D ·
þE·
d
d
ð10Þ
0:453
FC
1þ
3600
100
ð3Þ
k4 ¼ k40 þ k41 C z þ k42 C 2z
ð4Þ
k5 ¼ k50 þ k51 C z þ k52 C 2z
ð5Þ
ð12Þ
ffst ¼ ffst0 þ ffst1M þ ffst2M 2
ð13Þ
where F (kg/s) fuel flow, FC (per cent) fuel flow correction.
This correction is consequence of engine characteristic
degradation. Approximation of realistic parameters of fuel
flow of aircraft engine (The Boeing Company, 1985), by use
of interdependence presented by equations (11)-(13) can be
calculated by square of coefficient of correlation from 0.96.
Aircraft climb model
In order to set climb parameters, we applied basic climb equals,
in which height altitude of 1,500 ft QNH to TOC climb altitude
was divided to i (i ¼ 1, . . . , j) segments. The aircraft mass in
first climb segment is mcl1. For each climb segment ith, we define
spent time tcli, fuel gcli and range Xcli. Equations which describe
climb flight are (Jenkinson et al., 1999):
pffiffiffi
pffiffiffi
dh
¼ M · asl · u sin g < M · asl · ug
dt
pffiffiffi
pffiffiffi
dX
¼ M · asl · u cos g < M · asl · u
dt
dmcl
¼ 2F cl
dt
Rz ¼ g mcl
pffiffiffi
dðM · asl · uÞ
· mcl ¼ Tncl 2 RX 2 g · mcl g
dt
By analyzing interdependence of characteristics of turbo-fan
engines (Kahayas, 2007; Raymer, 2006; Mair and Birdsall,
1992) and realistic characteristics of engine (The Boeing
Company, 1985), we notice that equation (6) of
approximation of realistic parameters of engine’s
parameters with square of coefficient of correlation from
0.998:
ð14Þ
ð15Þ
ð16Þ
ð17Þ
ð18Þ
Limitations on which basis we calculate climb flight are:
.
available thrust is equal to the maximum climb thrust:
T max cl ¼ Tncl
.
.
Tn
¼ T o þ A · M þ B · M2
d
N1
N1 2
A ¼ a1 þ pffiffiffiffi a2 pffiffiffiffi
ut
ut
N1
N1 2
B ¼ b1 pffiffiffiffi þ b2 pffiffiffiffi
ut
ut
N1
N1 2
T o ¼ t1 pffiffiffiffi þ t2 pffiffiffiffi
ut
ut
ð11Þ
2
E ¼ e0 þ e1M þ e2M
where DC (per cent) is aircraft drag correction as a
consequence of aerodynamic degradation. Where coefficients
are k3, k4 and k5:
k3 ¼ k30 þ k31 C z þ k32 C 2z
D ¼ do þ d1M þ d2M 2
ð6Þ
.
ð7Þ
.
;hi ; i ¼ 1; . . . ; j
ð19Þ
fuel flow is function of flight altitude, Mach number and
thrust during climbing;
climb can be considered up to height which represents
operative top of the flight, and which is defined by
ROCmax ¼ 2.54 m/s (500 ft/min);
flight is straight, without turns or change of flight
direction; and
the change of climb angle is small g_ ¼ 0.
ð9Þ
Parameters needed to define climb parameters for flight in ith
segment height intervals, with constant M number in
troposphere are: ROCi, for used fuel, ðROCi =F i Þ and for
flight range in climb it is necessary to first define climb angle gi:
pffiffiffiffi
Group of functions (7)-(9) present influence N1= ut on
parameters of corrected thrust Tn/d (N), where d is
relationship between pressures of ambient air and pressure
ð20Þ
mcl i ¼ mcl i21 2 g cl i21 ; ði ¼ 2; . . . ; jÞ
pffiffiffiffi
ðM i ·asl · ui ðTncl i 2RXi ÞÞ=ðg ·mcl i Þ
; ði ¼ 1; ...;jÞ ð21Þ
ROCi ¼
120:133M 2i
ð8Þ
336
The turbo fan aircraft minimum cost climb technique
Aircraft Engineering and Aerospace Technology: An International Journal
P. Mirosavljević, S.Gvozdenović and O.Čokorilo
Volume 81 · Number 4 · 2009 · 334 –342
ROCi
Fi
pffiffiffiffi
ð22Þ
ðM i ·asl · ui ðTncl i 2RXi =g ·mcl i ÞÞ=ð120:133M i Þ
;
¼
Fi
ði ¼ 1; ...; jÞ
ðTncli 2 RXi Þ=ðg · mcl i Þ
; ði ¼ 1; . . . ; jÞ ð23Þ
gi ¼ ArcSin
1 2 0:133M 2i
Unit climb costs can be formulated by linking unit climb time
costs that are function of climb time and unit climb fuel costs,
which are function of fuel spent while climbing. Based on
equation (33) it is possible to calculate speed of costs change
with height. By multiplication of unit time costs with integral
function of climb time, we get change of unit time costs with
altitude in form:
CT
1
1 USD 1
USD
!
ð34Þ
3; 600 ROCi 3600 Fhr ðm=sÞ
m
For basic ith segment of climb, it is possible to define time
needed in order to climb from height hi2 1 to height hi:
Z hi
dh
; ði ¼ 1; . . . ; jÞ
ð24Þ
t cl i ¼
ROC
i
hi
Z hi
Fi
g cl i ¼
dh; ði ¼ 1; . . . ; jÞ
ð25Þ
hi ROCi
pffiffiffi
ð26Þ
X cl i ¼ t cl i ðcos gi ÞM i asl ui ; ði ¼ 1; . . . ; jÞ
and by multiplication of unit fuel costs with sub integral
function of climb fuel we get change of unit fuel costs with
altitude in the form of:
Fi
USD ðkg=sÞ
USD
!
ð35Þ
CF
kg ðm=sÞ
m
ROCi
After we arrange this redefined condition of minimum climb
costs on ith climb segment, we can calculate the definition of
costs change with altitude RCCi (USD/m) (rate of climb cost):
Climbing parameters from i ¼ 1, which fits to segment of height
at the beginning of climbing to i ¼ j, which fits to segment of
height TOC are total climb time, fuel and range:
t cl ¼
j
X
t cl i
ð27Þ
g cl i
ð28Þ
i¼1 X cl i
1; 000
ð29Þ
RCCi ¼
j
X
i¼1
Pj
Rcl ¼
Optimization of Mach’s climb number under
minimum costs criterion
For each segment of climbing, it is possible to define it by
method of logarithmic differential, optimal Mi number for
minimum climb costs. Total operation climb costs on ith climb
segment Ccl i ðUSDÞ, consist of climb fuel cost and climb time
cost. We can develop function of climb costs as follows:
C cl i ¼
ðCT · t cl i Þ
3; 600 þ CF · g cl i
þ ln Gcli þ ln K i
100 CI
þ ln CF F i
þ1
3; 600 F i
ð30Þ
ð38Þ
The condition for minimizing RCC i is achieved by
logarithmic differential lnRCCi by lnMi. The result of
differential is, then, equalized with 0 and solved by Mi.
Condition for minimum climbs costs, in case if we optimize
by Mi, for the representative engine performance, can be
presented as:
› ln RCCi
d ln M i ¼ 0
ð39Þ
d ln RCCi ¼
› ln M i
Minimizing climb cost Ccl i is possible if we minimize sum of
costs of fuel spent and time spent on ith climb segment.
In operative conditions of exploitation, it is necessary to express
costs Ccl i as function CI (McLean, 2006; Root, 2001), so to
define which criteria stresses more importance:
CT
¼ CI
CF · 100
ð36Þ
Aim of optimizing is minimizing of RCCi. The aim of
optimizing can be achieved if we define optimal distribution of
M number with climb altitude. Technique used to find
condition of optimal Mi for each ith climb segment is
logarithmic differential.
Lets logarithm RCCi on ith climb segment:
1
CT
ln RCCi ¼ ln
þ CF Fi
ROCi 3; 600
Ki
pffiffiffiffi
¼ ln
ð37Þ
M i asl ui ððTncli 2 Rxi Þ=ðGcli ÞÞ
CT
þ CF F i
þ ln
3; 600
pffiffiffiffi
ln RCCi ¼ 2ln M i 2 ln asl ui 2 lnðTncli 2 Rxi Þ
i¼1
g cl ¼
CT
1
Fi
þ CF
3; 600 ROCi
ROCi
ð31Þ
As a solution of this dilemma, it is possible to set up function of
costs Ccl i as a sum of time and fuel costs on ith segment:
› ln K i › lnðTncli 2 Rxi Þ › ln F i
2
þ
› ln M i
› ln M i
› ln M i
› lnðð100=3600ÞðCI=F i Þ þ 1Þ
þ
› ln M i
0 ¼ 21 þ
ðCT · t cl i Þ
ð32Þ
3; 600 þ CF · g cl i
Z hiþ1
Z hiþ1
CT
0:3048
Fi
dh þ CF
0:3048
dh ð33Þ
C cl i ¼
3;600 hi ROCi
ROCi
hi
C cl i ¼
ð40Þ
Condition for calculation of Mi on ith climb segment for
minimum climb costs Ccli:
337
The turbo fan aircraft minimum cost climb technique
Aircraft Engineering and Aerospace Technology: An International Journal
P. Mirosavljević, S.Gvozdenović and O.Čokorilo
Volume 81 · Number 4 · 2009 · 334 –342
CI 100
M ›K i
› lnðTncli 2 Rxi Þ
212
0¼
þ1 ·
› ln M i
F i 3600
K i ›M
ð41Þ
› ln F i
þ
› ln M i
Figure 1 The change of DCTOC with TOC, for minimum cost climb
technique, mcl1 ¼ 41,000 kg
100
0
It can be develop tree cases, CI ¼ min, CI ¼ (any value
between min and max) and CI ¼ max (Figure 5). After
aircraft reaches TOC, we calculate fuel, time and distance
needed to speed up to Mcr. After set up of optimal Mach
number distribution in minimum cost climb, it can be defined
second phase of optimization, which is determination optimal
TOC. This optimization task has goal minimization of en route
flight profile total operational costs.
In the end, we can calculate sum of results: total fuel spent
in all flight phases gt (kg), time tt (Fhr) and costs CTOC (USD)
in all flight phases:
g t ¼ g des þ g cr þ g acc þ g cl
ð42Þ
t t ¼ t des þ t cr þ t acc þ t cl
ð43Þ
DCTOC (USD)
–100
TOC
Min cost climb CI5 deg
Min cost climb CI20 deg
Min cost climb CI30 deg
Min cost climb CI50 deg
Min cost climb CI90 deg
Min cost climb CI5
Min cost climb CI20
Min cost climb CI30
Min cost climb CI50
Min cost climb CI90
–500
–600
–700
20,000
22,000
ð44Þ
24,000
26,000
28,000
30,000
TOC (ft)
where CT(USD/Fhr) denotes to the cost of time and
CF(USD/kg) denotes to the cost of fuel. As we said, the
second optimization task is to find TOC to minimize total
operating costs CTOC:
CTOC ! min
–300
–400
Total expenses during en route flight are CTOC represented by
equation (44):
tt
CT þ g t CF
C TOC ¼
3; 600
–200
Figure 2 The change of DCTOC with TOC, for minimum cost climb
technique, mcl1 ¼ 51,000 kg
50
ð45Þ
0
Optimization of TOC under minimum costs
criterion
–50
–100
DCTOC (USD)
There have been several experiments on presented model en
route flight profile, in order to notice advantages and
disadvantages of climbing on different TOC.
In order to research influence of climb technique, we took
into consideration following three factors that influence time,
fuel and costs of total flight: CI (from 5 to 90)., aircraft mass
at the beginning of climb mcl1 (41,000, 51,000 and 61,000 kg)
and altitude TOC (from 20,000 to30,000 ft). All analysis are
done for short-haul range R ¼ 840 km. Aircraft flight model
and presented model of aerodynamic and engine
characteristics is used for making software OPTCLIMB on
platform Mathematicaw5.1. Results which are acquired by
software, can help in process of defining optimal TOC
(Figures 1-3) for given CI. It can be determined measure of en
route cost deviation DC TOC from minimal en route cost. That
deviation must be minimized, which is criterion for defining
optimal TOC for is given aircraft mass, range and CI:
–150
–200
Min cost climb CI5 deg
Min cost climb CI20 deg
Min cost climb CI30 deg
Min cost climb CI50 deg
Min cost climb CI90 deg
Min cost climb CI5
Min cost climb CI20
Min cost climb CI30
Min cost climb CI50
Min cost climb CI90
–250
–300
–350
–400
–450
20,000
22,000
24,000
26,000
TOC (ft)
28,000
30,000
total operating cost can be achieve by application of minimum
climb cost technique to TOCopt, with or without presence
of dynamic disturbance caused by aircraft performance
degradation. In this, particular case is assumed aerodynamic
degradation DC ¼ þ 3 per cent and fuel-flow degradation
FC ¼ þ 3 per cent.
In case of climb with mcl1 ¼ 41,000 kg, Figure 1, we found
TOCopt ¼ 30,000 ft for range of CI from 5 to 50. For CI value
from 50 to 90, it can be determined decrease of TOCopt from
30,000 to 26,000 ft. In case of climb with mcl1 ¼ 51,000 kg,
DC TOC ¼ minðC TOC ; TOC ¼ 20; 000 ft; . . . ; 30; 000 ftÞ
2 C TOC
ð46Þ
DC TOC ! 0
From analysis conducted at fixed range R ¼ 840 km, it was
determined DC TOC change (Figures 1-3) with TOC.
The criterion DC TOC ¼ 0 define optimal TOC or TOCopt.
The mark deg on Figures 1-3 denote to case when is present
aircraft performance degradation. Minimization of en route
338
The turbo fan aircraft minimum cost climb technique
Aircraft Engineering and Aerospace Technology: An International Journal
P. Mirosavljević, S.Gvozdenović and O.Čokorilo
Volume 81 · Number 4 · 2009 · 334 –342
Figure 3 The change of DCTOC with TOC, for minimum cost climb
technique, mcl1 ¼ 61,000 kg
For lower altitudes, climbing is performed with constant
CAS, and for higher altitudes with const M, because of shape
of optimum M number curve change, as it is shown on
Figure 4. Finding optimal distribution of change of M
number with altitude, for minimum costs climb technique,
can be calculated based on iterative procedure Jenkinson et al.
(1999). In the procedure, first, we numerically define by
Newton-Ramphson method (Bressound, 2006), values of
optimal M numbers, for different altitudes h, then, acquired
result is analytically described with function M from h. By
iterative procedure for array of j segments of altitudes h, we
acquire optimal M numbers for minimum costs climb
technique for defined CI. Those are solutions results of
equation (30). Afterwards, we proceed to approximation of
numeric data for altitude h and optimal M number, by
method of smallest square. Let us name this approximate
function fopt. For defined function fopt, we define surface
below fopt in function of altitude h. In order to define speed
CASopt we should define minimum differences between
functions fopt and M opt0 which is function CASopt and h.
Functional connection M opt0 ¼ f ðCASopt ; hÞ is defined by
equation (47):
50
0
–50
DCTOC (USD)
–100
–150
Min cost climb CI5 deg
Min cost climb CI20 deg
Min cost climb CI30 deg
Min cost climb CI50 deg
Min cost climb CI90 deg
Min cost climb CI5
Min cost climb CI20
Min cost climb CI30
Min cost climb CI50
Min cost climb CI90
–200
–250
–300
–350
–400
20,000
22,000
24,000
26,000
TOC (ft)
28,000
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2 0 0
11=ð7=2Þ 3
1
u
2 !7=2
u6
1
CAS
7
opt
M opt0 ¼ u
1þ0:2
21A þ1A
215
t45@ @d
asl kt
30,000
Figure 2, we found TOCopt form 28,000 to 30,000 ft for range
of CI from 5 to 50. For CI value from 50 to 90, it can be
determined decrease of TOCopt from 28,000 to 22,000 ft.
In case of climb with mcl1 ¼ 61,000 kg, Figure 3, we found
TOCopt form 27,000 to 24,000 ft for range of CI from 5 to
30. For CI value from 30 to 90, it can be determined decrease
of TOCopt from 24,000 to 20,000.
ð47Þ
where d presents relative ambient pressure in troposphere.
We can notice a very important fact, that Mach’s number
Mopt special case of CASopt on altitude hco. The function that
links Mopt number and CASopt speed, on altitude hco for
troposphere can be calculated by equation (48):
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2 0 0
11=ð7=2Þ 3
1
u
2 !7=2
u6
1
CAS
7
opt
M opt ¼ u
1þ0:2
21A þ1A
215
t45@ @d
asl kt
co
The sub-optimal climb law
Once distribution of optimal M number values, with altitude h is
defined, for minimum costs climb technique, it is necessary to
enable operative use of achieved result, so that it can be applicable
in FMS or flight management and guidance system (FMGS).
That presents new optimization problem that is: adjustment of
theoretic results, to practical use, in the way that optimization
results are approximated in form of constant speed CAS and
constant M number, (climb law), so to be entered before take-off
in FMS or FMGS. Post-optimal results adjustment to operative
use requires solving of new optimization task.
The aim of post-optimal adjustment is:
.
to define constant speed CAS and up on what altitude hco
(cross-over altitude) it takes to apply const CAS, with least
differences from optimal change of M number with h,
(defined for minimum costs);
.
to define hco, up to which constant CAS is applied and
from what we begin to apply climb with const M number;
.
to find M number that is constant during climb, and that
is applied from altitude hco and which has least deviation
from optimum change of M number with altitude h
(defined for minimum costs).That is achieved by
satisfying the condition described in equation (41); and
.
constant CAS and constant M represent optimum
combination (climb law), connected to altitude hco, on
which it comes to cross-over from CAS to M.
ð48Þ
Figure 4 The approximation of optimal Mach number function with
altitude h, form minimal operating cost
35,000
Const M which is the
best approximation of
optimal M change
with altitude h
30,000
h (ft)
25,000
Const CAS
which is the best
approximation
of optimal M
change with h
altitude
M number change
with h for min
climb cost
20,000
15,000
Altitude hco at
which climb law
change from
const CAS to
const M
10,000
5,000
0
0.4
339
0.5
0.6
M
0.7
0.8
The turbo fan aircraft minimum cost climb technique
Aircraft Engineering and Aerospace Technology: An International Journal
P. Mirosavljević, S.Gvozdenović and O.Čokorilo
Volume 81 · Number 4 · 2009 · 334 –342
where dco is relative ambient pressure in troposphere on
altitude hco, asl kt (kt) speed of sound on SL, CAS(kt).
General condition can be defined, as a minimization of sum
of absolute difference surfaces, shown in equation (49). This
general condition is satisfied under condition given in
equation (50). Solution of optimization task is in finding
two variables CASopt and hco. By using numerical Newton
method of minimization we get solution of these two variables
in range of given conditions (50). In equation (49) in square
brackets, the sum of absolute differences of surfaces is shown:
.
First one below function of fopt (from starting altitude at
beginning of climb h0 to altitude hco) and another surface
below curve defined by equation (47) (from altitude of
beginning of climb altitude h0 to altitude hco).
.
Another surface, second in square bracket in equation
(49) is below function fopt (from altitude of cross-over hco
to altitude at the end of climb hj) and surface below curve
defined in equation (48) (from altitude of cross-over hco to
altitude at the end of climb hj):
Company, 2003b) proposed climb law 280/0.74, cruise at
Mcr ¼ 0.74, descent law 074/250. Analyzes show reduction of
total operating cost from 4.8 to 3.2 per cent, as shown on
Figure 6, for mcl0 ¼ 51,000 kg and R ¼ 840 km:
Z hco
Minimize
CASopt ;hco
Z hj
M opt2f opt dh
jM opt0 2f opt jdhþ
h0
St ref ¼
ð49Þ
subject to
ð50Þ
M opt # M MO and M opt0 # M MO ; and
h0 # h # hj ; and h0 # hco # hj
The minimization of equation (49) under condition (50)
obtain two variables hco and CASopt on which base can be
develop third variable Mopt. In this way can be obtained
solution with minimal differences between function fopt and
climb law consist of CASopt and Mopt and pressure altitude
hco. On Figure 5 is represented operative adoption of function
fopt to climb law, which provide minimum cost climb.
Figure 6 The percent of total en route operating costs savings
The application of proposed climb method
5.00
TOC = 30,000ft
Proposed method can be also applicable if we do not have FMS
on board in aircraft and have a goal to minimize operating costs
with climb technique. We can compare cost from proposed
method CTOC and cost Cref , generated with FPPM (The Boeing
4.80
CI
M
0.6
CAS
TOC = 22,000ft
TOC = 25,000ft
4.40
Stref [%]
0.7
x
=ma
TOC = 20,000ft
4.60
Figure 5 The comparison of optimal climb law 310/0.652 with optimal
function M ¼ f (h) for minimum time climb(CI ¼ max), minimum cost
climb (CI ¼ 40) and minimum fuel climb(CI ¼ min)
0.65
ð51Þ
The second way of application of proposed climb technique is for
aircraft equipped with FMS, but without PDB update option, for
aircraft where is detected performance degradation as a dynamic
disturbance. In the paper, presented method accounts
disturbance of aircraft performance, expressed in form of
variables DC and FC, for correction of climb law and TOCopt in
order to achieve minimum en route total operating costs.
If we have case mcl1 ¼ 51,000 ft, R ¼ 840 km, CI ¼ 40 with
no aircraft performance degradation, presented method of
minimum climb technique obtain as result climb law 311/0.664
and pressure altitude hco ¼ 19,183 ft, TOCopt ¼ 27,000 ft
selected from Figure 2, with total operating costs
CTOC ¼ 6,977 USD.
In case of aircraft performance degradation in presented by
variables DC ¼ 3 per cent and FC ¼ 3 per cent, for same
other condition, presented method correct optimal climb law
309/0.662 and pressure altitude hco ¼ 19,346 ft.
If we take TOCopt ¼ 27,000 ft from previous case we
achieve total operating cost CTOC ¼ 7,107 USD. It is evident
cost jump for 1.829 per cent (130 USD) as a consequence of
aircraft performance degradation.
The major goal of method is minimization of total operating
cost in new conditions it can be shown if is introduce new
TOCopt ¼ 28,000 ft we can obtain new result CTOC ¼ 7,105
USD and new climb law 309/0.666 and hco ¼ 19,727 ft. By
conducted correction we achieve minimization of total operating
costs of en route flight profile under dynamic disturbance.
hco
M opt ¼ f ðCASopt ; hco Þ; and
M opt0 ¼ f ðCASopt ; hÞ; and
CASMIN # CASopt # CASMO ; and
C ref 2 C TOC
£ 100
C ref
0 M=0.652
=31
0
CI=4
TOC = 27,000ft
4.20
4.00
3.80
3.60
0.55
min
3.40
CI=
0.5
0.45
3.20
0.4
3.00
0
5,000
10,000
15,000
20,000
25,000
0
h (ft)
20
40
60
CI
340
80
100
The turbo fan aircraft minimum cost climb technique
Aircraft Engineering and Aerospace Technology: An International Journal
P. Mirosavljević, S.Gvozdenović and O.Čokorilo
Volume 81 · Number 4 · 2009 · 334 –342
Conclusion
Calise, A. (1977), “Extended energy management method for
flight performance optimisation”, AIAA Journal, Vol. 15
No. 3, pp. 314-21.
Cook, A., Tanner, G., Williams, V. and Meise, G. (2007),
“Dynamic cost indexing”, paper presented at 6th
EUROCONTROL Innovative Research Workshops &
Exhibition, EUROCONTROL Experimental Centre,
Brétigny sur Orge, 4-6 December.
Houghton, E.L. and Brock, A.E. (1970), Aerodynamics
for Engineering Students, 2nd ed., Edward Arnold,
New York, NY.
IATA (2004), Guidance Material and Best Practices for Fuel and
Environmental Management, IATA, Montreal, Ref. No:
9093-01, ISBN 92-9195-444-6, available at: www.iata.org/
NR/rdonlyres/66A62927-D1CB-474C-821E16F68FB6F7C7/0/Fuel_Action_Plan.pdf/ (accessed 8
December 2008)
Jenkinson, L.R., Simpkin, P. and Rhodes, D. (1999), Civil Jet
Aircraft Design, Arnold Publishing, London.
Kahayas, N. (2007), “Aeronautical technology for future
subsonic civil transport aircraft”, Aircraft Engineering &
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McLean, D. (2006), “The operational efficiency of passenger
aircraft”, Aircraft Engineering & Aerospace Technology,
Vol. 78, pp. 32-8.
Maido, S. (2006), Aircraft Performance, Wiley, New York, NY.
Mair, W.A. and Birdsall, D.L. (1992), Aircraft Performance,
Cambridge University Press, Cambridge.
Ojha, K.O. (1993), “Fastest climb of a turbojet aircraft”,
Journal of Aircraft, Vol. 30 No. 1, pp. 127-9.
Raymer, D. (2006), Airplane Design: A Conceptual Approach,
4th ed., AIAA, Reston, VA.
Root, R. (2001), Cost Index, Performance Engineer Operations,
Flight Operations Engineering, Seattle, WA, November.
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of optimum flight profiles for short-haul Routes”, Journal of
Aircraft, Vol. 12 No. 8, pp. 669-74.
In the paper, it has been solved tree optimization problem in
presence of dynamic disturbance, expressed in form of aircraft
performance degradation:
.
it was defined climb technique for minimum climb cost;
.
it was defined TOCopt which in combination with climb
technique for minimum climb cost minimize total
operating cost of en route flight profile; and
.
achieved results is standardized for application in FMS or
it can be apply in any aircraft which is equipped with
altimeter, air speed indicator and Mach meter.
In the paper, it was defined new approach for defining
conditions for minimum cost climb technique and defined
also new parameter for achieving optimization goal,
minimization of RCC. In the process of defining optimal
TOC it was shown influence of choice of TOCopt on en route
total operating costs. Recalling assumptions of constant cruise
Mach number and constant descent which isolated climb
influence on total operating costs it is obvious existence of
further development of cost optimization in cruise and
descent phase. In the paper, it was also shown condition for
TOCopt determination, for given CI, as a extension of
minimum cost-climb technique. Presented technique is
especially applicable on short-haul flights, where climb
range can have length of about 40 per cent of total range.
The practical benefit from proposed method, climb for
minimum cost, for air operator can be synthesized in
reduction of total cost from 3 to 5 per cent compared to
AFM proposed flight profile. The second benefit is correction
of climb law and TOCopt for achieving minimum en route
operating costs in presence of aircraft performance
degradation. Indirect benefit can be obtaining form
information how much aircraft degradation costs.
References
Airbus Industry (2002), Getting to Grips with Aircraft
Performance Monitoring, Airbus, Customer Service, Flight
Operations Support and Line Assistance, Toulouse,
December, pp. 98-134, available at: www.smartcockpit.
com/pdf/flightops/aerodynamics/22/ (accessed 8 December
2008)
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trajectories for short-haul aircraft”, Journal of Aircraft,
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Further reading
Asselin, M. (1997), Introduction to Aircraft Performance, AIAA,
Reston, VA.
About the authors
P. Mirosavljević received his BS in Air Transport
Engineering from The Faculty of Transport and Traffic
Engineering, University of Belgrade, in 1996, and receive his
MSc in from the same faculty in 2001. He was accepted as
PhD student in 2003. He has been working since 1997 at The
Faculty of Transport and Traffic Engineering, as Teaching
and Research Assistant at lectures in courses: Flight
Mechanic (1997-present), The Transport Aircraft
(1997-2003) and The Transport Aircraft Performance
(2008-present). His research interests are in the areas of
aircraft performance optimization, aircraft cost management,
flight mechanics and airplane appraisal process.
P. Mirosavljević is the corresponding author and can be
contacted at: perami@sf.bg.ac.rs
S. Gvozdenović, PhD, received his BS in Air Transport
Engineering from The Faculty of Transport and Traffic
341
The turbo fan aircraft minimum cost climb technique
Aircraft Engineering and Aerospace Technology: An International Journal
P. Mirosavljević, S.Gvozdenović and O.Čokorilo
Volume 81 · Number 4 · 2009 · 334 –342
Engineering, University of Belgrade, in 1975. He was working
from 1975 to 1978 as Chief Engineer in aircraft factory
UTVA, on general aviation maintenance. He received his
MSc from The Faculty of Transport and Traffic Engineering
in 1990. He was accepted as PhD student in 1990 and
graduated in 1993. He has been working since 1978 at The
Faculty of Transport and Traffic Engineering, the first as
Teaching and Research Assistant at lectures in courses: Flight
Mechanic (1978-1994), The Transport Aircraft (1978-1994)
and than as Professor at lectures in courses: Flight Mechanic
(1994-till now), The Transport Aircraft (1994-till now) and
The Transport Aircraft Performance (1994-till now). He is
the Dean of The Faculty of Transport and
Traffic Engineering. His research interests are in areas of
aircraft performance optimization, aircraft cost management,
flight mechanics and airplane appraisal process.
O. Čokorilo received her BS in Air Transport Engineering
from The Faculty of Transport and Traffic Engineering,
University of Belgrade, in 2002, and receive her MSc from the
same faculty in 2007. She was accepted as PhD student in 2008.
She is working from the 2002 at The Faculty of Transport and
Traffic Engineering, as Teaching and Research Assistant at
lectures in courses: The Transport Aircraft (2003-till now). Her
research interests are in areas of safety management system, risk
management and aircraft cost management.
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