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

Combustor HPC LPC Fan

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 + 0.2𝑀𝑜2)𝑇𝑎𝑚𝑏 (1)

𝑃𝑡2 = (1 + 0.2𝑀𝑜2)𝑃𝑎𝑚𝑏𝑃𝑅𝐹 (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 = 𝑤𝑎2(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,

𝑤𝑎2 = 𝑀𝐹𝑃2𝛿2

√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 = 𝑤𝑎13(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,

𝑤𝑎13 =𝑀𝐹𝑃13𝛿13

√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 = 𝑤𝑎25(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,

𝑤𝑎25 = 𝑀𝐹𝑃25𝛿25

√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⁄ )]𝐻𝑡4−𝐻𝑡3

(𝐹𝐴⁄ )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,

∆𝑃𝑡,𝑐𝑜𝑚𝑏

𝑃𝑡3= 𝐶 (

𝑤𝑎3√𝑇𝑡3

𝑃𝑡3)

2

(11)

Where C is obtained from the design condition as,

0 1 2 25 3 4 45 5

LPT HPT


𝐶 = (

∆𝑃𝐶𝑜𝑚𝑏.𝑃𝑡3

(𝑤𝑎3√𝑇𝑡3

𝑃𝑡3)

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 = (𝑤𝑎25. [1 − 𝑝𝑐𝑤𝑏2]. 𝐻𝑡3 + 𝑤𝑓 . 𝐿𝐻𝑉.𝑏

)/𝑤𝑔4 (14)

𝑤𝑔4 = 𝑤𝑓 + 𝑤𝑎25(1 − 𝑝𝑐𝑤𝑏2) (15)

𝐹

𝐴⁄ = 𝑤𝑓

𝑤𝑔4−𝑤𝑓 (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 = 𝑤𝑔4(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

(CHth) 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 (CHth)ch, side are

given by the following relations,

(𝑇𝐹𝑇𝐻)𝑐ℎ,𝑠𝑖𝑑𝑒 = 𝑤𝑔4√𝑇𝑡4

𝑃𝑡4105 (19)

(𝐶𝐻𝑡ℎ)𝑐ℎ,𝑠𝑖𝑑𝑒 = 𝑤𝑎25(𝐻𝑡3−𝐻𝑡25)

𝑤𝑔4𝑇𝑡4 (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 = 𝑤𝑔45. (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

(CHtl) 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 (CHtl)cl,side are given by the

following relations,

(𝑇𝐹𝑇𝐿)𝑐𝑙,𝑠𝑖𝑑𝑒 = 𝑤𝑔45√𝑇𝑡45

𝑃𝑡45105 (23)

(𝐶𝐻𝑡𝑙)𝑐𝑙,𝑠𝑖𝑑𝑒 = [𝑤𝑎2(𝐻𝑡13−𝐻𝑡2)+𝑤𝑎13(𝐻𝑡25−𝐻𝑡13)

𝑤𝑔45𝑇𝑡45] (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 = √2𝑛

(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

𝑤𝑎19 = 𝑤𝑎2 − 𝑤𝑎13 (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.

𝜋𝑚𝑜𝑑𝑒𝑙 = [𝜋𝑑𝑒𝑠,𝑚𝑜𝑑𝑒𝑙−1

𝜋𝑑𝑒𝑠,𝑚𝑎𝑝−1] [𝜋𝑚𝑎𝑝 − 1] + 1 (27)

𝑤𝑚𝑜𝑑𝑒𝑙 = [𝑤𝑑𝑒𝑠,𝑚𝑜𝑑𝑒𝑙

𝑤𝑑𝑒𝑠,𝑚𝑎𝑝] 𝑤𝑚𝑎𝑝 (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

𝑬𝟏 = 𝑤𝑎3−𝑤𝑎25

𝑤𝑎3 (30)

b- The second error represents the continuity mismatch

between HPT flow function TFTH and its amount

calculated from the compressor side

𝑬𝟐 = (𝑇𝐹𝑇𝐻)𝑐ℎ,𝑠𝑖𝑑𝑒−𝑇𝐹𝑇𝐻

(𝑇𝐹𝑇𝐻)𝑐ℎ,𝑠𝑖𝑑𝑒 (31)

c- The third error represents the failure to satisfy the power

balance between HPT and HPC

𝑬𝟑 = (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

𝑬𝟒 = (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,

𝑬𝟓 = (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

𝑬𝟔 = 𝑃𝑡9−𝑃𝑡8

𝑃𝑡9 (35)

g- The seventh error represents the continuity mismatch

between gas generator and cold nozzle

𝑬𝟕 = 𝑃𝑡19−𝑃𝑡18

𝑃𝑡19 (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,

𝜕𝐸𝑖

𝜕𝑣𝑗= ∑

𝜕𝐸𝑖,𝑗

𝜕𝑣𝑗

𝑛𝑗=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 = ∑𝜕𝐸𝑖,𝑗

𝜕𝑣𝑗∆𝑣𝑗

𝑛𝑗=1 (38)

Where 𝜕𝐸𝑖,𝑗

𝜕𝑣𝑗 is approximately equal to

∆𝐸𝑖,𝑗

∆𝑣𝑗 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 − 𝐸𝐵𝑖, where 𝐸𝐵𝑖 is the ith base-error generated from

the 1st run or iteration. For zero error, 𝐸𝑖 equals to

∆𝐸𝑖 = −𝐸𝐵𝑖 and the equation (38) can be written as follow,


−𝐸𝐵𝑖 = ∑∆𝐸𝑖,𝑗

∆𝑣𝑗∆𝑣𝑗

𝑛𝑗=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 ∆𝐸𝑖,𝑗

∆𝑣𝑗 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

(CHth)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

ENG

INE

CO

RR

EEC

TED

NET

TH

RU

ST F

Nc

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

CO

RR

EEC

TED

FU

EL F

LOW

RA

TE w

fc


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

GA

S G

ENR

ATO

R P

RES

SUR

E R

ATI

O

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

ENG

INE

BY

PASS

RA

TIO


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

SPEC

IFIC

FU

EL C

ON

SUM

PTI

ON

sfc

(K

g/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

HP

C R

ESSU

RE

RA

TIO

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

(K

g.N

/Hr)

Thrust (N)

Figure (12) Thrust vs SFCcase 1:NASA-TM-78653 Computer predictioncase2: NASA-TM-78653 Engine specificationcase3: 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

icfd12-eg-5044_final

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