ICFD12-EG-5044_final

1 Copyright © 2016 by ICFD12
Proceedings of ICFD12:
Twelfth International Conference of Fluid Dynamics
19-20 December, 2016, Le Méridien Pyramids Hotel, Cairo, EGYPT
ICFD12-EG-5044
Steady State Off-Design Performance of Double Spool Turbofan Engine
Using SIMULINK®
Bassam E. Saleh
Egyptian Armed Force, Corresponding author
Mohamed R. Shaalan Ahmed F. AbdelGawad Mohamed H. Gobran
Mech. Power Eng. Dept., Faculty of Engineering, Zagazig University, Zagazig, Egypt
ABSTRACT
SIMULINK®
platform was used to predict the steady
state off-design performance of a separate flow double spool
turbofan engine (GE-CF6-50) as well as with design-point.
Engine performance characteristics were obtained. A numerical
but not realistic engine components maps presented to fulfill
the matching balance between engine components; thus scaling
these maps to the design point data were done. Block modules
of the program were built in SIMULINK®
using readymade
program library or user-defined functions. Initial guessing of
seven dependent parameters were set. The program continued
execution based on solver iterating until balancing was
achieved between the dependent parameters. On the other hand,
other independent parameters (Mach number, altitude) and one
base-line parameter were chosen separately. After balancing
was achieved, all performance characteristics were ready and
corrected to the inlet conditions. Results were introduced in
several conditions (cruse, take-off and SLS static ground run
up). Each case was studied in various high-pressure -
compressor corrected speeds. The main outcome of this study is
to explore that SIMULINK®
is an easy and effective tool in
turbofan modeling and performance estimation.
KEYWORDS:
Turbofan modeling, Engine off-design performance, Simulink.
INTRODUCTION
Off-design performance of the turbofan engine is one of the
most systematic analysis that turbofan is undergone through the
design process. Thus, many methods were introduced to predict
this type of performance.
In the present work, SIMULINK®
was used as a design tool
to analyze the performance using seven dependent parameters,
namely: corrected fan-speed CNf, fan scaled pressure-ratio Zf,
low-pressure compressor scaled pressure-ratio Zcl, high-
pressure compressor scaled pressure-ratio Zch, fuel flow-rate wf,
high-pressure turbine flow-function TFTH, low-pressure
turbine flow-function TFTL) and one base-line parameter (
high-pressure compressor corrected-speed CNch) with varying
the flight conditions (Altitude and Mach number).Several
SIMULINK®
blocks also named mask were established using
either the readymade library toolbox or were built by
interpreted Matlab function.
This study dealt with the steady state off-design
performance of separate flow double spool turbofan engine
with the aid of design point of the GE-CF6-50 turbofan.
A numerical but not realistic engine components maps were
presented to fulfill the matching balance between engine
components. Thus, scaling these maps to the design point data
were done to assure the reality of the used maps. The method of
solution could be either serial nested loops or matrix iteration
(MI). This study uses the (MI) to solve the partial differential
equations by the solver. After the balancing was achieved, the
performance characteristics were tabulated referring to input
conditions.
LITERATURE SURVEY
H. Fishbach and W, Koenig[10](1972) introduced a
GENENG II program to calculate the design and off-design
performance iteratively of several types of turbofans, J.R.
Szuch, Et. AL. [8](1982) make an advanced way to deal with
turbofan simulation using hybrid analog-digital computers, C.
K. Drummond Et. Al.[4](1992) introduce a different way to
deal with the computer programs, they used the object oriented
programming instead of mathematical languages, Ping Zhu and
H saravanamuttoo[13](1992) gave a method for doing the
matching calculations starting from the turbine (hot) rather than
from the compressor operating. B. Curnock Et. Al. [3](2001)
introduce a new method to model high bypass double spool
turbofan depending on its radial profile, Philip P. Walsh and
Paul Fletcher[12](2004) published their 2nd. Edition for the
Gas turbine performance book discusses the possible ways of

solution of the off-design performance analysis which is either
by serial nested loop or matrix iteration, Ya-tien
Chiu[19](2004) investigate the effect of using isothermal
combustion inside the high pressure turbine (HPTB) instead of
the afterburner as a way of augmentation and increasing the
performance, A. Alexiou K. Mathioudakis[1](2005) discuss an
OOP with a readymade components library using drag & drop
technique for model creation, they also discussed
implementation of engine dynamics and frequency response,
S.L. Yang Et. Al. [7](2005) introduced a report presents a
performance of steady state, dual spool, separate exhaust
turbofan engine with interstage turbine burner also which is a
relatively new concept in increasing the specific thrust and
pollutant emissions reduction, J. S. M. Camporeale Et. Al.
[9](2006) submitted a paper discuss the real-time dynamics for
two cases of gas turbine , single shaft heavy duty gas turbine
engine and double shaft aero-derivative engine, they used the
SIMULINK®/MATLAB® platform to run the code based on
lumped non linear representation of the gas turbine engine
components, Sonny Martin Et. Al. [18](2008),introduces a
paper on development and validation of an aero-engine
simulation model for advanced controller design, Model
implementation is in the Matlab/Simulink environment, Full
flight-envelope validation of both the model and controller has
been performed with the assistance of Alstom Aerospace, with
the exception of engine start-up as this is outside the validity of
this model. The model is also compatible with the Real-Time-
Workshop. R. Andriani and U.Ghezzi[14](2009) introduced a
technique to recover the thermal enthalpy in the exhaust by the
principle of regeneration which here consists of two addition
cycles, Santosh Yarlagadda[15](2010) issued a report discuss
the Performance Analysis of J85 Turbojet Engine Matching
Thrust with Reduced Inlet Pressure to the Compressor using
SIMULINK® platform, Simulink model for the J85 turbojet
engine was verified for performance accuracy with available
test data of the engine. S.M. EASTBOURN[17](2012), also
introduced a report dealt with modeling and simulation of a
dynamic of a turbofan engine using MATLAB/ SIMULINK®
,
The new engine model is then integrated with the full “Tip-to-
Tail” aircraft model, then compared to the previous “Tip-to-
Tail” aircraft model to confirm accuracy, F. Schur [4](2013),
Issued a paper discuss a transient model of a turbofan engine in
SIMULINK®
, showing that thermal efficiency of the high
pressure compressor and high pressure turbine are mostly factor
affecting the performance. A transient model of the high
pressure system of an IAE V2500 is therefore developed,
Hamid Asgari Et. Al. [6](2013), issued a paper focuses on
major research activities of modeling and simulation of gas
turbines. Discussing the white-box model which is used when
there is enough knowledge about the physics of the system, and
black-box model which is used when no or little information is
available about the physics of the system (Jelali & Kroll 2004).
Artificial neural network (ANN) is one of the most significant
methods in black-box modeling. S. C. UYSAL[16](2014),
issued a report discusses the high bypass turbofan engines
aerothermodynamics and optimization, based on building an
(EDM) ENGINE DESIGN MODEL with the aid of
optimization tool box in SIMULINK® taking into account
Variable Specific Heat Model and the Flow Property
Calculations as a blocks modeling.
ENGINE MODELING
Methodology
The program established under SIMULINK®
consists of
four main blocks, namely: off-design module block, error loop
block, errors due to variable change block, and solver block as
well as two other supplementary results blocks (performance
block and data tables block) in which all resulted data were
obtained. The main idea is to use the matching constrains and
balancing technique with suitable initial guess to raise the
errors inside the off-design module block. The Matrix Iteration
method was used to alter these values until balance.
In matrix iteration, the equations are solved simultaneously.
This requires a numerical method that utilizes partial
derivatives, which are the effect of changing each matching
guess individually on the errors in all the matching constraints.
The basic steps in this methodology are as follows:
1. Choose initial values of matching guesses, vj.
2. Complete one iteration through the off-design module of
the engine.
3. Calculate the base error EBi between calculated values of
matching constraints and values from maps.
4. Make a small change in each matching guess vj in turn
and repeat the last two steps.
5. From the error values obtained, evaluate the partial
derivatives of the errors in each matching constraint with
respect to each matching guess. This step produces the
matrix of partial derivatives EMAT.
6. Invert the matrix of partial derivatives using LU
decomposition.
7. Multiply the inverted matrix of partial derivatives by the
base error vector.
8. The new results of (vj) are multiplied by a relaxation factor
of 0.1
9. Simultaneously, change all matching guesses by the
amounts given in the previous step.
10. Repeat the above steps until the errors between calculated
values of the matching constraints and the values looked up
from the component maps are within an allowable tolerance,
0.3%.
Engine Components and governing equations
13 19
CombustorHPCLPCFan
Cold Nozzle
Engine
Inlet
Ambient
Condition
0 1 2 25 3 4 45 5 9
HPT LPT Hot Nozzle

1. Engine Inlet
This component is modeled by two blocks. The first is a
readymade block from SIMULINK®
library (ISA Model). This
block has the altitude as an input and results in the inlet
conditions (temperature, pressure). The second is the ram block
which is built using interpreted Matlab function and has the
inlet total temperature, total pressure and Mach number
resulting in the fan-inlet conditions (Tt2, Pt2) which are
functions in inlet conditions. M and PRF for subsonic intakes
are always unity. The engine inlet conditions are modeled by
the following equations:
𝑇𝑡2 = (1 + 0.2𝑀𝑜
2)𝑇𝑎𝑚𝑏 (1)
𝑃𝑡2 = (1 + 0.2𝑀𝑜
2)𝑃𝑎𝑚𝑏 𝑃𝑅𝐹 (2)
The fan-inlet total enthalpy and entropy are calculated using
gas properties relations .
2. Engine Fan
The air passes through the fan and is compressed
adiabatically by means of pressure difference between fan
upstream and downstream. The power consumed in the fan
which is derived by the low pressure turbine spool is given by
(𝑃𝑤) 𝑓 = 𝑤 𝑎2(𝐻𝑡13 − 𝐻𝑡2) (3)
and by knowing Zf and CNf , operating point in the fan map can
be developed. Thus ,fan mass flow parameter (MFP), pressure
ratio and efficiency are determined from map lookup tables and
by using aero-thermodynamic relations including gas properties
which are embedded in single block. All the fan outlet
conditions are known (Pt13,Tt13,S13,Ht13) and thus the inlet fan
mass flow rate wa2 is given by,
𝑤 𝑎2 =
𝑀𝐹𝑃2 𝛿2
√𝜃2
(4)
Where 𝛿2 and 𝜃2 are inlet reference conditions.
3. Low-Pressure Compressor
The air is then forced to the low-pressure compressor (LPC)
which is derived by the low-pressure turbine spool. The air is
adiabatically compressed to higher levels in the LPC. The
power delivered to LPC is given by
(𝑃𝑤) 𝑐𝑙 = 𝑤 𝑎13(𝐻𝑡25 − 𝐻𝑡13) (5)
Because LPC has the same speed of the fan, then its corrected
speed is given by
𝐶𝑁𝑐𝑙 = 𝐶𝑁𝑓√
𝜃2
𝜃13
(6)
Knowing both CNcl and Zcl, the operating point was determined
on LPC map. Low-pressure compressor MFP, pressure ratio
and efficiency were developed from map lookup tables. Thus
using aero-thermodynamic relations which are also embedded
in a single block, All LPC outlet conditions are known
(Pt25,Tt25,S25,Ht25) and thus the inlet LPC mass flow rate wa13 is
given by,
𝑤 𝑎13 =
𝑀𝐹𝑃13 𝛿13
√𝜃13
(7)
Where 𝛿13 and 𝜃13 are fan reference conditions.
4.High-Pressure Compressor
The air is then discharged to combustion pressure by high-
pressure compressor which is derived separately by a high-
pressure turbine spool and the power consumed in it is
evaluated by the following formula,
(𝑃𝑤) 𝑐ℎ = 𝑤 𝑎25(𝐻𝑡3 − 𝐻𝑡25) (8)
Knowing both CNch and Zch , the operating point was
determined on the HPC map. High-pressure compressor MFP,
pressure ratio and efficiency were developed from map lookup
tables and by using aero-thermodynamic relations which are
also embedded in a single block, All HPC outlet conditions are
known(Pt3,Tt3,S3,Ht3) and thus the inlet HPC mass flow-rate
wa25 is given by,
𝑤 𝑎25 =
𝑀𝐹𝑃25 𝛿25
√𝜃25
(9)
Where 𝛿25 and𝜃25 are LPC reference conditions.
5. Combustor
When the pressure reaches the combustion pressure, and
with addition of fuel to the combustor, a flame ignition occurs
and the fuel is burned stoichiometry. The product of
combustion is then expelled out the combustor with maximum
permissible turbine inlet temperature (TIT), which also depends
on the turbine material durability.
Major factors that affect the combustion process are its
thermal efficiency b, which is defined as the ratio of actual
energy supplied to the air to energy in the fuel consumed.
Thermal efficiency depends on type of the combustor, fuel-to-
air ratio (F/A), combustor inlet and outlet conditions (Tt3, Pt3,
Tt4, Pt4), and fuel type (LHV). The combustor efficiency could
be given by the following formula
b =
[1+( 𝐹
𝐴⁄ )]𝐻 𝑡4−𝐻 𝑡3
( 𝐹
𝐴⁄ )𝐿𝐻𝑉
(10)
Another problem raised to surface is the pressure drop
across the combustor as it affects the fuel consumption and the
output power. According to Knoing and Fishback [10] , the
total pressure loss is directly proportional to combustor inlet
mass flow parameter and is given as follow,
∆𝑃 𝑡,𝑐𝑜𝑚𝑏
𝑃 𝑡3
= 𝐶 (
𝑤 𝑎3√𝑇𝑡3
𝑃 𝑡3
)
2
(11)
Where C is obtained from the design condition as,

𝐶 = (
∆𝑃 𝐶𝑜𝑚𝑏.
𝑃 𝑡3
(
𝑤 𝑎3√𝑇 𝑡3
𝑃 𝑡3
)
2 )
𝐷𝑒𝑠.
(12)
and thus the combustor outlet pressure is given by the
following formula,
𝑃𝑡4 = 𝑃𝑡3 − ∆𝑃𝑡,𝑐𝑜𝑚𝑏 (13)
The stage outlet enthalpy is derived by the following formula,
𝐻𝑡4 = (𝑤 𝑎25. [1 − 𝑝𝑐𝑤 𝑏2]. 𝐻𝑡3 + 𝑤𝑓. 𝐿𝐻𝑉.  𝑏
)/𝑤 𝑔4 (14)
𝑤 𝑔4 = 𝑤𝑓 + 𝑤 𝑎25(1 − 𝑝𝑐𝑤 𝑏2) (15)
𝐹
𝐴⁄ =
𝑤 𝑓
𝑤 𝑔4−𝑤 𝑓
(16)
while the combustor outlet temperature and entropy are
obtained from cycle iteration of the stage total pressure and
enthalpy.
6.High-Pressure Turbine
The high-pressure turbine is the stage that delivers power to
the high-pressure compressor through the high-pressure spool.
The map of the turbine discussed here is of the format turbine
total enthalpy drop and the turbine efficiency vs turbine
corrected speed, at specified turbine flow functions (TFF). The
power delivered by HPT to high-pressure compressor is given
by the following formula,
(𝑃𝑤) 𝑡ℎ = 𝑤 𝑔4(𝐻𝑡4 − 𝐻𝑡45) (17)
and since the HPC corrected speed as a base-line parameter is
the only known and there is no value of the HPT corrected
speed. Thus, a relation should be introduced to connect the
HPC corrected speed (CNch) with HPT corrected speed (CNth),
which is as follows,
𝐶𝑁𝑡ℎ = 𝐶𝑁𝑐ℎ (√
𝜃25
𝜃25,𝐷𝑒𝑠
)
100
√𝑇𝑡4
(18)
and with values of TFTH and CNth, which are used to locate
operating point on HPT map, thus HPT corrected enthalpy drop
(CHth) and efficiency (th) should be determined.
Now, data of the HPT from turbine side is known from the
map. Thus, it is time to calculate the same values from HPC
side and examine how the turbine should satisfy the balance or
generate errors. Where, (TFTH)ch, side and (CHth)ch, side are
given by the following relations,
(𝑇𝐹𝑇𝐻) 𝑐ℎ,𝑠𝑖𝑑𝑒 =
𝑤 𝑔4√𝑇𝑡4
𝑃 𝑡4
105
(19)
(𝐶 𝐻𝑡ℎ) 𝑐ℎ,𝑠𝑖𝑑𝑒 =
𝑤 𝑎25(𝐻 𝑡3−𝐻 𝑡25)
𝑤 𝑔4 𝑇𝑡4
(20)
Once HPT corrected enthalpy-drop was known, the total
enthalpy of the next stage (Ht45) is determined. By knowing
both (Ht45) and (F/A) and by iteration of thermodynamic
relations, (Tt45) should be determined and thus the remaining
characteristics of the stage (Pt45) and (S45).
7.Low-Pressure Turbine
The hot gases are then discharged to the LPT and all
upstream characteristics are known from the previous stage.
LPT is the component responsible for driving both the fan and
LPC by single spool called low-pressure spool. The power
delivered from LPT to those components is given by the
following formula,
(𝑃𝑤) 𝑡𝑙 = 𝑤 𝑔45. (𝐻𝑡45 − 𝐻𝑡5) (21)
A relation should be introduced to connect LPC corrected-
speed (CNcl) with LPT corrected-speed (CNtl) which is as
follows,
𝐶𝑁𝑡𝑙 = 𝐶𝑁𝑐𝑙 (√
𝜃13
𝜃13,𝐷𝑒𝑠
)
100
√𝑇𝑡45
(22)
With values of TFTL and CNtl, which are used to locate
operating point on LPT map, LPT corrected enthalpy-drop
(CHtl) and efficiency (tl) should be determined.
Now data of LPT from turbine side is known from the map.
Thus, it is time to calculate the same values from LPC side and
examine how the turbine should satisfy the balance or generates
errors. Where,(TFTL)cl,side and (CHtl)cl,side are given by the
following relations,
(𝑇𝐹𝑇𝐿) 𝑐𝑙,𝑠𝑖𝑑𝑒 =
𝑤 𝑔45√𝑇𝑡45
𝑃 𝑡45
105
(23)
(𝐶 𝐻𝑡𝑙) 𝑐𝑙,𝑠𝑖𝑑𝑒 = [
𝑤 𝑎2(𝐻 𝑡13−𝐻 𝑡2)+𝑤 𝑎13(𝐻 𝑡25−𝐻 𝑡13)
𝑤 𝑔45 𝑇𝑡45
] (24)
Once LPT corrected enthalpy-drop was known, the total
enthalpy of the next stage (Ht5) is determined and by knowing
both (Ht5) and (F/A) and by iteration of thermodynamic
relations, (Tt5) should be determined and thus the remaining
characteristics of the stage (Pt5) and (S5).
8. Hot Nozzle
In the present model, a convergent nozzle is considered in
which the remaining of the pressure potential energy resulting
from the turbine is transformed to a kinetic energy resulting in a
change of momentum and produce engine thrust. Two possible
conditions may exist:
a. when the static pressure at exit is higher than the critical
pressure, the flow is said to be a subsonic flow.
b.when the static pressure at the exit is lower than or equal
to the critical pressure, the flow is said to be sonic flow or
chocked flow (Mexit) = 1
The nozzle jet velocity is expressed as follows,

𝑉𝑗 = √2 𝑛
(𝐻𝑡,ℎ𝑛 − 𝐻) = 𝑀9√ 𝛾𝑅𝑇𝑡9 (25)
9.Cold Nozzle
The cold nozzle in case of separate flow nozzles may be
subsonic or chocked nozzle. Thus, this condition should also be
examined by comparing the static exit pressure with critical
pressure. Generally, it is dealt like the hot nozzle except that
mass flow-rate across the cold nozzle is given by
𝑤 𝑎19 = 𝑤 𝑎2 − 𝑤 𝑎13 (26)
Components map scaling
As the real maps of the engine were not available and by
using the numerical data maps mentioned in ref.[10], scaling
law is applied to obtain the required data for the components
maps. This is done by comparing the design point of the given
engine component with corresponding design point of the
available map.
𝜋 𝑚𝑜𝑑𝑒𝑙 = [
𝜋 𝑑𝑒𝑠,𝑚𝑜𝑑𝑒𝑙−1
𝜋 𝑑𝑒𝑠,𝑚𝑎𝑝−1
] [𝜋 𝑚𝑎𝑝 − 1] + 1 (27)
𝑤 𝑚𝑜𝑑𝑒𝑙 = [
𝑤 𝑑𝑒𝑠,𝑚𝑜𝑑𝑒𝑙
𝑤 𝑑𝑒𝑠,𝑚𝑎𝑝
] 𝑤 𝑚𝑎𝑝 (28)
 𝑚𝑜𝑑𝑒𝑙
= [
 𝑑𝑒𝑠,𝑚𝑜𝑑𝑒𝑙
 𝑑𝑒𝑠,𝑚𝑎𝑝
]  𝑚𝑎𝑝
(29)
After map scaling is done, each map data were tabulated in
table format and saved as a “.mat” file in the MATLAB®
workspace. All the maps were grouped together and saved. In
starting the program, those maps should be initialed before
running the program, otherwise an error will be generated.
Matching constraints and balancing technique
1. Matching constraints
The method for determining the equilibrium run points of
the turbofan engine is to search for the fan running point which
in turn match with the LPC running point. Thus locate point of
HPC which matches with LPC. Simultaneously search for the
point of the HPT that match the HPC point and also the LPT
point that match with LPC point. All these matches should
have constraints to connect them together and hence introduce
the full capable engine in all off-design regimes.
These matching constraints are summarized as follow:
a-Continuity across the gas generator components and across
the gas generator-nozzles combinations.
b-Power balance between HPT and its related HPC, and the
LPT and its related (fan, LPC) combination.
c-Mixer static pressure balance which is not applicable here
for separate flow nozzles.
During the simulation process, if these constraints are
satisfied then the engine is said to be balanced. However, if not
then errors will be generated related to the number of the
dependent variables. These errors can be summarized as follow:
a- The first error represents the failure to satisfy the
continuity between LPC and HPC
𝑬 𝟏 =
𝑤 𝑎3−𝑤 𝑎25
𝑤 𝑎3
(30)
b- The second error represents the continuity mismatch
between HPT flow function TFTH and its amount
calculated from the compressor side
𝑬 𝟐 =
(𝑇𝐹𝑇𝐻) 𝑐ℎ,𝑠𝑖𝑑𝑒−𝑇𝐹𝑇𝐻
(𝑇𝐹𝑇𝐻) 𝑐ℎ,𝑠𝑖𝑑𝑒
(31)
c- The third error represents the failure to satisfy the power
balance between HPT and HPC
𝑬 𝟑 =
(𝐶∆𝐻) 𝑐ℎ,𝑠𝑖𝑑𝑒−(𝐶∆𝐻) 𝑡ℎ
(𝐶∆𝐻) 𝑐ℎ,𝑠𝑖𝑑𝑒
(32)
d- The fourth error represents the failure to satisfy continuity
mismatch between LPT flow function TFTL and its
amount calculated from the compressor side
𝑬 𝟒 =
(𝑇𝐹𝑇𝐻) 𝑐𝑙,𝑠𝑖𝑑𝑒−𝑇𝐹𝑇𝐿
(𝑇𝐹𝑇𝐻) 𝑐𝑙,𝑠𝑖𝑑𝑒
(33)
e- The fifth error represents the failure to satisfy the power
balance between LPT and its corresponding LPC and fan
and is given by,
𝑬 𝟓 =
(𝐶∆𝐻) 𝑐𝑙,𝑠𝑖𝑑𝑒−(𝐶∆𝐻) 𝑡𝑙
(𝐶∆𝐻) 𝑐𝑙,𝑠𝑖𝑑𝑒
(34)
f- The sixth error represents the continuity mismatch
between gas generator and hot nozzle
𝑬 𝟔 =
𝑃 𝑡9−𝑃 𝑡8
𝑃 𝑡9
(35)
g- The seventh error represents the continuity mismatch
between gas generator and cold nozzle
𝑬 𝟕 =
𝑃 𝑡19−𝑃 𝑡18
𝑃 𝑡19
(36)
2. Matrix iteration balancing technique
Initially, the guessed dependent parameters (7 variables) are
checked whether they satisfy the matching constraints or not. If
they do then the engine is said to be balanced. If not then the
engine is failed to satisfy its matching constraints and a set of 7
errors will be generated. These errors represent the amount of
which the engine fails to satisfy the constraints as mentioned in
the previous section. Those errors are function of the dependent
parameters (7 variables) and expressed as a set of partial
differential equations. With neglecting second and higher order
terms of these equations, the linearized form can be written as
follows,
𝜕𝐸 𝑖
𝜕𝑣 𝑗
= ∑
𝜕𝐸 𝑖,𝑗
𝜕𝑣 𝑗
𝑛
𝑗=1 (37)
Where
𝑖 = 1  𝑛 … … 𝑛 is the number of generated errors
𝑗 = 1  7 is the number of dependent parameters
Simplifying the last equation, it can be written as follows,
∆𝐸𝑖 = ∑
𝜕𝐸 𝑖,𝑗
𝜕𝑣 𝑗
∆𝑣𝑗
𝑛
𝑗=1 (38)
Where
𝜕𝐸 𝑖,𝑗
𝜕𝑣 𝑗
is approximately equal to
∆𝐸 𝑖,𝑗
∆𝑣 𝑗
and represents the
sensitivity of the error (i) due to the variation in the variable (j).
Since the equation is really non-linear, LHS term ∆𝐸𝑖 is
given by
∆𝐸𝑖 = 𝐸𝑖 − 𝐸 𝐵𝑖, where 𝐸 𝐵𝑖 is the ith
base-error generated from
the 1st
run or iteration. For zero error, 𝐸𝑖 equals to
∆𝐸𝑖 = −𝐸 𝐵𝑖 and the equation (38) can be written as follow,

−𝐸 𝐵𝑖 = ∑
∆𝐸 𝑖,𝑗
∆𝑣 𝑗
∆𝑣𝑗
𝑛
𝑗=1 (39)
The above equation is solved for ∆𝑣𝑗 in which the new
values of the dependent parameters (variables) is corrected by
the following correlation,
𝑣𝑗,𝑛𝑒𝑤 = 𝑣𝑗,𝑜𝑙𝑑 + ∆𝑣𝑗 (40)
For the non-linearity of the system, the equations (39), and
(40) should be run several iterations until balance is reached.
For every iteration, the amount
∆𝐸 𝑖,𝑗
∆𝑣 𝑗
is updated. Also,
a relaxation factor of 0.1 is multiplied by ∆𝑣𝑗 to avoid the
overshooting of the results and make the iteration runs
smoothly.
When the iteration does not reach balance after specified
number of iterations, the matching initial guessed parameters
should be changed and the cycle is repeat again.
Steady State off-design performance in SIMULINK®
1.Off-Design Module Block
It is the main program block in which all engine
components and their corresponding thermodynamic relations
are introduced and set, (Fig. 1). The block has 10 input
terminals and 10 output terminals. This block initially generates
base-errors. If the balance is not satisfied, one more iteration is
carried out to alter all the seven dependent parameters. This
gives another error if not balanced. This cycle is repeated
several iterations until the errors are within certain limit. In
such case, the system is balanced. And the condition signal
comes true and permits the run of the two blocks (performance
and data tables) to calculate the engine performance and record
in data tables. The other three block inputs are altered manually
according to flight régime (SLS with zero Mach, Take-off with
0.5 Mach, Cruse flight with 0.85 Mach) and at which, corrected
high-pressure speed is chosen.
2.Error-loop block
This block has a fourteen input ports, eight output ports, and
two jobs done every iteration, (Fig. 2). First, it is a mixer in
which the seven base-errors EBi are combined in one
concatenate vector. Second, the seven dependent parameters are
altered into base-incremental amount Vj.
3.Error due to vj block
The objective of this block is to alter each dependent
parameter by a small increment in each iteration separately and
show the resulting errors from this change, (Fig. 3). These
resulting errors are the base-constitute of the error matrix
EMAT developed in the next section. It is almost about seven
identical blocks similar to the off-design module block in all its
input and output ports except that in each block of these seven
blocks, it has only one input port that its value changes
separately ( vj + vj ). Also, these blocks have no output ports
for the performance, data tables, or condition signal.
4. Solver block
This is the major subroutine block, (Fig. 4). It is the solver
that solves the partial differential equations by the matrix
iteration balance technique. The block collects all parameters
needed for solving, then manipulates those inputs with matrix
operations to give the amount of variable increment vj needed
for the next iteration step.
This block consists of a major EMAT block and some
other blocks. EMAT block collects the following inputs(7 errors
due to vj – 7 base variable increments vj) and builds EMAT
matrix using equation (39) and the matrix inversion block.
Solving for Vj as in equation (39) using matrix multiply, the
initial variables should be altered by the amount of Vj. Using
equation (40), the new value of Vj is developed and a new
iteration cycle carried out until the errors reach a specified limit
(balance).
5. Performance and data-tables blocks
These two blocks are conditioned blocks that were
established using the embedded Matlab function property in
program library, (Fig. 5). The two blocks almost run after the
system reaches balance and all variables are settled. In the first
block, all performance relations are given with the inputs of all
data necessary from the Off-design module.
The other block is for storing these data and additional data
referenced to the inlet conditions (2,2) in tabulated form that
are used, later on, in figures handling.
The outputs of the performance block are: net thrust,
corrected net thrust, corrected fuel flow rate, specific thrust,
specific fuel consumption, bypass ratio, engine pressure ratio.
These outputs are needed for exploring the performance of the
engine in different flight regime. While the data tables block
outputs all the stages outlet conditions referenced to the engine
inlet conditions (2,2)
RESULTS AND DISCUSSION
The results of this study are related to CF6-50 double spool
turbofan engine with separate exhausts. The high-pressure
compressor speed CNCH is taken as a base-line parameter.
Thus, three sets of different flight configurations, corrected
to flight inlet conditions, are developed. These sets are :
a.The steady state performance at SLS (Altitude= 0 m) and
Mach number (Mo=0).
b. The steady state performance at take-off (Altitude = 0 m)
and Mach number (Mo=0.5).
c.The steady state performance at cruise flight (Altitude =
10670 m) and Mach number (Mo=0.85).
Figures (6)-(9) show the corrected net thrust CFt, corrected
fuel flow-rate cwf, gas generator pressure ratio G.G, and bypass
ratio , respectively, as function of CNCH. Figure (10) shows
the relation between the specific fuel consumption SFC and
specific thrust FS. Figure (11) shows the engine operation-line
in high-pressure compressor map. Figure (12) shows a
comparison between this study and another study given by a
NASA-TM-78653[11] in case of specific fuel consumption SFC
with thrust.

CONCLUSIONS
Steady state off-design performance is single step in
the modeling and simulation of the turbo fan engine, followed
by transient response and finally the controller design.
In step of the steady state under study, SIMULINK®
showed a good estimation of the performance characteristics
regards the other programming languages or any other
readymade software, the results are accurate, clear and almost
the same of some other studies.
Further study will be established for the transient
response and controller design using SIMULINK®
ACKNOWLEDGMENTS
I hereby pray to Allah to bless me. Thanks are extended to
my family for their support; my professors for their continuous
help, and finally, to anyone prays to Allah for my support.
NOMENCLATURE
Abbreviations
CNch = corrected HPC speed
CNcl = corrected LPC speed
CNf = corrected fan speed
(CHth)ch, side= corrected enthalpy drop in HPT from HPC side
EMAT= errors matrix
HPC = high-pressure compressor
HPT = high-pressure turbine
LPC = low-pressure compressor
LPT = low-pressure turbine
LHV = lower heat value
LU = lower upper decomposition
MFP = mass flow parameter
PRF = pressure recovery factor
TFTH = high-pressure turbine flow-function
TFTL = low-pressure turbine flow-function
Zcl = LPC scaled pressure-ratio
Zch = HPC scaled pressure-ratio
Zf = fan scaled pressure-ratio
Symbols
EBi = base-error
Ei = generated error number i
F/A = fuel to air ratio
ht = total enthalpy
H = hot nozzle outlet static enthalpy
Ht,hn = total enthalpy across the hot nozzle
M = Mach number
Pcwb2 = percent of the bleed air mass flow rate from HPC
Pt = total pressure
Pw = power
Pt,comb= total pressure drop across the combustor
Tt = total temperature
vj = dependent variable
vj = change in dependent variable
Vj = jet exit velocity
wa = air flow-rate
wg = gas flow-rate
wf = fuel flow-rate
δ = corrected total temperature
b = combustor efficiency
n = nozzle efficiency
GG = gas generator pressure ratio
 = corrected total pressure
REFERENCES
[1] A. Alexiou K. Mathioudakis “Development of Gas Turbine
Performance Models using a Generic Simulation Tool”
Laboratory of Thermal Turbo machines, National Technical
University of Athens,2005
[2] A. Elzahaby ”Research Bulletin on the determination of
double spool turbofan engine flight performance” University of
Helwan engineering research bulletin, Volume 4, 1992.
[3] B. Curnock, J. Yin, R. Hales, P. Pilidis “High-bypass
turbofan model using a fan radial-profile performance map “
Aircraft Design 4 (115–126),2001
[4] Colin K. Drummond, Gregory J. Follen, and Charles W.
Putt “Gas Turbine System Simulation: An Object-Oriented
Approach “ NASA-TM-106044,1992
[5] F. Schur, “ A transient Model of a turbofan engine in
SIMULINK”, Deutscher Luft- und Raumfahrt kongress.
ID( 301478), 2013.
[6] Hamid Asgari, XiaoQi Chen, Raazesh Sainudiin, “
Modeling and Simulating of Gas Turbines” international
journal of modeling, identification and control, vol.20, No. 3,
2013.
[7] J.D. Mattingly, C.J. Marek, K.H.Liew, E.Urip, S.L. Yang,
"Performance Cycle Analysis for turbofan engine with
interstage turbine burner" , NASA-TM-213659,2005.
[8] John R. Szuch, Susan M. Krosel, and William M. Bruton
“An automated procedure for developing hybrid computer
simulations of turbofan engines” NASA-TP-1851, 1982
[9] J S. M. Camporeale, B. Fortunato and M. Mastrovito,
"modular code for real time dynamic simulation of gas turbines
in SIMULINK®", ASME Journal of Engineering for Gas
Turbines and Power, vol.128, issue 3, 2006
[10] Laurence H. Fishbach and Robert W, Koenig “A Program
for calculating design and off-design performance of two and
three spool turbofans with as many as three nozzle”,
NASA TN D:6553,1972.
[11] Morris, S. J. “Computer Program for the Design and Off-
Design Performance of Turbojet and Turbofan Engine Cycles”,
NASA-TM-78653,1978.
[12] Philip P. Walsh and Paul Fletcher, GAS TURBINE
PERFORMANCE, 2nd.edition, Blackwell Science publishing,
Oxford, ISBN 0-632-06434-X, 2004.
[13] Ping Zhu and H saravanamuttoo “Simulation of an
Advanced Twin-Spool Industrial Gas Turbine” ASME Journal
of Engineering for Gas Turbines and Power,1992.

[14] R. Andriani and U.Ghezzi "performance analysis of high
by pass jet engine with intercooling and regeneration” AIAA
2009-4800, 2009.
[15] Santosh Yarlagadda, " Performance Analysis of J85
Turbojet Engine Matching Thrust with Reduced Inlet Pressure
to the Compressor", The University of Toledo ,2010.
[16] S. C. UYSAL, “ High Bypass Ratio Turbofan Engines
Aerothermodynamics Design and Optimization”, Middle East
Technical University, Ankara,2014.
[17] S.M. Eastbourn, ”Modeling and Simulation of a dynamic
turbofan engine using MATLAB/SIMULINK”, Wright State
University,2012.
[18] Sonny Martin, Iain Wallace and Declan G. Bates,
“Development and Validation of an Aero-engine Simulation
Model for advanced Controller Design” American Control
Conference, Seattle, Washington, USA, 2008.
[19] Ya-tien Chiu, " A Performance Study of a Super-cruise
Engine with Isothermal Combustion inside the Turbine ",
Blacksburg, Virginia , 2004

Fig.1.Off-Design Module block

Fig.2.Error-Loop block.

Fig.3.Error Due to vj block.

Fig.4.Solver block

Fig.5.Performance and data-tables blocks.

0
50000
100000
150000
200000
250000
300000
350000
400000
0.6 0.7 0.8 0.9 1 1.1 1.2
ENGINECORREECTEDNETTHRUSTFNc
HPC RELATIVE CORRECTED SPEED CNCH
Figure (6) ENGINE CORRECTED NET THRUST vs HPC CORRECTED SPEED
Mo=0.85 Alt. = 10670
Mo=0 Alt.=0
Mo=0.5 Alt.=0

0
1
2
3
4
5
6
0.6 0.7 0.8 0.9 1 1.1 1.2
CORREECTEDFUELFLOWRATEwfc
Figure (7) CORRECTED FUEL FLOW RATE vs HPC CORRECTED SPEED
Mo=0.85 Alt.=10670
Mo=0 Alt.= 0
Mo=0.5 Alt.=0

0
0.5
1
1.5
2
2.5
3
0.6 0.7 0.8 0.9 1 1.1 1.2
GASGENRATORPRESSURERATIOG.G
Figure (8) GAS GENERATOR PRESSURE RATIO vs HPC CORRECTED SPEED
Mo=0.85 Alt.=10670
Mo=0 Alt.=0
Mo=0.5 Alt.=0

2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
0.6 0.7 0.8 0.9 1 1.1 1.2
ENGINEBYPASSRATIO
Figure (9) BYPASS RATIO vs HPC CORRECTED SPEED
Mo=0.85 Alt.=10670
Mo=0 Alt.=0
Mo=0.5 Alt.=0

0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 100 200 300 400 500
SPECIFICFUELCONSUMPTIONsfc(Kg/N.Hr)
SPECIFIC THRUST FS (N/(Kg/sec))
Figure (10) SPESIFIC FUEL CONSUMPTION vs SPESIFIC THRUST
Mo=0.85 Alt.=10670
Mo=0 Alt.=0
Mo=0.5 Alt.=0

0
2
4
6
8
10
12
14
16
18
20
22
24
26
20 30 40 50 60 70 80 90 100
HPCRESSURERATIO
HPC CORRECTED MASS FLOW RATE
Figure (11) HPC OPERATING LINE corr. speed 0.5662
corr. speed 0.674
corr. speed 0.787
corr. speed 0.899
corr. Speed 1.0
corr. Speed 1.034
corr. Speed 1.067
corr. Speed1.124
corr. Speed 1.236
corr. Speed 1.292
surge line
Operalting line Mo=05 Alt=0.
Operating line Mo=0.85 Alt.=10670
operating line Mo=0 Alt=0

0.03
0.04
0.05
0.06
0.07
0.08
0 50000 100000 150000 200000 250000
SFC(Kg.N/Hr)
Thrust (N)
Figure (12) Thrust vs SFC
case 1:NASA-TM-78653 Computer prediction
case2: NASA-TM-78653 Engine specification
case3: Off-Design results with SIMULINK
M=0.8 Alt.=25000ft CAE
M=0.5 Alt.=25000ft CAE
M=0.4 Alt.=0 CAE
M=0 Alt.=0 CAE
M=0.8 Alt.=25000ft Engine Spec.
M=0.5 Alt.=25000ft EngineSpec.
M=0.4 Alt.=0 Engine Spec.
M=0 Alt.=0 Engine Spec.
M=0.8 Alt.=25000ft SIMULINK
M=0.5 Alt.=25000ft SIMULINK
M=0.4 Alt.=0 SIMULINK
M=0 Alt.=0 SIMULINK
M=0.8,Alt.=25000, computer
M=0.8,Alt.=25000, Engine Spec.
M=0.8,Alt.=25000, simulink
M=0.5,Alt.=25000, computer
M=0.5,Alt.=25000, Engine spec.
M=0.5,Alt.=25000, simulink
M=0.4,Alt. =0, simulink
M=0.4,Alt. =0, Engine spec.
M=0.4,Alt. =0, computer
M=0,Alt. 0, Engine spec.
M=0,Alt. 0, computer
M=0,Alt.= 0, simulink

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