Numerical, but not realistic, engine components maps were presented to fulfill the matching balance between engine components, thus, scaling these maps to ...
Zagazig University Faculty of Engineering Department of Mechanical Power Engineering
Modeling and Simulation of A Double Spool Turbofan Engine Using SIMULINK® A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical Power Engineering
by
Eng. Bassam Elsayed Saleh Supervisors
Prof. Dr. Mohamad Rafaat Ahmad Shalaan Prof. Dr. Ahmed Farouk Abdel Gawad Assoc. Prof. Dr. Mohamed Hasan Gobran Mechanical Power Engineering Department Faculty of Engineering Zagazig University Zagazig 2017
TABLE OF CONTENTS ABSTRACT
III
ACKNOWLEDGEMENT
IV
LIST OF FIGURE CAPTIONS
V
LIST OF TABLES
VIII
NOMENCLATURES
IX
CHAPTER (1) INTRODUCTION
1
1.1 Classification of turbofan engine
1
1.2 Case study
3
1.3 Parametric cycle and performance analysis
4
1.4 SIMULINK® and MATLAB® platform
5
1.5 Objectives
5
CHAPTER (2) LITERATURE REVIEW
6
2.1 Gas turbine engine modeling
6
2.2 Gas turbine engine performance
9
2.3 Gas turbine engine simulation with SIMULINK®
12
2.4 Gas turbine engine dynamics and control.
14
CHAPTER (3) ENGINE MODELING
16
3.1 Design point and data processing
16
3.1.1 Engine station numbering
16
3.1.2 Components maps
16
3.1.3 Components maps scaling
17
3.1.4 Methodology
18
3.2 Double-spool turbofan engine modeling
21
3.2.1 Engine components and governing equations
21
3.2.2 Aerothermodynamics processes
27
3.2.3 Building up engine components block in SIMULINK
30
CHAPTER (4) SIMULATION OF ENGINE OFF-DESIGN PERFORMANCE 37 4.1 Matching constraints
37
4.2 Matrix iteration balancing technique
38 I
4.3 Steady state off-design performance in SIMULINK®
40
4.3.1 Off-design module block
40
4.3.2 Error loop block
43
4.3.3 Errors due to Vs block
43
4.3.4 Solver block
45
4.3.5 Performance and data tables blocks
46
CHAPTER (5) STEADY-STATE RESULTS AND DISCUSSIONS
47
5.1 Flight configuration discussions
47
5.2 Results graph and associated curves
49
CHAPTER (6) ENGINE TRANSIENT OFF-DESIGN PERFORMANCE
53
6.1 Engine non-linear dynamic modeling.
53
6.1.1 Dynamics assumptions.
53
6.1.2 Mathematical modeling.
54
6.2 Transient off-design performance in SIMULINK®.
56
6.2.1 Balancing technique in transient response.
56
6.2.2 SIMULINK® blocks in transient response.
58
6.3 Open-loop transient response.
60
6.4 Results and discussions.
63
CHAPTER (7) CONCLUSIONS AND SUGGESTION FOR FUTURE WORK
74
REFRENCES Appendix (A.1) Thermodynamic properties equations
75 80
Appendix (A.2) Interpreted Matlab functions of engine component blocks 81 Appendix (A.3) Engine station thermodynamic reference conditions Appendix (A.4) Engine components numeric map data
II
99 100
ABSTRACT SIMULINK® platform was used to predict the steady-state off-design performance of a separate flow double-Spool turbofan engines. At the design point of GE-CF6-50 engine , the performance characteristics were obtained. 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 carried out. Block modules for the program were built in SIMULINK® using readymade program library or built with the aid of user-defined functions. Initial guessing of seven dependant parameters
were chosen. The program
balanced these parameters due to solver iteration until balance was achieved. Other independent parameters (Mach number and altitude) and one base-line parameter were chosen separately. With balancing achieved, all performance characteristics were ready and corrected to the inlet conditions. Results were obtained under several conditions (cruise, take off and SLS static ground run up). Each case was studied in multiple high-pressure compressor corrected speeds. Further study for the transient behavior of the turbofan engine in case of open loop scheme was carried out as a proof of model integrity and model verification. The main benefit of this study is to explore how the SIMULINK® is an easy tool in turbofan modeling performance prediction and analysis.
III
ACKNOWLEDGMENTS In the name of Almighty, Allah, the cherisher and sustainer of the world. I would like to express my deepest and greatest appreciation to my supervisors, Professor Dr. Mohamed Rafaat Ahmad Shalaan, Professor Dr. Ahmed Farouk AbdelGawad and Assoc. Professor Dr. Mohamed Hassan Gobran for their valuable advice and continuous
support throughout this
research. Their guidance, encouragement, advice and constructive criticisms resulted in the appearance of this thesis. My deepest thanks to my lovely wife for her love, patience, encouragement, and support. I am indebted to my mother, and my sisters for their support that they offered me over my studying years. And here in, I wish my father, God rests his soul, to be proud of me. Last but not least, I would like to thank my colleague, Dr. Ahmed Azooz from Military Technical College for his help in the initial stages of this work.
IV
LIST OF FIGURE CAPTIONS Figure (1-1) scheme of Single spool turbofan engine
1
Figure (1-2) scheme of double spool turbofan engine
1
Figure (1-3) scheme of three spool turbofan engine
1
Figure (1-4) scheme of Aft-Fan turbofan engine
2
Figure (1-5) scheme of mixed flow turbofan engine
2
Figure (1-6) scheme of separate flow turbofan engine
2
Figure (1-7) scheme of Low Bypass Ratio vs High Bypass
3
ratio turbofan engine Figure (1-8) scheme of CF6-50 turbofan engine Figure (3-1) separate flow double spool turbofan engine
3 16
model station numbering Figure (3-2) Flow chart of matrix iteration balancing technique
20
Figure (3-3) Brayton cycle of separate flow double spool
27
turbofan engine Figure (3-4) ISA model
30
Figure (3-5) Ram block
30
Figure (3-6) Fan block and Fan mask
31
Figure (3-7) LPC block and LPC mask
32
Figure (3-8) HPC block and HPC mask
32
Figure (3-9) Fan block and Fan mask
33
Figure (3-10) LPC block and LPC mask
34
Figure (3-11) HPC block and HPC mask
34
Figure (3-12) Hot nozzle block
35
Figure (3-13) Cold nozzle block
35
Figure (3-14) Error blocks
36
Figure (4-1) Off-Design Module block
40
Figure (4-2) Off-Design Module block breakdown
42
Figure (4-3) Error loop block and mask
43
Figure (4-4) Error due to vj mask
43
Figure (4-5) Error due to vj block breakdown
44
V
Figure (4-6) solver block and mask
45
Figure (4-7) Performance and data tables blocks
46
Figure (5-1) Engine corrected net thrust vs HPC corrected speed
49
Figure (5-2) Engine corrected fuel flow rate vs HPC corrected speed
49
Figure (5-3) Gas generator pressure ratio vs HPC corrected speed
50
Figure (5-4) Bypass ratio vs HPC corrected speed
50
Figure (5-5) Specific fuel consumption vs specific thrust
51
Figure (5-6) HPC operating lines
51
Figure (5-7) Specific fuel consumption vs thrust for comparison cases
52
Figure (6-1) Solver block layout in transient operation
59
Figure (6-2) A layout shows CNch, CNf are altered via performance block 59 transient response Figure (6-3) Performance block in steady-state response
59
Figure (6-4) Details of the performance block in transient operation
60
Figure (6-5)The basic representation of open-loop and closed-loop scheme 61 Figure (6-6) Relative Core spool speed NH1, Cruise flight Mo=0.85,
66
Alt. = 10670 m Figure (6-7) Relative Core spool speed NH1, SLS take off Mo=0.5,
66
Alt. = 0 Figure (6-8) Relative Core spool speed NH1, SLS run up Mo=0,
67
Alt. = 0 Figure (6-9) Relative low spool speed NL1, Cruise flight Mo=0.85,
67
Alt. = 10670 m Figure (6-10) Relative low spool speed NL1, SLS take off Mo=0.5,
68
Alt. = 0 Figure (6-11) Relative low spool speed NL1, SLS run up Mo=0,
68
Alt. = 0 Figure (6-12) Relative Max. cycle temperature Tt41, Cruise flight
69
Mo=0.85, Alt. = 10670 m Figure (6-13) ) Relative Max. cycle temperature Tt41, SLS take off Mo=0.5, Alt. = 0 VI
69
Figure (6-14) Relative Max. cycle temperature Tt41, SLS run up
70
Mo=0, Alt. = 0 Figure (6-15) Net thrust Ft , cruise flight Mo=0.85, Alt. = 10670 m
70
Figure (6-16) Net thrust Ft , SLS take off Mo=0.5, Alt. = 0
71
Figure (6-17) Net thrust Ft, SLS run up Mo=0, Alt. = 0
71
Figure (6-18) Fan speed at high altitude based on SQP
72
Figure (6-19) High pressure Turbine inlet temperature based on SQP
72
Figure (6-20) HPC Steady-state and transient operating line in Cruise flight 73
VII
LIST OF TABLES Table (6-1) Dependent variables and their related generated errors
56
used in transient response. Table (6-2) Steady state values of dependent variables used in transient response.
VIII
62
NOMENCLATURE A.1 Symbols a
stage speed of sound (m/sec.)
CFt
corrected net thrust
CNf
corrected fan speed
CNcl
corrected low-pressure compressor speed
CNch
corrected high-pressure compressor speed
CNtl
corrected low-pressure turbine speed
CNth
corrected high-pressure turbine speed
Cp
specific heat at constant pressure (j/kg.K)
Cv
specific heat at constant volume (j/kg.K)
Cwf
corrected fuel flow rate
EBi
matching constraints base errors
Ei
matching constraints errors
Ei,j
error of matching constraints “I” due to matching variable “j”
F/A
fuel to air ratio
Fs
specific thrust (N/kg/sec.)
Ft
engine net thrust(N)
Ht
total enthalpy (j/kg)
I
mass moment of inertia (kg.m2)
Mo
flight mach number
M0
inlet mach number
M9
hot nozzle exit mach number
P
static pressure (N/m2)
Pcri
critical pressure (N/m2)
Pex
excess power of turbine (j/sec.)
Pt
total pressure (N/m2)
Pw
power (j/sec.)
R
gas constant (j/kg.k)
S
entropy (j/kg.k)
T
static temperature (k) IX
Tt
total temperature (k)
TFth
high-pressure turbine flow function
TFtl
low-pressure turbine flow function
vj
dependent matching variables
Vj
engine jet velocity (m/sec.)
wa
air mass flow rate (kg/sec.)
wf
fuel flow rate (kg/sec.)
wg
gas mixture mass flow rate (kg/sec.)
Zf
fan scaled pressure ratio
Zcl
low-pressure compressor scaled pressure ratio
Zch
high-pressure compressor scaled pressure ratio
engine bypass ratio
dimensionless total pressure
gas heat capacity ratio or gas ratio of specific heats
b
burner efficiency
th
high-pressure turbine efficiency
tl
low-pressure turbine efficiency
n
hot nozzle efficiency
isen
compression process isentropic efficiency
turbine spool angular speed (rad/sec.)
compressor pressure ratio
G.G
gas generator pressure ratio
dimensionless total temperature
A.2 Abbreviations CAD
computer aided design
CAE
computer aided engineering
CFD
computational fluid dynamics
EDM
engine design model
EMAT
error matrix of partial derivatives
EPR
engine pressure ratio X
FTP
full throttle performance
GTE
gas turbine engine
GUI
graphical user interface
HPC
high-pressure compressor
HPT
high-pressure turbine
HPTB
high-pressure turbine burner
IPC
intermediate pressure compressor
IPT
intermediate pressure turbine
ITB
interstage turbine burner
LHV
fuel latent heat value
LPC
low-pressure compressor
LPT
low-pressure turbine
LU
lower upper decomposition
MFP
mass flow parameter
MI
matrix iteration
OOP
object oriented programming
pcwb2
percent of mass air bleed from high-pressure compressor
PRF
pressure recovery factor
PTP
partial throttle performance
RTM
real-time modeling
SFC
specific fuel consumption
SLS
sea level standard
SNL
serial nested loop
TFE
turbofan engine
TIT
turbine inlet temperature
TSFC
thrust specific fuel consumption
XI
CHAPTER (1) INTRODUCTION 1.1
Classification of Turbofan engines The turbofan engine could be classified according to either the number
of its rotation shaft, or according to the place of the fan, or finally according to the type of air and gas flow. 1.1.1 Classification according to the number of rotation shafts. The turbofan engine could be single-spool, double-spool or three spools. In the single-spool engine, the fan, compressor and the turbine are all mounted on one shaft. In the double-spool engine, the fan (and may be the Low Pressure Compressor LPC), the Low Pressure Turbine LPT are mounted on one shaft while the High pressure Compressor HPC and High Pressure Turbine HPT are mounted on the other shaft. In the three-spool engine, the fan and LPT are mounted on one shaft, the IPC and IPT are mounted on the second shaft while the HPC and HPT are mounted on the last shaft.
Fig.(1-1)Single spool rotor (courtesy UTHM)
Fig.(1-2)Double spool rotor (courtesy wikipedia)
Fig.(1-3) Three spool rotor (courtesy UTHM)
1
1.1.2 Classification according to the location of the fan The fan could be placed in the rear of the Turbofan Engine TFE and is called Aft-Fan turbofan otherwise it’s called Fan.
Fig.(1-4) Aft-Fan TFE(courtesy of GE CF700)
1.1.3 Classification according to the type of air and gas flow Turbofan engine could expel the exhaust separate from the secondary air and is referred to as “Separate Flow TFE” or mixes them together before expelling and in this case is referred to as “Mixed Flow TFE”.
Fig.(1-5) Mixed Flow.(courtesy of wikipedia) Fig.(1-6) Seperate Flow.(courtesy of wikipedia) 1.1.4 Classification according to the bypass ratio Two types of TFE exist according to the Bypass ratio, either low-bypass ratio or high-bypass ratio. Always low-bypass engines are used in military applications, while the high bypass ones are used in civilian applications.
2
Fig.(1-7) Low-Bypass Ratio vs High-Bypass Ratio.(courtesy of
1.2
NASA GRC)
Case study CF6 engine family of GE aviation group first entered commercial revenue
service in 1971. Certified to power more than 13 different aircraft types, the CF6 has accumulated over 100 million flight cycles in service[41]. CF6-50 which is categorized as double spool, separate flow, high bypass ratio TFE is selected to power the DC-10 series aircraft, and later selected to power the Airbus A300 and Boeing 747. The CF6-50 is a 46,000-54,000 pound (206,000–240,000 Newton) thrust . Many studies were made to model and simulate the CF6-50 off-design performance. In the present work, SIMULINK® package under MATLAB® platform is used to do this. In order to simulate the engine model, some data should be available. However, due to the KNOW-HOW restrictions, any available data will be used and must be referenced to the design point of the selected engine.
Fig.(1-8) CF6-50 double-spool turbofan engine.(courtesy of 3
GE Aviation)
1.3
Parametric cycle and performance analysis Cycle analysis is concerned with the thermodynamic changes of the
working fluid (air and products of combustion in most cases) as it passes through the engine. It is divided into two types of analysis: parametric cycle analysis (design point) and/or engine performance analysis (off-design)[26]. Design point determines the performance of the engine at different flight conditions and values of design choice and design limits during design phase. Off-design determines the performance of a specific engine (fixed structure) at flight conditions and throttle settings. The study takes the off-design performance as the main target to start with as it is a necessary step before production assuming that the design point of the appropriated engine is known. When the engine is installed in an aircraft, its performance varies with flight conditions and throttle setting and is limited by the engine control system. In flight, the pilot controls the engine through the throttle and also by changing flight conditions and so the thrust and fuel consumption will change. So it is important to know how the engine will act in such condition by simulation to avoid the undesired regime and to build the flight envelope with respect to engine operation limits. Off-design performance of separate flow double spool turbofan engine needs at least seven dependent parameters and one base-line parameter and three other independent parameters to ascertain the full performance characteristics. The dependent parameters determine each component’s operating point on the component map and its matched point with the other components which is called “Matching Technique”. These dependent parameters are initially guessed and by iterative methods. The correct values are determined
by convergence. In such case, the
performance
characteristics data are obtained, processed, and tabulated, and by using a graphic software, the required curves are established.
4
1.4
SIMULINK® and MATLAB® platform MATLAB® is one of the most famous computer aided engineering CAE
software used in the 20th. century and at present. In MATLAB® any written code (functions and commands) can be executed under that platform.[42] SIMULINK® is a software package embedded within MATLAB®. It has the ability to transform any relation to a certain block with inputs and outputs. These blocks are really a transformation of a MATLAB® code that permits any one to deal with something like WINDOWS without the need to learn how to write a code in MATLAB® but just how to deal with it. SIMULINK® is a powerful tool in control design and has a library filled with many toolboxes for different engineering displinces such as “communication toolbox, control toolbox, simscape toolbox, SIMULINK® design optimization, etc.” . SIMULINK® also has the ability to generate a code for the model in several codes like “C/C++, HDL, and PLC code”. The present work used SIMULINK® to establish the engine model blocks and run them to obtain the performance results. Some of these blocks are already found in the SIMULINK® library and ready to use in the present model and some other blocks were built using MATLAB® code ”interpreted MATLAB® function”. Each block is a subassembly and has a low level block which may or may not be a subassembly either. The higher assembly block was named “Double spool TFE CF6-50 modeling”.
1.5
Objectives The goal of this work is to use SIMULINK® as a design tool for modeling
and simulation of the turbofan engine. In addition, the results of SIMULINK® software are validated with other language codes like QuickBasic “QB”. The
program
which
developed
in
SIMULINK® under
MATLAB®
framework gives the simplest way to model the double-spool turbofan engine with object oriented programming OOP and graphical user interface GUI which is state of the art, in addition to the affordable run-time minimizing.
5
CHAPTER (2) LITERATURE REVIEW
2.1 Gas turbine engine modeling Macmillan and Palmer (1974) [39] developed a modular type of computer program for calculation of Gas Turbine Off-Design Performance using FORTRAN IV. Their simulations on series/parallel engines showed that the choice of design point component parameters and airflow distribution to be a critical point of the engine design if a successful transition between cycles is to occur. The flight conditions at transition, compressor or fan pressure ratio, and internal air flow distribution are shown to be totally interconnected. Szuch et al. (1982) [14] made an advanced way to deal with turbofan simulation using hybrid analog-digital computers combination with program written in FORTRAN. The program uses a main host program run under the digital computer and a target analog and digital programs run under the analog computer with user interaction. They offered that methodology to combine the advantages of the analog and digital programs and computers. Drummond et al. (1992) [4] introduced a different way to deal with the computer programs. They used the object oriented programming OOP instead of mathematical languages programming like FORTRAN or C. This OOP gave the simulations code reliability, maintainability, and manageability. Their framework explained some obstacles that OOP can overcome. They introduced the principal of gas turbine engine hierarchically and modularity which was just a subroutine in the ex-languages. Curnock et al. (2001) [3] introduced a new method to model high-bypass fan applied to double-spool turbofan with separate flow depending on its radial profile with comparison to the other ways of modeling either by single fan map or by two fan maps one for bypass and one for core. They concluded that both fan models based on high radial profile or two separate fan maps are more accurate than modeling fan with only one map.
6
LI et al. (2003) [22] discussed the performance modeling of low-bypass double-spool turbofan with radial flow profiles to achieve more realistic averaged properties of the flow at the downstream components. Fan performance data for engine performance simulation was obtained as fan performance characteristic maps by rig testing. In certain cases, low-bypass ratio fans displayed a behavior where the overall fan non-dimensional performance was dependent on the operating bypass ratio at a fixed nondimensional overall flow and corrected speed. They described a fan model that addressed the deviation from the rig test schedule of bypass ratio versus corrected speed that occurs when simulating the engine . Alexiou, and Mathioudakis (2005) [1] discussed a generic simulation tool for modeling gas turbine performance. They introduced an OOP with a readymade components library using drag & drop technique for model creation. They applied their tool to create a turbofan engine model based on OOP and compared it with RTM. They found it’s identical in the steady state performance at SLS and also for transient cases. They also discussed implementation of engine dynamics and frequency response. Martin et al. (2008) [33] introduced the development and validation of an aero-engine simulation model for advanced controller design of full aerothermodynamic modular model of a two-spool, high-bypass turbofan engine with an unmixed exhaust together with a switched gain-scheduled aero engine controller with pump less transfer and anti-windup. Model implementation was in the Matlab/Simulink environment. Full flight-envelope validation of both the model and controller was performed with the assistance of Alstom Aerospace, with the exception of engine start-up as this was outside the boundary of validity of their model. The model was also compatible with the Real-TimeModeling. Connolly et al. (2009) [16] discussed the modeling of turbofan volume dynamics for investigations of Aero-Propulsive-Servo-Elastic effects in a supersonic approach
commercial was
used
transport. whereby
A each 7
one-dimensional component
lumped
(fan,
volume
high-pressure
compressor, combustor, etc.) was represented as a single volume using characteristic performance maps and conservation equations for continuity, momentum
and
energy.
The
simulation
was
developed
in
the
MATLAB/SIMULINK environment in order to facilitate controls development, and ease of integration with a future aero-servo-elastic vehicle model being developed at NASA. The complete simulation demonstrated steady-state results that closely matched a proposed engine suitable for a supersonic business jet at the cruise condition. Preliminary investigation of the transient simulation revealed expected trends for fuel flow disturbances as well as upstream pressure disturbances. A framework for system identification enabled development of linear models for controller design. Utilizing their framework, a transfer function modeling an upstream pressure disturbance’s impacts on the engine speed was developed as an illustrative case of the system identification. Their work would eventually enable an overall vehicle aero-propulsive-servo-elastic model. Asgari et al. (2013)[9] focused on major research activities which were carried out in the field of modeling and simulation of gas turbines. It covered main white-box model which was used when there is enough knowledge about the physics of the system. Mathematical equations regarding dynamics of the system were utilized to make a model, and black-box model which was used when no or little information was available about the physics of the system. Their aim was to disclose the relations between variables of the system using the obtained operational input and output data from performance of the system. They stated artificial neural network (ANN) is one of the most significant methods in black-box modeling.
8
2.2 Gas turbine engine performance Fishbach and Koenig (1972)[18] introduced a GENENG II program to calculate the design and off-design performance iteratively of several types of turbofans including double-spool turbofan. They gave a numeric unrealistic maps for the different engine components which were a helpful data to follow on and to start with. Zhu and saravnamuttoo (1992)[27] gave a new method for doing the matching calculations in off-design performance. They started from the turbine (hot) end rather than from the compressor operating point. as no data were available from the manufacturer other than sales brochures giving any design and off-design performance data. Elzahaby (1992)[2] discussed the determination of double-spool turbofan engine flight performance using an approximate method and verified on two real engines, LARZAK, F404 turbofans that power Alfa jet and F/A-18 Hornet. Walsh and Fletcher (2004)[26] published their 2nd edition of “Gas turbine performance” book. It deals with all practical and theoretical aspects of GTE performance. Likewise engine types passing through operational envelope, design and off- design performance, transient performance, starting, performance testing until the performance of economics of GTE. Also they discussed the possible ways of solution of the off-design performance analysis which is either by serial nested loop or matrix iteration. For serial nested loops, the matching guesses and matching constraints are paired and solved in a nested sequence, whereby for each pass through an outer iteration loop each iteration loop within it is repeated until convergence. In matrix iteration, the overall interaction is recognized and the equations are solved simultaneously. This requires a numerical method utilizing partial derivatives, which are the effect of changing each matching guess individually on the errors in all the matching constraints. The last one is the method of solution that is introduced in the present work. Chiu (2004)[39] investigated the effect of using isothermal combustion inside the high-pressure turbine (HPTB) instead of the afterburner as a way of 9
augmentation and increasing the performance. He viewed that in off-design regimes, the new engine technique not only satisfies the thrust and efficiency requirement at the design cruise point, but also provides enough thrust and comparable or better efficiency in all other flight regimes . Liew et al. (2005)[17] presented the performance of steady-state, dualspool, separate exhaust turbofan engine with interstage turbine burner. It was a relatively new concept in increasing the specific thrust and pollutant emissions reduction. They used a code written in Microsoft EXCEL macrocode with Visual Basic applications. They discussed the parametric cycle analysis, then, the performance analysis and made the analysis in two cases, the first was FTP over Mach number and Altitude, the second was PTP. Mattingely et al. (2006)[12] gave aerothermodynamics cycle analysis of a dual-spool separate exhaust turbofan engine with an interstage turbine burner. With the encouraging results from parametric cycle analysis, a detailed performance cycle analysis of the identical engine was also conducted for steady-state engine performance prediction. The results from off-design cycle analysis showed that the ITB engine at full throttle setting enhanced the performance over baseline engine. Furthermore, ITB engine operating at partial throttle settings exhibits higher thrust at lower specific fuel consumption and improved thermal efficiency over the baseline engine. Ryck et al. (2008)[11] calculated the performance characteristics for a real turbofan engine using the PERF v3.11 computer program. Andriani and Ghezzi (2009)[28] introduced a technique to recover the thermal enthalpy in the exhaust by the principle of regeneration which consisted of two addition cycles. First is using exhaust enthalpy to pre-heat air line before entering combustion chamber which is a method used wildly in the industrial gas turbine engine but not used widely in aero-engines due to weight considerations. Second cycle is intercooling process to cool the exit air from the LPC before it enters the HPC in order to minimize the compression work. Their study showed that although the principle of regeneration increases 10
the performance and the efficiencies but it also increases the fuel consumption likewise the afterburner. Also it recovers heats in the exhaust but decreases the enthalpy levels inside the core resulting in reduction in the exit velocity and so the thrust. Although considering the high values of turbine temperature it shows better results at TSFC. Zachos (2010)[25] introduced a performance modeling for a gas turbine engine in the Sub-Idle operating region. Because the data obtained from the rig tests is usually insufficient in low speeds, there is a need for further research about components behavior within the Sub-Idle regime before any whole engine relight performance prediction is attempted. The Sub-Idle compressor map generation methodologies were pushed a step forward by the definition of the zero-speed curve, that is the lowest speed line of a compressor map. In this way, the sub-Idle characteristic could be interpolated between the zero-speed line and the above-idle given speed lines. Consequently, the generation of the characteristic within the whole range of operation was allowed. Tsoutsanis et al. (2010)[6] introduced a performance adaptation of gas turbine for power generation applications. As the limitations of components map were exclusive manufacturer's property, compressor maps imposed at off-design performance prediction of a gas turbine were overcome by the development of a novel off-design performance adaptation method. Their proposed adaptation method initially generated a series of compressor maps, which in turn provided the performance of the engine model at off-design conditions. Hence, from a family of possible solutions, the best set of compressor map coefficients was determined through a genetic algorithm optimizer. The genetic algorithm optimization was based on a maximum fitness criterion between the engine model simulated measurements and the target measurements of the adaptation, which were available from the service engine. Schnell et al. (2013)[29], gave a detail study of the performance analysis of the integrated V2527-engine fan at ground operation, including 11
both guide vanes CAD ,CFD and fan CT and CFD survey. The result data ware validated with that obtained from the “EEC” during the last shop visit. Furthermore a parametric study characterizing the fan inlet flow at the nacelle entry at varying boundary conditions (e.g. cross flows, different fan mass flow rates etc.) was carried out, allowing to quantify the nacelle induced inlet distortions potentially influencing the fan performance. Rodrigues et al.(2015)[5] introduced an off-design performance prediction of the CFM56-3 aircraft engine using the GASTURB software. The model was first created for its design point, where its geometry was defined and the off-design performance of the engine was then modeled using data from engine test bed. The model was useful to predict the data that the engine manufacturers did not reveal.
2.3 Gas turbine engine simulation with SIMULINK® Camporeale et al.(2006)[34] discussed the real-time dynamics for two cases of gas turbines, single-shaft heavy duty gas turbine engine and doubleshaft 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. The engines were modeled by set of algebraic equations and ordinary differential equations “ aero-thermal model” which was solved by means of forward substitution procedures characterized by the following key features: -
The mathematical model and the numerical scheme were specially developed in order to obtain the same high fidelity and computational efficiency.
-
The code was modular and could be applied to any GTE
Yarlagadda (2010)[37] reported discussion on the performance analysis of J85 turbojet engine matching thrust with reduced inlet pressure to the compressor using SIMULINK® platform. The model for the J85 turbojet engine was verified for performance accuracy with available test data of the engine, developed a real-time turbojet engine integrating aerothermodynamics of 12
engine components. Software programs SmoothC and SmoothT were used to derive the data from characteristic rig test performance maps for the compressor and turbine, respectively. Dynamic Look-up tables in Simulink were used to interpolate the real-time performance of the engine from rig-test data. Furthermore a flow control mechanism that produced a pressure drop across inlet was assumed and the analysis was carried out with reduced compressor inlet pressure for matching thrust. Performance parameters were analyzed with the increase in compressor pressure ratio and shaft rpm. Easrbourn (2012)[35] introduced also a report deal with modeling and simulation of a dynamic of a turbofan engine using MATLAB/SIMULINK®. The new engine model was then integrated with the full “Tip-to-Tail” aircraft model, then, compared to the previous “Tip-to-Tail” aircraft model to confirm accuracy and quantify computational time improvements. The new “Tip-to-Tail” aircraft model was then used for a simple design trade study of a critical component of the cooling system. Schur (2013)[8] discussed a transient model of a turbofan engine in SIMULINK®, showing that thermal efficiency of the high-pressure compressor and high-pressure turbine were mostly factor affecting the performance. A transient model of the high-pressure system of an IAE V2500 was therefore developed. It consisted of the high-pressure compressor, combustion chamber and high-pressure turbine which were modeled by their respective component maps. Also to further increase the models accuracy, the gas properties dependency on temperature and chemical composition of the fluid was taken into account. The combination of advanced map read-out methods and variable gas properties lead to a model in Simulink which showed the transient behavior of the high-pressure system and converged to a reliable steady state. UYSAL
(2014)[30]
reported
the
high-bypass
turbofan
engines
aerothermodynamics and optimization. It gave a new analytical approach to high-bypass turbofan engine crossing the SIMULINK®, which was used after
13
in turbomachinery design, based on building an (EDM) with the aid of optimization tool box in SIMULINK®.
2.4 Gas turbine engine dynamics and control Reberga et al. (2005)[20] introduced linear parameter varying modeling of turbofan engine by using either classical Jacobian linearization or velocitybased linearization. The most promising control technique may be the gain scheduling achieved by interpolating controllers synthesized at different linearized operating points throughout the flight envelope. Menon et al. (2006)[24] described a nonlinear control of high performance aircraft engine from a real time simulation model. A numerical design method was used to automatically generate the controller C-code. Also robustness comparison was given with the previous gain scheduled linear control law. Gaudet (2007)[36] introduced a development of a dynamic modeling and control system design methodology for gas turbines. The resulting dynamic model was also used as a virtual test bed to assess engine performance at its operating limits. It started with the controller selection, the method then detailed how to translate control system requirements into engine protection limiters and fuel schedules. The resulting control system design was comprised of two sections “startup sequencing and engine control”. It allowed gas turbine control throughout the entire operating regime. Martin (2008)[33] described the development and validation of aeroengine simulation model for advanced controller design. A comprehensive nonlinear dynamic model of a turbofan engine was developed and validated against real industrial data. A switched gain scheduled feedback control system was designed for the engine model and implemented incorporated rumples transfer and anti-windup. According to current industrial practice, full flight envelope validation of the model was performed by analyzing the resulting closed-loop performance properties for a range of different pilot thrust demands against the type of responses required from a real turbofan engine. 14
Zhao and Ding (2009)[40] introduced A Novel Optimization Control for aero-engine, based on Sequential Quadratic Programming (SQP) which was a global optimal method and was outlined by two phases, long course optimization and short course optimization. Madarász et al. (2010)[19] reported intelligent technologies in modeling and control of turbojet engines. They discussed the various types of jet engine controllers such like electronic limiters, partial authority flight control augmentation “PAFCA”, high integration digital electronic control “HIDEC”, digital engine control “DEC” and full authority digital engine control “FADEC”. They concluded an ideal test bed for research of methods in the areas of non-linear dynamic systems modeling and design of advanced control algorithms. Further research was intended to be done in the area of situational modeling that would be headed towards broadening of input parameters in the situational model of the engine and further refinement situational classes. Correa et al. (2012)[7] discussed dynamic modeling of nonlinear and control system for a truboshaft used for electric power generation. The control strategy in that model seek the operation in the regions of best performance. It operated to keep the desired reference speed to protect the engine operation from surge, flameout, overspeed, and overheat. The engine performance should attend specifications of the transient conditions. It is subjected to three main constraints “time to peak (tp), percent overshoot (Mp) and stabilization time (ts)”. They discussed controlling these constraints by two control loops. The PID controller gain were obtained by optimization technique. Schur and Blenk (2013)[8] gave a transient model for turbofan engine in SIMULINK. A transient model of high-pressure system was developed using each component respective map. The map was scaled to the design point as the original maps were not available. To increase the model accuracy, the gas properties dependency temperature and chemical composition of the fluid was taken into account. The model was simulated under the SIMULINK platform, which showed the transient behavior of the high-pressure system and converged to a reliable steady-state. 15
CHAPTER (3) ENGINE MODELING 3.1 Design point and data processing 3.1.1 Engine station numbering Unmixed subsystems
double-spool
turbofan
engine
is
partitioned
into
two
(cold section and hot section) in which a bypass separator is
introduced in front of the low-pressure compressor. Stations are numbered in Fig.(3-1). 13
19
Fan
COLD NOZZLE
0
1
2
HPC
LPC
25
HPT
C.Ch.
3
4
HOT NOZZLE
LPT
45
5
9
Fig.(3-1) Separate flow double-spool turbofan engine model station numbering 0 1 2 25 3 4
free stream air (ambient) inlet condition fan inlet condition HPC inlet condition combustor inlet condition HPT inlet condition
45 5 9 13 19
LPT inlet condition hot nozzle inlet condition hot section exit condition fan outlet condition cold section exit condition
3.1.2 Components maps Fishbach and Koenig[18] gave a numeric but unrealistic data for the components map that are used here as dummy maps to start with which are shown in appendix (B.1). Each map consists of certain numbers of speed line for compressors and certain numbers of turbine flow functions lines. Each line has multiple points data. Due to the inequality of number of points in each line, a linear interpolation is done to reach all lines with the same dimensions. Finally, all maps were subdivided into a group of scaled values of pressure ratios “ beta lines” in compressors and a group of indices in turbines. This step 16
is vital in building up the maps lookup tables in the SIMULINK® blocks. These lookup tables have three inputs and a single output that differ by the type of map. After doing the data processing for the numeric maps, EXCEL software was used to draw all the components maps to assure the homogeneity of the maps. 3.1.3 Components map scaling As the real maps of the engine were not available and by using the numeric data maps mentioned at Fishbach and Koenig[18], 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 the corresponding design point of the available map.
𝛑
𝛑
,
= [ 𝛑 = [
=[
Where 𝛑
, 𝐚
, ,
, 𝐚
, 𝐚
,
−
−
]
] [𝛑
]
efficiency respectively.
𝐚
𝐚
,
𝐚
− ]+
(3.1) (3.2) (3.3)
are pressure ratio, mass flow rate and
After map scaling is done, each map data was tabulated in table format and saved as a “.mat” file in the MATLAB® workspace and all the maps together were grouped and saved. When starting the program, those maps should be initiated before running the program, otherwise error messages will be generated.
17
Methodology According to Walsh and Fletcher[26], off-design matching calculations are always done by computer programs. Those programs use component station numbering to identify the inlet or outlet conditions where the outlet ones resulted from some aerothermodynamics relations governing the component based on the upstream characteristics. Those calculations are done either by Serial Nested Loops (SNL) or by Matrix Iteration (MI). For serial nested loops the matching guesses and matching constraints are paired and solved in a nested sequence. Whereby for each pass though higher loop, each loop within it is repeated until convergence. In matrix iteration, the overall interaction is recognized and the equations are set and solved simultaneously. This requires a numerical method utilizing partial derivatives. By the effect of changing each matching guess individually and reporting the effect on the errors in all the matching constraints. In this work , (MI) method was used knowing all matching guesses and trying to find the matching constraints within specified tolerance. This was done using the high-pressure compressor corrected speed as base line parameter, and a group of seven dependent parameters which specify the operating point. These matching guesses parameters are:1- CNf 234-
corrected fan speed
z =
𝛑 −𝛑 , 𝛑 , 𝐚 −𝛑 ,
z = z
5- w
=
6- TFth=
7- TFtl=
𝛑 −𝛑 , 𝛑 , 𝐚 −𝛑 ,
𝛑 −𝛑 , 𝛑 , 𝐚 −𝛑 , √𝐓
𝐏
𝐏
√𝐓
Fan scaled pressure ratio
(3.4)
LPC scaled pressure ratio
(3.5)
HPC scaled pressure ratio
(3.6)
fuel flow rate HPT flow function
(3.7)
LPT flow function
(3.8)
18
In matrix iteration shown in Fig.(3-2), the equations are solved simultaneously. This requires a numerical method utilizing 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 dependent matching guesses (vj). 2- Complete one run through the off-design module of the engine. 3- Calculate the base error (EBi) between calculated values of the matching constraints and values looked up from the maps. 4- Make a small change in matching guess (vj) 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 error matrix of partial derivatives (EMAT). 6- Invert the matrix of partial derivatives using lower upper decomposition(LU). 7- Multiply the inverted matrix of partial derivatives by the base error vector. 8- The results are new values of (vj) which are multiplied by a 0.1 relaxation factor 9- Simultaneously change all matching guesses by the amounts given in the previous step 10-
Repeat the above processes until the errors between calculated values
of the matching constraints and values looked up from the component maps are within an allowable tolerance band, 0.3%, these tolerance was examined for different values (0.1%, 0.2%, 0.3% and 0.5%) and 0.3% was the optimum choice.
19
Fig.(3-2) Flow chart of matrix iteration balancing technique.
20
3.2 Double-spool turbofan engine modeling 3.2.1 Engine components and governing equations 3.2.1.1 Engine Inlet For any given altitude (Alt.), the ambient condition could be evaluated (Pamb, Tamb) or also named (Po, To). These ambient conditions are given by readymade block in SIMULINK® “ ISA model ” or by the following formulas. for Alt. ≤ 11000 (m) Tamb = 288 – 0.0065 * Alt.
(3.9)
Pamb = 101325(1-2.2569e-5Alt.)
5.259
(3.10)
for Alt.> 11000 (m) Tamb = 216.5
(3.11)
Pamb = 22339.4e(11000-Alt.)/6339.6
(3.12)
For the turbofan engine understudy, it has a subsonic intake. To demonstrate the efficiency of the inlet, care should be taken for what so called Total Pressure Recovery Factor PRF which is the indication of the inlet efficiency. It has an empirical formula defined as follows: ≤M ≤
PRF = 1 PRF = 1 – 0.075 PRF =
( o+
)
M −
.
< 𝑀O ≤ 5
50.00001 Tt13s=Tt13s+DeltaH13/Cp13; end end S13=s(Tt13s)-R*log(Pt13); deltaSf=S13-S2; if abs(deltaSf) > abs(0.0005*S2) Tt13s=Tt13s/exp(deltaSf/Cp(Tt13s)); end end Ht13s=h(Tt13s); Ht13=Ht2+((Ht13s-Ht2)/etaf); Tt13=Ht13/Cp13; for k=1:25 Ht13ss=h(Tt13); Cp13=Cp(Tt13); DeltaHsf=(Ht13-Ht13ss); errorHsf=DeltaHsf/Ht13ss; if abs(errorHsf)>0.00001 Tt13=Tt13+DeltaHsf/Cp13; end end S13=s(Tt13)-R*log(Pt13); yout=[Tt13 Pt13 S13 Ht13]; function outs=s(t) Y = 1.8*t; outs= 4.184* ((0.25020051*log(Y))+(1.4450767E-26*Y^7)-(2.4211288E22*Y^6)+(1.5243153E-18*Y^5)-(3.782064E-15*Y^4)-(2.239279E12*Y^3)+(3.2759743E-8*Y^2)-(5.1576879E-5*Y)+0.0454323); function outh=h(t) Y = 1.8*t; outh=0.55555*4.184*((1.2644425E-26 * Y^8)-(2.0752522E-22*Y^7)+(1.270263E18*Y^6)-(3.0256518E-15*Y^5)-(1.6794594E-12*Y^4)+(2.1839826E-8*Y^3)(2.575844E-5*Y^2)+(0.2502*Y)-1.7558886); function outCp=Cp(t) Y = 1.8*t;
81
outCp=4.184*(((1.01554E-25*Y^7)-(1.452677E-21*Y^6)+(7.6215767E-18*Y^5)(1.5128259E-14*Y^4)-(6.717836E-12*Y^3)+(6.5519486E-8*Y^2)-(5.1536079E5*Y))+0.25);
2- LPC block adiabatic compression function yout=Adiabatic_compression_LPC (Tt13,Pr,Pt13,Ht13,S13,etacl) Mwa=28.97; Ru=8.31416; R=Ru/Mwa; Pt25=Pr*Pt13; Tt25=Tt13*Pr^0.28572; for i=1:20 Cp25=Cp(Tt25); Ht25s=h(Tt25); Tt25s=Ht25s/Cp25; for j=1:25 Ht25=h(Tt25s); DeltaHcl=(Ht25s-Ht25); errorHcl=DeltaHcl/Ht25; if abs(errorHcl)>0.00001 Tt25s=Tt25s+DeltaHcl/Cp25; end end S25=s(Tt25s)-R*log(Pt25); deltaScl=S25-S13; if abs(deltaScl) > abs(0.0005*S13) Tt25s=Tt25s/exp(deltaScl/Cp(Tt25s)); end end Ht25s=h(Tt25s); Ht25=Ht13+((Ht25s-Ht13)/etacl); Tt25=Ht25/Cp25; for k=1:25 Ht25ss=h(Tt25); Cp25=Cp(Tt25); DeltaHscl=(Ht25-Ht25ss); errorHscl=DeltaHscl/Ht25ss; if abs(errorHscl)>0.00001 Tt25=Tt25+DeltaHscl/Cp25; end end S25=s(Tt25)-R*log(Pt25); yout=[Tt25 Pt25 S25 Ht25]; function outs=s(t) Y = 1.8*t; outs= 4.184* ((0.25020051*log(Y))+(1.4450767E-26*Y^7)-(2.4211288E22*Y^6)+(1.5243153E-18*Y^5)-(3.782064E-15*Y^4)-(2.239279E12*Y^3)+(3.2759743E-8*Y^2)-(5.1576879E-5*Y)+0.0454323); function outh=h(t) Y = 1.8*t; outh=0.55555*4.184*((1.2644425E-26 * Y^8)-(2.0752522E-22*Y^7)+(1.270263E18*Y^6)-(3.0256518E-15*Y^5)-(1.6794594E-12*Y^4)+(2.1839826E-8*Y^3)(2.575844E-5*Y^2)+(0.2502*Y)-1.7558886); function outCp=Cp(t) Y = 1.8*t;
82
outCp=4.184*(((1.01554E-25*Y^7)-(1.452677E-21*Y^6)+(7.6215767E-18*Y^5)(1.5128259E-14*Y^4)-(6.717836E-12*Y^3)+(6.5519486E-8*Y^2)-(5.1536079E5*Y))+0.25);
3- HPC block adiabatic compression function yout=Adiabatic_compression_HPC (Tt25,Pr,Pt25,Ht25,S25,etach) Mwa=28.97; Ru=8.31416; R=Ru/Mwa; Pt3=Pr*Pt25; Tt3=Tt25*Pr^0.28572; for i=1:20 Cp3=Cp(Tt3); Ht3s=h(Tt3); Tt3s=Ht3s/Cp3; for j=1:25 Ht3=h(Tt3s); DeltaHch=(Ht3s-Ht3); errorHch=DeltaHch/Ht3; if abs(errorHch)>0.00001 Tt3s=Tt3s+DeltaHch/Cp3; end end S3=s(Tt3s)-R*log(Pt3); deltaSch=S3-S25; if abs(deltaSch) > abs(0.0005*S25) Tt3s=Tt3s/exp(deltaSch/Cp(Tt3s)); end end Ht3s=h(Tt3s); Ht3=Ht25+((Ht3s-Ht25)/etach); Tt3=Ht3/Cp(Tt3s); for k=1:25 Ht3ss=h(Tt3); Cp3=Cp(Tt3); DeltaHsch=(Ht3-Ht3ss); errorHsch=DeltaHsch/Ht3; if abs(errorHsch)>0.00001 Tt3=Tt3+DeltaHsch/Cp3; end end S3=s(Tt3)-R*log(Pt3); yout=[Tt3 Pt3 S3 Ht3 Cp3]; function outs=s(t) Y = 1.8*t; outs= 4.184* ((0.25020051*log(Y))+(1.4450767E-26*Y^7)-(2.4211288E22*Y^6)+(1.5243153E-18*Y^5)-(3.782064E-15*Y^4)-(2.239279E12*Y^3)+(3.2759743E-8*Y^2)-(5.1576879E-5*Y)+0.0454323); function outh=h(t) Y = 1.8*t; outh=0.55555*4.184*((1.2644425E-26 * Y^8)-(2.0752522E-22*Y^7)+(1.270263E18*Y^6)-(3.0256518E-15*Y^5)-(1.6794594E-12*Y^4)+(2.1839826E-8*Y^3)(2.575844E-5*Y^2)+(0.2502*Y)-1.7558886); function outCp=Cp(t)
83
Y = 1.8*t; outCp=4.184*(((1.01554E-25*Y^7)-(1.452677E-21*Y^6)+(7.6215767E-18*Y^5)(1.5128259E-14*Y^4)-(6.717836E-12*Y^3)+(6.5519486E-8*Y^2)-(5.1536079E5*Y))+0.25);
4- Combustor thermo function yout=thermo_combustor(Pt4,Ht4,F_A,Cp3) Tt4=Ht4/Cp3; for i=1:25 Cp4=((Cp(Tt4)+(F_A)*CCp(Tt4))/(1+F_A)); Ht4s=((h(Tt4)+(F_A)*hh(Tt4))/(1+F_A)); Phi4=((s(Tt4)+(F_A)*sss(Tt4))/(1+F_A)); deltaHcch=Ht4-Ht4s; errorHcch=deltaHcch/Ht4s; if abs(errorHcch) > 0.00001 Tt4=Tt4+(deltaHcch/Cp4); end Mw4 = 28.97 - 0.946186 * F_A; Rg = 8.31416/ Mw4; S4=Phi4-(Rg*log(Pt4)); end Cp4=((Cp(Tt4)+(F_A)*CCp(Tt4))/(1+F_A)); yout=[Tt4 S4 Cp4 Mw4];
function outs_air=s(t) Y = 1.8*t; outs_air= 4.184* ((0.25020051*log(Y))+(1.4450767E-26*Y^7)-(2.4211288E22*Y^6)+(1.5243153E-18*Y^5)-(3.782064E-15*Y^4)-(2.239279E12*Y^3)+(3.2759743E-8*Y^2)-(5.1576879E-5*Y)+0.0454323); function outs_fuel=sss(t) Y = 1.8*t; outs_fuel = 4.184* ((0.073816638 * log(Y))+(1.038267E-25 * Y^7)(2.22261188D-21* Y^6)+(2.0425826D-17* Y^5)-(1.0512776D-13* Y^4)+( 3.3228928D-10* Y^3)-(6.8859505E-7* Y^2)+(1.225863E-03* Y)+ 0.688595); function outh_air=h(t) Y = 1.8*t; outh_air=0.55555*4.184*((1.2644425E-26 * Y^8)-(2.0752522E22*Y^7)+(1.270263E-18*Y^6)-(3.0256518E-15*Y^5)-(1.6794594E12*Y^4)+(2.1839826E-8*Y^3)-(2.575844E-5*Y^2)+(0.2502*Y)-1.7558886); function outh_fuel=hh(t) Y = 1.8*t; outh_fuel=0.55555*4.184*((9.0848388D-26*Y^8)-(1.9050949D21*Y^7)+(1.7021525D-17*Y^6)-(8.4102208D-14*Y^5)+(2.4921698D-10*Y^4)(4.5906332E-7*Y^3)+(6.129315E-04*Y^2)+(0.073816638*Y)+30.058153); function outCp_air=Cp(t) Y = 1.8*t; outCp_air=4.184*(((1.01554E-25*Y^7)-(1.452677E-21*Y^6)+(7.6215767E-18*Y^5)(1.5128259E-14*Y^4)-(6.717836E-12*Y^3)+(6.5519486E-8*Y^2)-(5.1536079E5*Y))+0.25); function outCp_fuel=CCp(t)
84
Y = 1.8*t; outCp_fuel = 4.184*(((7.26787E-25 * Y^7)-(1.3335668D-20*Y^6)+(1.0212913D16*Y^5)-(4.2051104D-13*Y^4)+(9.968792E-10*Y^3)-(1.3771901E6*Y^2)+(1.225863E-03*Y))+0.073816638);
5- Compression Ram function yout=compression_ram (Tt2,Pt2) Mwa=28.97; R=8.31416/Mwa; Cp2=Cp(Tt2); Ht2s=h(Tt2); Tt2s=Ht2s/Cp2; Cp2=Cp(Tt2s); Ht2=h(Tt2s); S2=s(Tt2s)-R*log(Pt2); yout=[Ht2 S2 Cp2]; function outs=s(t) Y = 1.8*t; outs= 4.184* ((0.25020051*log(Y))+(1.4450767E-26*Y^7)-(2.4211288E22*Y^6)+(1.5243153E-18*Y^5)-(3.782064E-15*Y^4)-(2.239279E12*Y^3)+(3.2759743E-8*Y^2)-(5.1576879E-5*Y)+0.0454323); function outh=h(t) Y = 1.8*t; outh=0.55555*4.184*((1.2644425E-26 * Y^8)-(2.0752522E-22*Y^7)+(1.270263E18*Y^6)-(3.0256518E-15*Y^5)-(1.6794594E-12*Y^4)+(2.1839826E-8*Y^3)(2.575844E-5*Y^2)+(0.2502*Y)-1.7558886); function outCp=Cp(t) Y = 1.8*t; outCp=4.184*(((1.01554E-25*Y^7)-(1.452677E-21*Y^6)+(7.6215767E-18*Y^5)(1.5128259E-14*Y^4)-(6.717836E-12*Y^3)+(6.5519486E-8*Y^2)-(5.1536079E5*Y))+0.25);
6- If Z- condition function y=if_Z(u) if u=1 Z=1; else Z=u; end y=Z;
7- Fuel to air ratio function yout=Fuel_to_air_ratio(w3,Ht3,Ht4) F/A = 0; while F/A == 0
85
Q=43115.800; etab=0.985; wf=w3*(Ht4-Ht3)/((Q*etab)-Ht4); F/A=wf/w3; end yout=[F/A wf];
8- HPT isentropic expansion function yout=HPT1(Ht45s,Ht45_old,Pt4,Cp4,F_A,S4) Pr=0.5; Pt45=Pr*Pt4; for i=1:25 Tt45s=Ht45s/Cp4; for j=1:25 Cp45=((Cp(Tt45s)+(F_A)*CCp(Tt45s))/(1+F_A)); Ht45sc=((h(Tt45s)+(F_A)*hh(Tt45s))/(1+F_A)); Mw45 = 28.97 - 0.946186 * F_A; Rg = 8.31416/ Mw45; Phi45s=((s(Tt45s)+(F_A)*sss(Tt45s))/(1+F_A)); deltaHth1=Ht45s-Ht45sc; errorHth1=deltaHth1/Ht45sc; if abs(errorHth1)> 0.00001 Tt45s=Tt45s+(deltaHth1/Cp45); end S45_old=Phi45s-(Rg*log(Pt45)); end DeltaSth1=S45_old-S4; if abs(DeltaSth1)>abs(0.0005*S4) Pt45=Pt4*exp(DeltaSth1*(Mw45/8.31416)+log(Pt45/Pt4)); end end Tt45=Ht45_old/Cp45; for k=1:25 Cp45=((Cp(Tt45)+(F_A)*CCp(Tt45))/(1+F_A)); Ht45ss=((h(Tt45)+(F_A)*hh(Tt45))/(1+F_A)); Phi45=((s(Tt45)+(F_A)*sss(Tt45))/(1+F_A)); deltaHth2=Ht45_old-Ht45ss; errorHth2=deltaHth2/Ht45ss; if abs(errorHth2)>0.00001 Tt45=Tt45+(deltaHth2/Cp45); end S45=Phi45-(Rg*log(Pt45)); end
yout=[Pt45 Tt45 S45 Cp45];
function outs_air=s(t) Y = 1.8*t; outs_air= 4.184* ((0.25020051*log(Y))+(1.4450767E-26*Y^7)-(2.4211288E22*Y^6)+(1.5243153E-18*Y^5)-(3.782064E-15*Y^4)-(2.239279E12*Y^3)+(3.2759743E-8*Y^2)-(5.1576879E-5*Y)+0.0454323);
86
function outs_fuel=sss(t) Y = 1.8*t; outs_fuel = 4.184* ((0.073816638 * log(Y))+(1.038267E-25 * Y^7)(2.22261188D-21* Y^6)+(2.0425826D-17* Y^5)-(1.0512776D-13* Y^4)+( 3.3228928D-10* Y^3)-(6.8859505E-7* Y^2)+(1.225863E-03* Y)+ 0.688595); function outh_air=h(t) Y = 1.8*t; outh_air=0.55555*4.184*((1.2644425E-26 * Y^8)-(2.0752522E22*Y^7)+(1.270263E-18*Y^6)-(3.0256518E-15*Y^5)-(1.6794594E12*Y^4)+(2.1839826E-8*Y^3)-(2.575844E-5*Y^2)+(0.2502*Y)-1.7558886); function outh_fuel=hh(t) Y = 1.8*t; outh_fuel=0.55555*4.184*((9.0848388D-26*Y^8)-(1.9050949D21*Y^7)+(1.7021525D-17*Y^6)-(8.4102208D-14*Y^5)+(2.4921698D-10*Y^4)(4.5906332E-7*Y^3)+(6.129315E-04*Y^2)+(0.073816638*Y)+30.058153); function outCp_air=Cp(t) Y = 1.8*t; outCp_air=4.184*(((1.01554E-25*Y^7)-(1.452677E-21*Y^6)+(7.6215767E-18*Y^5)(1.5128259E-14*Y^4)-(6.717836E-12*Y^3)+(6.5519486E-8*Y^2)-(5.1536079E5*Y))+0.25); function outCp_fuel=CCp(t) Y = 1.8*t; outCp_fuel = 4.184*(((7.26787E-25 * Y^7)-(1.3335668D-20*Y^6)+(1.0212913D16*Y^5)-(4.2051104D-13*Y^4)+(9.968792E-10*Y^3)-(1.3771901E6*Y^2)+(1.225863E-03*Y))+0.073816638);
87
9- HPT adiabatic expansion function yout=turbo(Pt45,Cp45_old,F_A,Ht45_old,Wg4,Wg45,wach,Ht3) Ht45=((Wg4*Ht45_old)+(0.1*wach*Ht3))/Wg45; Tt45=Ht45/Cp45_old; for m=1:25 Cp45=((Cp(Tt45)+(F_A)*CCp(Tt45))/(1+F_A)); Ht45se=((h(Tt45)+(F_A)*hh(Tt45))/(1+F_A)); Mw45 = 28.97 - 0.946186 * F_A; Rg = 8.31416/ Mw45; Phi45=((s(Tt45)+(F_A)*sss(Tt45))/(1+F_A)); deltaHth3=Ht45-Ht45se; errorHseth3=deltaHth3/Ht45se; if abs(errorHseth3)> 0.00001 Tt45=Tt45+deltaHth3/Cp45; end S45=Phi45-(Rg*log(Pt45)); end yout=[Cp45 Tt45 S45 Ht45 F_A]; function outs_air=s(t) Y = 1.8*t; outs_air= 4.184* ((0.25020051*log(Y))+(1.4450767E-26*Y^7)-(2.4211288E22*Y^6)+(1.5243153E-18*Y^5)-(3.782064E-15*Y^4)-(2.239279E12*Y^3)+(3.2759743E-8*Y^2)-(5.1576879E-5*Y)+0.0454323); function outs_fuel=sss(t) Y = 1.8*t; outs_fuel = 4.184* ((0.073816638 * log(Y))+(1.038267E-25 * Y^7)(2.22261188D-21* Y^6)+(2.0425826D-17* Y^5)-(1.0512776D-13* Y^4)+( 3.3228928D-10* Y^3)-(6.8859505E-7* Y^2)+(1.225863E-03* Y)+ 0.688595); function outh_air=h(t) Y = 1.8*t; outh_air=0.55555*4.184*((1.2644425E-26 * Y^8)-(2.0752522E22*Y^7)+(1.270263E-18*Y^6)-(3.0256518E-15*Y^5)-(1.6794594E12*Y^4)+(2.1839826E-8*Y^3)-(2.575844E-5*Y^2)+(0.2502*Y)-1.7558886); function outh_fuel=hh(t) Y = 1.8*t; outh_fuel=0.55555*4.184*((9.0848388D-26*Y^8)-(1.9050949D21*Y^7)+(1.7021525D-17*Y^6)-(8.4102208D-14*Y^5)+(2.4921698D-10*Y^4)(4.5906332E-7*Y^3)+(6.129315E-04*Y^2)+(0.073816638*Y)+30.058153); function outCp_air=Cp(t) Y = 1.8*t; outCp_air=4.184*(((1.01554E-25*Y^7)-(1.452677E-21*Y^6)+(7.6215767E-18*Y^5)(1.5128259E-14*Y^4)-(6.717836E-12*Y^3)+(6.5519486E-8*Y^2)-(5.1536079E5*Y))+0.25); function outCp_fuel=CCp(t) Y = 1.8*t;
88
outCp_fuel = 4.184*(((7.26787E-25 * Y^7)-(1.3335668D-20*Y^6)+(1.0212913D16*Y^5)-(4.2051104D-13*Y^4)+(9.968792E-10*Y^3)-(1.3771901E6*Y^2)+(1.225863E-03*Y))+0.073816638);
10- Condition Nth function yout=if_Nth(u1,Nth) if u1
