ISTANBUL TECHNICAL UNIVERSITY  GRADUATE SCHOOL 

PhD. THESIS 

DECEMBER 2023 
 

ADVANCED ENERGY AND EXERGY ANALYSIS ON 

AIRCRAFT JET ENGINES 

Sara FAWAL 

Department of Aeronautics and Astronautics 

 

Aeronautics and Astronautics Engineering Programme 

 



 

  



 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Department of Aeronautics and Astronautics 

 

Aeronautics and Astronautics Engineering Programme 

 

DECEMBER 2023 

ISTANBUL TECHNICAL UNIVERSITY  GRADUATE SCHOOL 

ADVANCED ENERGY AND EXERGY ANALYSIS ON 

AIRCRAFT JET ENGINES 

PhD. THESIS 

Sara FAWAL 

(511212113) 

Thesis Advisor: Prof. Dr. Ali KODAL 

 



 

  



 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Uçak ve Uzay Mühendisliği Anabilim Dalı 

 

Uçak ve Uzay Mühendisliği Programı 

 

ARALIK 2023 

ISTANBUL TEKNİK ÜNİVERSİTESİ  LİSANSÜSTÜ EĞİTİM ENSTİTÜSÜ 

HAVACILIK JET MOTORLARINDA İLERİ 

ENERJİ VE EKSERJİ ANALİZİ 

DOKTORA TEZİ 

Sara FAWAL 

(511212113) 

Tez Danışmanı: Prof. Dr. Ali KODAL 

 



 

 

 



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Thesis Advisor :  Prof. Dr. Ali KODAL              .............................. 

 Istanbul Technical University  

Jury Members :  Prof. Dr. Bahri ŞAHİN   ............................. 

Istanbul Gelişim University 

Prof. Dr. İbrahim ÖZKOL   .............................. 

Istanbul Technical University 

Assis. Prof. Dr. Hayri ACAR  .............................. 

Istanbul Technical University 

 

   Prof. Dr. Muammer KALYON  .............................. 

Istanbul Ticaret University 

 

Sara FAWAL, a Ph.D. student of İTU Graduate School student ID 511212113, 

successfully defended the thesis/dissertation entitled “ADVANCED ENERGY AND 

EXERGY ANALYSIS ON AIRCRAFT JET ENGINES”, which she prepared after 

fulfilling the requirements specified in the associated legislations, before the jury 

whose signatures are below. 

 

 

Date of Submission : 15 Ekim 2023 

Date of Defense : 8 December 2023 
 



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Elhamdüllilah. 

My thesis is dedicated to all pursuers of knowledge, truth and justice. 

 

 

 

 



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ix 

FOREWORD 

The chapters in this text are organized essentially along an advanced energy and 

exergy performance analysis for the folowing types of aircraft engines: Turbojet 

without an Afterburner, Turbojet with an Afterburner, Ramjet and finally the 

Turboramjet. Starting with the most simplest application of the Brayton cycle this text 

provides a comprehensive study and design approach towards more intellignet 

engines; the text concludes on the discussion of Turbie Based Combined Cycles 

(TBCC). 

I express my sincere appreciation to all my profesors who have contributed to my 

academic progress and professional development. I would especilly like to thank Dr. 

Marius PARASCHIVOIU and Dr. Hoi Dick NG for all their support and due diligence, 

without them I would not have become a PhD. Student and Istanbul Technical 

University. I would also like to thank my professor and mentor Dr. Ali KODAL at İTÜ 

for guiding and supporting me every step of the way thoughout my whole PhD. 

experience. In addition I would like to thank İTÜ for being a home away from home 

and all its wonderful staff memebers who are always supprotive and helpful and great 

you with a smile and warm heart. 

Finally, I thank both my Mom and Dad for being there for me all my life.  

In closing I would like to share a passage by an anonymus writer: 

What is Truth?  

A question that is sometimes cynical, sometimes sincere, but always important. Truth 

is real; it is honesty with what is. Truth is learned, it is not something to create, but 

something to discover. Truth is omniscient; it is above, beyond, around and in us all. 

Truth is elusive; it can be denied, covered, rejected, ignored and even hated. Truth is 

noble; it is worth guarding from misguided attempts to wrap truth into half truth. Truth 

is powerful; it is a friend if embraced, an enemy if denied. Truth is unswerving; it is 

the straight path to lasting success. 

 

 

 

 

December 2023 

 

Sara FAWAL 

(Aerospace Engineering) 

 

 

 

 

 

 

 

 



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xi 

TABLE OF CONTENTS 

Page 

 

FOREWORD ............................................................................................................. ix 
TABLE OF CONTENTS .......................................................................................... xi 

ABBREVIATIONS ................................................................................................. xiii 
SYMBOLS ................................................................................................................ xv 

LIST OF TABLES ................................................................................................. xvii 
LIST OF FIGURES ................................................................................................ xix 
SUMMARY ........................................................................................................... xxiii 
ÖZET ....................................................................................................................... xxv 

1. INTRODUCTION .................................................................................................. 1 
1.1 Purpose of Thesis ............................................................................................... 3 
1.2 Literature Review ............................................................................................... 5 

1.3 Conjecture ........................................................................................................ 11 

2. THEORETICAL MODEL OF TURBOJET: NO AFTERBURNER ............. 13 
2.1 Objective Formulations and Established Variables .......................................... 13 

2.2 Mass Flow, Engine Speed and Shaft Force Models and Altitude .................... 17 
2.3 Algebraic Optimization Methodology ............................................................. 19 

2.4 Outcomes and Considerations .......................................................................... 20 

2.5 Prospects of Turbojets: No Afterburner ........................................................... 54 

3. THEORETICAL MODEL OF TURBOJET: WITH AFTERBURNER ........ 57 
3.1 Objective Formulations and Established Variables ......................................... 57 
3.2 Outcomes and Considerations .......................................................................... 59 
3.3 Prospects of Turbojets: With Afterburner ........................................................ 65 

4. THEORETICAL MODEL OF RAMJET.......................................................... 67 
4.1 Objective Formulations and Established Variables ......................................... 67 
4.2 Outcomes and Considerations .......................................................................... 69 

4.2.1 Single oblique and normal shock solution ................................................ 70 
4.2.2 Multiple oblique and single normal shock solution .................................. 75 

4.3 Prospects of Ramjets ........................................................................................ 82 

5. TURBINE BASED COMBINED CYCLE (TBCC): THE TURBORAMJET 

ENGINE ............................................................................................................... 85 
5.1 Objective Formulations and Established Variables ......................................... 85 

5.2 Outcomes and Considerations .......................................................................... 86 
5.3 Prospects of Turboramjets .............................................................................. 101 

6. CONCLUDING REMARKS AND FUTURE WORKS ................................. 103 
6.1 Conclusion: Turbojet without and with an AB, Ramjet and Turboramjet 

Engines .......................................................................................................... 103 
6.2 Towards the Future ......................................................................................... 104 

APPENDICES ........................................................................................................ 117 
CURRICULUM VITAE ........................................................................................ 135 



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xiii 

ABBREVIATIONS 

AB : Afterburner 

ALT : Altitude 

CBSF : Carnot Brayton Shape Factor 

ECOL : Ecological Function 

ECOP : Ecological Coefficient of Performance 

𝐄𝐂𝐎𝐏̅̅ ̅̅ ̅̅ ̅̅  : Normalized ECOP 

EFECPOD : Effective Ecological Power Density 

EP : Effective Power 

EPD : Effective Power Density 

EPLOS : Effective Power Loss Parameter 

EXED : Exergy Destruction 

EXEF : Exergy Efficiency 

ft : Feet 

GB : Great Britain 

hPa : Hectopascals 

km : Kilometer 

LHV : Lower Heating Value 

MP : Maximum Power 

MPD : Maximum Power Density 

PLOS : Power Loss Parameter 

RAF : Royal Air Force 

RF : Radiative Forcing 

SFC : Specific Fuel Consumption 

TBCC : Turbine Based Combined Cycles 

TSFC : Thrust Specific Fuel Consumption 

UK : United Kingdom 

 

 



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xv 

SYMBOLS 

𝐴5   Cross-sectional Area at Station 5 (𝑚2) 

𝐶𝑎   Velocity of Air (m/s) 

𝐶4   Velocity at Station 4 (m/s) 

𝐶5   Velocity at Nozzle Exit (m/s) 

𝐹𝑠   Specific Thrust (N-s/kg) 

𝐹𝑠ℎ𝑎𝑓𝑡   Shaft Force (kN) 

g   Acceleration due to Gravity (𝑚 𝑠2⁄ ) 

ℎ   Height from Sea Level (m) 

𝐼𝑎   Air Specific Impulse (s) 

LHV   Lower Heating Value (kJ/kg) 

𝑚̇0   Mass flow rate (kg/s) 

𝑚̇𝑎   Mass flow of air (kg/s) 

𝑚̇𝑓   Mass flow of fuel (kg/s) 

N   Engine Speed (RPM) 

𝑃́   Non-Dimensional Pressure Difference at Inlet and Exit States 

𝑝𝑎   Inlet Air Pressure (kPa) 

𝑝5   Pressure at Station 5 (kPa) 

𝑄𝑅   Fuel Heating Value (kJ/kg) 

𝑄̇𝑓𝑢𝑒𝑙   Fuel Heat Release Rate (kJ/s) 

𝑄̇𝐻𝑇   Total Heat Transfer (kJ/s) 

𝑄̇𝐿𝑇   Total Heat Rejection (kJ/s) 

𝑄̇𝐿𝐾   Rate of Heat Leak (kJ/s) 

R   Gas Constant (J/kg-K) 

𝑆̇𝑔   Rate of Entropy Generation (kJ/K-s) 

TSFC   Thrust Specific Fuel Consumption (kg/N-s) 

𝑇̌   Non-Dimensional Temperature Difference at Inlet and Exit States 

𝑇0   Environment Temperature (K) 

𝑇𝐻   Temperature of Hot Reservoir (K) 

𝑇𝐿   Temperature of Cold Reservoir (K) 

𝑣5   Specific Volume at Station 5 (𝑚3 𝑘𝑔⁄ ) 



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𝑣𝑁𝑜𝑧𝑧𝑙𝑒   Nozzle Specific Volume (𝑚3 𝑘𝑔⁄ ) 

𝑊̇   Power produced by Real Brayton Cycle (kJ/s) 

𝑊̇𝐵𝑟𝑎𝑦   Power produced by Ideal Brayton Cycle (kJ/s) 

𝑊̇𝑑   Power Density (kJ/s) 

𝑊̇𝑙𝑘   Power Leaked from the System (kJ/s) 

𝑊̇𝑟𝑒𝑣   Reversible Power (kJ/s) 

𝑊̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒  Turbine Work (kJ/s) 

𝑊̇𝑢𝑠𝑒𝑓𝑢𝑙  Useful Power (kJ/s) 

𝑋̇𝐷𝐸𝑆   Rate of Exergy Destruction (kJ/s) 

𝛾   Specific Heat Ratio 

√𝜃 =  √𝑇𝑟𝑒𝑓 𝑇𝑎⁄  Temperature Correctıon Factor 

𝜂𝑏   Burner Efficiency 

𝜂𝑐   Compressor Efficiency 

𝜂𝑖   Intake/Diffuser Efficiency 

𝜂𝑗   Jet/Nozzle Efficiency 

𝜂𝑚   Mechanical Efficiency 

𝜂𝑝   Propulsive Efficiency 

𝜂𝑡   Turbine Efficiency 

𝜂𝑡ℎ   Thermal Efficiency 

𝑓   Fuel to Air Ratio 

𝜃𝑐   Compressor Pressure Ratio Parameter 

M∞   Flight Mach Number 

𝜉   Percentage of Internal Conductance for Heat Leak 

𝑇𝑆𝐹𝐶∗, 𝐼𝑎
∗, 𝑓∗  Optimum TSFC, 𝐼𝑎 and 𝑓  

𝛼  Cycle temperature ratio 

 

  



xvii 

LIST OF TABLES 

Page 

 
Table 2.1: Delegated variable inputs. .................................................................................................. 20 
Table 2.2: Thrust (kN) and TSFC (kg/N-s) at maximum ECOL, ECOP, MP and MPD for 

distinct quantities of altitude. ........................................................................................... 48 
Table 2.3: Thrust (kN) and TSFC (kg/N-s) at maximum ECOL, ECOP, MP and MPD for 

distinct quantities of 𝑀∞. ................................................................................................ 48 
Table 2.4: State Investigation at 𝑀∞0.5 and 0.9 for an Altitude of 10 km ......................................... 48 
Table 3.1: Additional delegated variable inputs for Afterburner condition. ........................................ 59 
Table 4.1: Additional delegated variable inputs for Ramjet condition. ............................................... 70 
Table 5.1: Delegated variable inputs for Turboramjet condition. ........................................................ 87 
 

 

 

  



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xix 

LIST OF FIGURES 

Page 

Figure 1.1: P-v diagram (left) and Part-section (right) of the Whittle jet engine. ................................. 3 
Figure 2.1: Engine arrangement (a) and T-s schematic representation of a turbojet cycle (b). ........... 13 
Figure 2.2: Various optimization functions vs. power at 𝜂𝑡 = 𝜂𝑐 = 0.95. ......................................... 22 
Figure 2.3: Non-dimentionalized power, power density, specific volume variance, ECOP, 

ECOL and 𝜂𝑡ℎ for variations of compressor pressure ratio parameter, 𝜃𝑐. ..................... 22 
Figure 2.4: Manifestations of power losses (a) and their non-dimentionalized forms (b) for 

variations of 𝜃𝑐 ................................................................................................................ 23 
Figure 2.5: Contours of PLOS for variations of (a) NG and (b) Shaft Force for distinct 

quantities of 𝜂𝑐 = 𝜂𝑡 ............................................................................................................ 25 
Figure 2.6: Contours of power and PLOS (a) and EPLOS for variations of 𝜃𝑐 for dsitinct 

quantities of 𝜂𝑐 = 𝜂𝑡 ......................................................................................................... 26 
Figure 2.7: Contours of power and PLOS (a) EPLOS for variaitons of 𝜃𝑐 for distinct 

quantities of 𝑀∞ .............................................................................................................. 26 
Figure 2.8: Contours of 𝐼𝑎and PLOS (a) and TSFC (b) 𝜃𝑐 for distinct quantities 𝑀∞ ...................... 27 
Figure 2.9: Contours of thrust and PLOS (a) and 𝜂𝑝 (b) for variaitons of 𝑀∞ for distinct 

quantities of 𝜃𝑐 ................................................................................................................. 27 
Figure 2.10: Contours of PLOS for variaitons of power for distinct quantities of 𝜂𝑐 = 𝜂𝑡 ................. 28 
Figure 2.11: Contours of PLOS for variaitons of power for distinct quantities of 𝑀∞ ...................... 28 
Figure 2.12: Contours of PLOS for variaitons of power for distinct quantities of heat leakage 

rates. ................................................................................................................................. 29 
Figure 2.13: Contours of PLOS for variations of power for distinct quantities of altitudes ................ 29 
Figure 2.14: Contours of 𝑇𝑆𝐹𝐶 ∗ (a), 𝑓 ∗ (b) and 𝐼𝑎 ∗  (c) for variations of 𝑀∞ at maximum 

power and minimum PLOS conditions ............................................................................ 30 
Figure 2.15: Optimum EPLOS contours for prescribed ALT and 𝑀∞. .............................................. 31 
Figure 2.16: Optimum 𝜂𝑡ℎ contours for prescribed ALT and 𝑀∞. .................................................... 32 
Figure 2.17: Optimum PLOS contours for prescribed ALT and 𝑀∞. ................................................ 32 
Figure 2.18: Optimum 𝐼𝑎 contours for prescribed ALT and 𝑀∞. ...................................................... 32 
Figure 2.19: Optimum Thrust contours for prescribed ALT and 𝑀∞................................................. 33 
Figure 2.20: Optimum 𝜃𝑐 contours for prescribed ALT and 𝑀∞. ...................................................... 33 
Figure 2.21: Optimum TSFC contours for prescribed ALT and 𝑀∞. ................................................ 33 
Figure 2.22: Optimum power contours for prescribed ALT and 𝑀∞. ................................................ 34 
Figure 2.23: Optimum CBSF contours for prescribed ALT and 𝑀∞. ................................................ 34 
Figure 2.24: Contours of 𝐼𝑎 (a) and 𝜂𝑡ℎ (b) for variaitons of 𝜃𝑐 for distinct 𝜂𝑐 = 𝜂𝑡 

quantities. ......................................................................................................................... 36 
Figure 2.25: Contours of 𝐼𝑎 (a) and 𝜂𝑡ℎ (b) for variations of 𝜃𝑐 for distinct quantities of 𝛼.............. 36 
Figure 2.26: Contours of 𝐼𝑎 (a) and 𝜂𝑡ℎ (b), TSFC (c) and Thrust (d) for variaitons of 𝜃𝑐 for 

distinct quantities of altitude. ........................................................................................... 37 
Figure 2.27: Contours of 𝜂𝑡ℎ (a), TSFC (b), and the specific volume variance (c) utilizing 

MP, MPD, MECOP and MECOL for varying 𝛼. ............................................................. 39 
Figure 2.28: Contours of 𝐼𝑎 (a), 𝜂𝑝 (b), 𝜂𝑡ℎ (c),and the specific volume variance (d) 

utilizing MP, MPD, MECOP and MECOL for varying 𝑀∞. .......................................... 40 
Figure 2.29: Contours of TSFC (a), f (b) and Thrust (c) utilizing MP, MPD, MECOP and 

MECOL for variations of 𝑀∞. ......................................................................................... 41 
Figure 2.30: Contours of Power (a), TSFC (b) and 𝐼𝑎 (c) for variaitons of 𝜃𝑐 for distinct 

quantities of 𝑀∞. ............................................................................................................. 42 
Figure 2.31: Contours of ∆𝜈 at the inlet and exit of the compressor and burner ................................. 44 
Figure 2.32: Contours of ∆𝜈 at the inlet and exit of the turbine and nozzle ........................................ 45 
Figure 2.33: Non-dimensional pressure (𝑃) across indicated engine component ............................... 46 
Figure 2.34: Non-dimensional temperature (𝑇) across the burner ...................................................... 46 



xx 

Figure 2.35: Component size variation at maximum ECOL, ECOP, MP and MPD for 

changes of altitude. ........................................................................................................... 50 
Figure 2.36: Varying Module dimensions at maximum ECOL, ECOP, MP and MPD for 

distinct vales of 𝑀∞.......................................................................................................... 51 
Figure 2.37: Length of the combustion chamber at maximum ECOL, ECOP, MP and MPD 

for distinct quantities of altitude. ...................................................................................... 52 
Figure 2.38: Length of the combustion chamber at maximum ECOL, ECOP, MP and MPD 

for distinct quantities of 𝑀∞. ............................................................................................ 53 
Figure 3.1: Engine arrangement (a) and T-s schematic representation of a turbojet cycle (b) 

with an afterburner ............................................................................................................ 57 
Figure 3.2: 𝜂𝑡ℎ, 𝜂𝑜 and 𝜂𝑝 efficiency for variations of altitude as a function of flight Mach 

number, 𝑀∞...................................................................................................................... 62 
Figure 3.3: 𝑓 and 𝑇𝑆𝐹𝐶 efficiency for distinct quantities of altitude for variaitons 𝑀∞. .................... 62 
Figure 3.4: 𝑇ℎ𝑟𝑢𝑠𝑡 (a), 𝐼𝑎 (c) and 𝜈𝑁𝑜𝑧𝑧𝑙𝑒 (d) for variations of altitude as a function of 𝑀∞ 

and 𝑇ℎ𝑟𝑢𝑠𝑡 (b) as a function of 𝑃𝑜𝑤𝑒𝑟. ........................................................................... 63 
Figure 3.5: 𝑃𝑜𝑤𝑒𝑟, 𝐸𝑃𝐿𝑂𝑆 and 𝑃𝐿𝑂𝑆 for distinct quantitis of 𝜂𝑐 for variations of 𝜃𝑐. ..................... 64 
Figure 3.6: 𝑃𝑜𝑤𝑒𝑟, 𝐸𝑃𝐿𝑂𝑆 and 𝑃𝐿𝑂𝑆 for distinct quantities of 𝑀∞ as a function of 𝜃𝑐. .................. 64 
Figure 3.7: Dimensional metamorphisis of respective engine modules at maximum MP for 

variations 𝑀∞. .................................................................................................................. 65 
Figure 4.1: Engine arrangement (a) and T-s schematic representation of a ramjet cycle (b) ............... 68 
Figure 4.2: f, and TSFC for variations of δ and for distinct quantities of 𝑀∞ ..................................... 71 
Figure 4.3: Thrust and 𝑊𝑎 for variaitons of δ and for distinct quantities of 𝑀∞ ................................ 72 
Figure 4.4: 𝜂𝑡ℎ, 𝜂𝑜 and 𝜂𝑝 efficiencies for variations of δ and for distinct quantities of 𝑀∞ ............ 73 
Figure 4.5: Representation of Oblique and Normal Shock angles for variations of wedge 

angles δ and Mach number 𝑀∞ ........................................................................................ 74 
Figure 4.6: Single Oblique and Normal Shock vs. Multiple Oblique and Single Shock 

solution. ............................................................................................................................. 75 
Figure 4.7: 𝜂𝑡ℎ, 𝜂0 and 𝜂𝑝 for turbojet with and without an AB vs. ramjet engine as a 

function of 𝑀∞. ................................................................................................................ 77 
Figure 4.8: 𝑓, TSFC and 𝐼𝑎 for turbojet with and without an AB vs. ramjet engine as a 

function of 𝑀∞. ................................................................................................................ 78 
Figure 4.9: Thrust (a) and 𝜈𝑁𝑜𝑧𝑧𝑙𝑒 (c) as a function of 𝑀∞ and Thrust vs. Power for 

turbojet with and without an AB vs. ramjet engine. .......................................................... 79 
Figure 4.10: 𝜂𝑡ℎ, 𝜂0 and 𝜂𝑝 for variations of LHV for ramjet engine as a function of 𝑀∞. .............. 80 
Figure 4.11: 𝑓, 𝑇𝑆𝐹𝐶 and 𝐼𝑎 for.variations of LHV for ramjet engine as a function of 𝑀∞. ............. 81 
Figure 4.12: Thrust (a) and 𝜈𝑁𝑜𝑧𝑧𝑙𝑒 (c) as a function of 𝑀∞ and Thrust vs. Power for 

variations in LHV.............................................................................................................. 82 
Figure 5.1: Wraparound (a), Over/Under (b) and Thermodynamic Cycle (c) of Turboramjet 

Engine ............................................................................................................................... 86 
Figure 5.2: Engine arrangement (a) and T-s schematic representation of a turboramjet cycle 

(b) with an afterburner ...................................................................................................... 87 
Figure 5.3: 𝜂𝑡ℎ, 𝜂0 and 𝜂𝑝 for variations of altitude for turboramjet engine as a function of 

𝑀∞. ................................................................................................................................... 91 
Figure 5.4: 𝑓, 𝑇𝑆𝐹𝐶 and 𝐼𝑎 for.variations of altitude for turboramjet engine as a function of 

𝑀∞. ................................................................................................................................... 92 
Figure 5.5: Thrust (a) and 𝜈𝑁𝑜𝑧𝑧𝑙𝑒 (c) as a function of 𝑀∞ and Thrust vs. Power for 

variations in altitude. ......................................................................................................... 93 
Figure 5.6: 𝜂𝑡ℎ for variations of inlet air mass flow at 10 km (a) and 20 km (b) for 

turboramjet engine as a function of 𝑀∞. .......................................................................... 97 
Figure 5.7: 𝜂0 for variations of inlet air mass flow at 10 km (a) and 20 km (b) for 

turboramjet engine as a function of 𝑀∞. .......................................................................... 97 
Figure 5.8: 𝜂𝑝for variations of inlet air mass at 10 km (a) and 20 km (b) flow for turboramjet 

engine as a function of 𝑀∞. ............................................................................................. 98 
Figure 5.9: f for variations of inlet air mass flow at 10 km (a) and 20 km (b) for turboramjet 

engine as a function of 𝑀∞. ............................................................................................. 98 
Figure 5.10: TSFC for variations of inlet air mass flow at 10 km (a) and 20 km (b) for 

turboramjet engine as a function of 𝑀∞. .......................................................................... 99 
Figure 5.11: 𝐼𝑎 for variations of inlet air mass flow at 10 km (a) and 20 km (b) for 

turboramjet engine as a function of 𝑀∞. .......................................................................... 99 



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Figure 5.12: Thrust for variations of inlet air mass flow at 10 km (a) and 20 km (b) for 

turboramjet engine as a function of 𝑀∞. ....................................................................... 100 
Figure 5.13: Thrust for variations of inlet air mass flow at 10 km (a) and 20 km (b) for 

turboramjet engine as a function of 𝑃𝑜𝑤𝑒𝑟. .................................................................. 100 
Figure 5.14: 𝜈𝑁𝑜𝑧𝑧𝑙𝑒 for variations of inlet air mass flow at 10 km (a) and 20 km (b) for 

turboramjet engine as a function of 𝑀∞. ....................................................................... 101 
 

 

 

 

 

 

 

 

 

 



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xxiii 

ADVANCED ENERGY AND EXERGY ANALYSIS ON AIRCRAFT JET 

ENGINES 

SUMMARY 

A comparative performance analysis for various optimization criterion functions is to 

be carried out for an irreversible Brayton cycle applicable to aircraft jet engines: 

Ramjet, Turbojet (No Afterburner), Turbojet (With Afterburner), Turbo-Ramjet. 

Newly defined parameters are introduced as power loss parameter (PLOS), effective 

power loss parameter (EPLOS) and Carnot-Brayton shape factor (CBSF) for a better 

assessment of the performance and power losses throughout the operation of the engine 

cycle. In addition, optimization functions, such as maximum power (MP), maximum 

power density (MPD), ecological coefficient of performance (ECOP) and ecological 

function (ECOL) are considered and their optimal operation conditions are compared 

with respect to each other. 

This research studied the effects on the prescribed optimization criterions targeted 

towards the aviation industry under variations of compressor pressure ratio 𝜃𝑐, 

compressor and turbine efficiencies (𝜂𝑐 and 𝜂𝑡 respectively), cycle temperature ratio / 

maximum cycle temperature, altitude and flight Mach number 𝑀∞ where applicable 

with respect to the jet engine being considered. Therefore, the classical irreversible 

Brayton cycle is extended and applied to airbreathing engines; which included effects 

of all the engine components (from free stream to inlet to outlet) as part of the 

thermodynamic cycle model. 

While many researchers have carried out performance analysis for internal combustion 

engines including gas turbine engine, this study is an extension of the available 

optimization functions such as MP, MPD, ECOP and ECOL for aircraft jet engines. 

As mentioned, power density is defined as the ratio of power to the maximum specific 

volume in the cycle. Whereas ECOP is defined as the ratio of power output to the loss 

rate of availability and ECOL as the power output minus the loss rate of availability. 

In order to extend the classical irreversible Brayton cycle to airbreathing engines 

applicable for aircrafts, further development studies must be carried out to obtain: 

higher propulsion efficiency and higher ratios of power output with respect to engine 

weight, volume, and frontal area. The objective is to obtain a larger power output to 

engine size (weight) in a more thermodynamically efficient manner for a real turbojet 

cycle where maximum ECOP, ECOL, power density and power conditions can be used 

as a basis for the determination of optimal operating conditions and preliminary design 

constraints for real turbojet engines at flight conditions. 

The comparative performance analysis for various optimization criterion functions 

used for the aircraft engine cycle will be applied to ramjet, turbojet without afterburner 

and tubojet with afterburner to reach the final intended application of turboramjet 

engine. The turboramjet engine cycle is identified as Turbine Based Combined Cycle 

Engines (TBCC). Such hybrid cycle engines can be applied to UAV’s, UCAV’s and 

powering future hypersonic flight vehichles. 



xxiv 

The software to be used for the comparative performance analysis for the irreversible 

Brayton cycle applicable to aircraft jet engine cycles is the academic version of 

MATLAB 2018b provided by the MathWorks group. 

The emissions and radiative forcing (RF) from the aviation industry and its effects on 

air pollution and the ecology are an important concern, where aviation ranks as one of 

the top ten emitters. The major greenhouse gas emitters that contribute to RF are: 

carbon dioxide CO2, carbon monoxide CO, water H2O, nitrous oxide NOX, sulphur 

oxides SOX and volatile organic compounds VOCs. 

Thus, performance evaluation of aircraft propulsion systems must be assessed with 

respect to environmental and ecological conditions as well as power and fuel 

consumption considerations. Therefore, various optimization criterion functions which 

can be used as tools by the aviation industry to design ‘new generation engines’ which 

are economically and ecologically favourable. 

It is anticipated that this research would provide valuable insight in the preliminary 

design of airbreathing engines (Ramjet, Turbojet: No Afterburner, Turbojet: With 

Afterburner and Turbo-Ramjet) and set a stage for exploration towards adaptive engine 

components and cycles for the conception of truly intelligent engines; an engine that 

can assess its current operating state and work under the most efficient power regime 

(ECOL or ECOP or MP or MPD) to achieve the designers and engine’s intended 

performance potential. 

 

 

 



xxv 

HAVACILIK JET MOTORLARINDA İLERİ ENERJİ VE EKSERJİ 

ANALİZİ 

ÖZET 

Uçak jet motorlarına uygulanabilen tersinmez bir Brayton çevrimi için çeşitli 

optimizasyon kriter fonksiyonları için karşılaştırmalı bir performans analizi 

yapılacaktır: Turbojet (art yakıcısız), Turbojet (art yakıcılı), Ramjet ve Turbo-Ramjet. 

Motor çevriminin çalışması boyunca performansın ve güç kayıplarının daha iyi 

değerlendirilmesi için yeni tanımlanan parametreler güç kaybı parametresi (PLOS), 

etkin güç kaybı parametresi (EPLOS) ve Carnot-Brayton şekil faktörü (CBSF) olarak 

tanıtıldı. Ayrıca maksimum güç (MP), maksimum güç yoğunluğu (MPD), ekolojik 

performans katsayısı (ECOP) ve ekolojik fonksiyon (ECOL) gibi optimizasyon 

fonksiyonları dikkate alınmakta ve optimum çalışma koşulları birbirleriyle 

karşılaştırılmaktadır. 

 

Bu araştırma, söz konusu jet motoruna göre, kompresör basınç oranı 𝜃𝑐, kompresör ve 

türbin verimlilikleri (𝜂𝑐 and 𝜂𝑡 sırasıyla), çevrim sıcaklığı oranı / maksimum çevrim 

sıcaklığı, yükseklik/irtifa ve uçuş Mach sayısı 𝑀∞ varyasyonları altında havacılık 

endüstrisine yönelik öngörülen optimizasyon kriterleri üzerindeki etkileri inceledi. Bu 

nedenle, klasik tersinmez Brayton çevrimi uzatılır ve hava soluyan motorlara 

uygulanır; termodinamik çevrim modelinin bir parçası olarak tüm motor bileşenlerinin 

(serbest akıştan girişe ve çıkışa) etkilerini içeriyordu. 

 

Birçok araştırmacı, gaz türbinli motor dahil içten yanmalı motorlar için performans 

analizi yapmış olsa da, bu çalışma, uçak jet motorları için MP, MPD, ECOP ve ECOL 

gibi mevcut optimizasyon fonksiyonlarının bir uzantısıdır. Belirtildiği gibi, güç 

yoğunluğu, gücün çevrimdeki maksimum spesifik hacme oranı olarak tanımlanır. 

ECOP, güç çıkışının kullanılabilirlik kayıp oranına oranı ve ECOL ise güç çıkışı eksi 

kullanılabilirlik kayıp oranı olarak tanımlanır. Klasik tersinmez Brayton çevrimini 

uçaklar için geçerli olan hava soluyan motorlara genişletmek için, motor ağırlığı, 

hacmi ve ön alana göre daha yüksek tahrik verimliliği ve daha yüksek güç çıkışı 

oranları elde etmek için daha fazla geliştirme çalışması yapılmalıdır. Amaç, 

maksimum ECOP, ECOL, güç yoğunluğu ve güç koşullarının optimal çalışma 

koşullarının belirlenmesi için bir temel olarak kullanılabileceği gerçek bir turbojet 

çevrimi için termodinamik olarak daha verimli bir şekilde motor boyutuna (ağırlığına) 

daha büyük bir güç çıkışı elde etmektir. ve uçuş koşullarında gerçek turbojet motorları 

için ön tasarım kısıtlamaları. 

 

Uçak motoru çevrimi için kullanılan çeşitli optimizasyon kriteri fonksiyonları için 

karşılaştırmalı performans analizi, turboramjet motorunun nihai amaçlanan 

uygulamasına ulaşmak için ramjet, art yakıcısız turbojet ve art yakıcılı tubojet'e 

uygulanacaktır. Turboramjet motor çevrimi, Türbin Tabanlı Kombine Çevrim 



xxvi 

Motorları (TBCC) olarak tanımlanır. Bu tür hibrit çevrimli motorlar, İHA'lara, 

UCAV'lere ve geleceğin hipersonik uçuş araçlarına güç sağlamak için uygulanabilir.  

 

Uçak jet motoru çevrimlerine uygulanan tersinmez Brayton çevrimi için 

karşılaştırmalı performans analizi için kullanılacak yazılım, MathWorks grubu 

tarafından sağlanan MATLAB 2018b'nin akademik versiyonudur. 

 

Havacılık endüstrisinden kaynaklanan emisyonlar ve ışınımsal zorlama (RF), hava 

kirliliği ve ekoloji üzerindeki etkileri, havacılığın ilk on yayıcıdan biri olarak yer aldığı 

önemli bir endişe kaynağıdır. RF'ye katkıda bulunan başlıca sera gazı emisyonları 

şunlardır: karbondioksit CO2, karbon monoksit CO, su H2O, azot oksit NOX, kükürt 

oksitler SOX ve uçucu organik bileşikler VOC'ler. 

 

Bu nedenle, hava taşıtı tahrik sistemlerinin performans değerlendirmesi, güç ve yakıt 

tüketiminin yanı sıra çevresel ve ekolojik koşullar açısından da değerlendirilmelidir. 

Bu nedenle, havacılık endüstrisi tarafından ekonomik ve ekolojik olarak uygun 'yeni 

nesil motorlar' tasarlamak için araç olarak kullanılabilecek çeşitli optimizasyon kriter 

fonksiyonları. 

 

Art yakıcısız turbojet, yeni tanımlanmış performans parametreleri (PLOS, EPLOS ve 

CBSF) kullanılarak incelendi ve maksimum güç, maksimum güç yoğunluğu, ECOP 

ve ECOL gibi mevcut amaç fonksiyonları ile birleştirildi. Ayrıca bu parametreler için 

MP, MPD, MECOP ve MECOL'ün optimal çalışma koşulları üzerindeki değişken 

etkileri ve bunların motor performansı üzerindeki etkisi gösterilmiştir.  

 

Art yakıcılı turbojet, maksimum güç (MP), PLOS ve EPLOS bazında değerlendirildi 

ve art yakıcısız turbojet ile karşılaştırıldı. Çalışan bir art yakıcıya sahip bir turbojet 

motor düşünüldüğünde, performans analizi ve karşılaştırması, öncelikle Mach sayısı 

ve irtifa olan daha kısıtlayıcı parametrelere doğru bir kayma alır. Ek olarak, art 

yakıcısız ve art yakıcılı motor bileşenleri olmayan turbojetin özgül hacmi MP bazında 

değerlendirilmiş ve karşılaştırılmıştır. 

 

Ramjet sadece maksimum güç (MP) bazında araştırılmış ve brülörsüz ve brülörlü ve 

brülörlü turbojet ile karşılaştırılmıştır. Ramjet, iki farklı yöntem kullanılarak 

değerlendirildi: Tek Eğik ve Normal Şok (SOSN) çözümü ile Çoklu Eğik Şok ve Tek 

Normal Şok çözümü. SOSN varsayımı, giriş için çok daha yüksek bir durgunluk basınç 

düşüşü sağlar ve gerçek gaz türbini giriş basıncı geri kazanımı (ram geri kazanımı) 

tasarım özelliklerinin bir temsili değildir. Bu nedenle, MIL-E-5007D spesifikasyonu 

artık difüzör durgunluk basınç oranının tanımı için kullanılmaktadır ve 1-5 arasındaki 

Mach sayıları için geçerlidir. 

 

Ramjet'e benzer şekilde, turboramjet motorunun performansı, Mach sayısının bir 

fonksiyonu olarak irtifa ve giriş hava kütle akışındaki değişiklikler için çift modlu 

çalışmada maksimum güç (MP) amaç fonksiyonu kullanılarak değerlendirildi. Çift 

modlu çalışma altında, turbojet motor art yakıcı çalışır durumda kabul edildi. 

 

Motor konfigürasyonları, termodinamik tahrik çevrimlerine ve denklemlerine göre 

kesinlikle değerlendirildi. Ne yazık ki, bu tür bir değerlendirme kullanıldığında, motor 

bileşenleri arasındaki bağlantı ve karşılıklı bağımlılık kaybolur. Herhangi bir motorun 

itici faktörü, yakıt akışındaki değişikliktir. Yakıttaki bir artış basınçları, sıcaklıkları, 



xxvii 

NG ve hava akışını yükseltir; bunun tersi de doğrudur. Bu nedenle, gerçek bir motor 

performansı değerlendirmesi için aşağıdakilerin karşılıklı bağımlılığı: ortam koşulları 

(serbest akış basıncı, sıcaklık ve Mach sayısı); giriş hava kütle akışı; kompresör, 

yanma odası, türbin ve çıkış nozulu giriş ve çıkış basınçları ve sıcaklıkları; yakıt akışı; 

ve gaz jeneratörü hızı birbirini etkilemelidir. Ancak, sevk denklemlerinin sınırlaması 

nedeniyle, bu ara bağlantı şekli görülemez; güncellenmedikçe. 

 

Bununla birlikte, sevk denklemleri ve amaç fonksiyonları, havacılık endüstrisi 

tarafından ön tasarım amaçları için bir rehber olarak kullanılabilecek araçlardır.Bu 

araştırmanın, hava soluyan motorların (Ramjet, Turbojet: No Afterburner, Turbojet: 

With Afterburner ve Turbo-Ramjet) ön tasarımında değerli bilgiler sağlayacağı ve 

adaptif motor bileşenlerine ve gerçekten akıllı motorlar; tasarımcıların ve motorun 

amaçlanan performans potansiyeline ulaşmak için mevcut çalışma durumunu 

değerlendirebilen ve en verimli güç rejiminde (ECOL veya ECOP veya MP veya 

MPD) çalışabilen bir motor. 

 

 

 



xxviii 



1 

1. INTRODUCTION 

It was the inception of the Brayton cycle that gave way to the inevitable invention of 

internal combustion engine and its use in the power generation, automative, 

aeronautical and marine industry. The Brayton cycle is defined as a four step process 

begining by an adiabatic compression, followed by an isobaric heat addition, which is 

succeeded by an adiabatic expansion and finally ending with an isobaric heat rejection.  

It was not long after that the Brayton cycle’s full adavantage was realized in the 

aviation industry through the development of gas turbine / jet engines; where now air 

is being used as the primary working fluid. For a jet engine the Brayton process 

corresponds to the compressor, combustion chamber, turbine and surrounding 

atmosphere respectively [1]. 

Jet propulsion is based on Sir Isaac Newton’s third law of motion: the principle of 

action and reaction,where the flow of air gets compressed and the addition of fuel 

converts mechanical and thermal energy into thrust. The thrust is then defined using 

Newton’s second law of motion: 𝐹𝑜𝑟𝑐𝑒 = 𝑀𝑎𝑠𝑠 𝑥 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛, where the force has 

a large enough acceleration to overcome the mass of the object itself [2][3].  

In 1928, as an Air Commodore at the Royal Air Force in Cranwell Sir Frank Whittle 

wrote his thesis on “Future Developments in Aircraft Design ”. In his thesis Whittle 

writes: 

“There are three ways of speculating on the future. There is the immediate 

future, the further future, and the far future. The object of the work is to discuss 

the “middle” future, with a certain amount of speculation which probably 

overlap the immediate future.  

 

Development will take place along the following lines.  

1. Increase of range, 

2. Increase of speed, 

3. Increase of reliability, 

4. Decrease of structural weight, 

5. More economical flight, 

6. Increase of ceiling, 

7. Increase of load carrying capacity, 



2 

8. Greater ability to withstand the elements. 

 

Many of these will be interdependent, for instance, a decrease in structural 

weight will result in increased range, etc (p. 2,3).” 

 

Whittle explains in his thesis that an increase in rage can be attainable through 

enhancement of an aircaft’s stramline, airfoil section and reduction in structural 

weight. Where as an increase in speed can be reached from an overall decrease in L/D 

(due to the increase of passive drag) and flying at higher altitudes. In adition, relability 

and decrease in structural weight can be sought via multi-powerplant usage and 

distributing engine weight over the span of the aircraft wings rather than being 

clustered near the fuselage. Moreover, flying at altitudes above the tropopause (33 000 

ft) where occurences such as depressions and convection currents do not exist, ensures 

that winds will be absolutely steady. However, limitations in structural weight depends 

on blade velocity and temperature of gases, this is especially true for the turbine. An 

increase in efficiency and ceiling can be obtained if an air driven turbine is used as the 

prime mover rather than a steam turbine; where an air turbine has the capability of 

giving back energy through supercharging, greater efficiency at higher altitudes and 

more flexibility in rate of rotation than current engines. Also, future power units would 

utilize the method of drive: forcing gases at high velocity through a nozzle where the 

reaction obtained being opposite in direction to the flow of the escaping gas [4][5]. 

It is only but two years after Sir Frank Whittle (accredited as the father of jet engines) 

had written his thesis that the combination of the Brayton cylce and Jet propulsion 

materialized into the first jet aircraft engine and patented on the 16th of January 1930, 

GB347206A “Improvements Relating to the Propulsion of Aircraft and other 

Vehicles”. The single shaft jet engine consisted of a centrifugal compressor, comustion 

chamber, a multi-stage axial turbine and a propelling nozzle. Air is used as the working 

fluid where a portion of the expanded air from the turbine is used to drive the 

compressor and the remaining portion passes throught the propelling nozzle further 

expanding to the atmosphere providing fluid reaction. The pressure drop during 

expansion takes place in two stages: the first pressure drop takes place through the 

nozzles of the turbine and the second occurs in the propelling nozzles. The controling 

means of the engine include: fuel control, gas flow/path control, mechanical control of 

the speed of the blower and final emission of the gas may be directionally controlled 



3 

for manoeuvering purposes. In additon, an auxiliary unit would be needed for starting, 

fuel injection, lubrication and the like. In his patenet Whittle claims that his invention 

will provide larger thrust in proportion to its weight, perform better at higher altitudes, 

attain higher velocities, operate using existing heavy oils or fuels and have a relatively 

low fuel consumption. In addition, the efficiency of the device as a propulsive engine 

is not to be reduced by the reduction of density of the atmosphere, rather, owing to the 

reduction of temperature of the upper atmosphere may be enhanced [6]. 

 

 

Figure 1.1: P-v diagram (left) and Part-section (right) of the Whittle jet engine. 

 

Almost a century later the thermal / energy cycle of preference utilized by the 

aerospace industry has not deviated away from that of the Brayton cycle and its 

essential engine components and control systems as described above. In addition, the 

objectives of the aeronautical industry as described by Sir Frank Whittle, to this day, 

are still being pursued: longer range; lower weight and fuel consumption; higher speed 

(Mach number), altitude, thrust and powerplant efficiency.  

The focuse of this thesis will be realated to the objectives of: fuel consumption 

reduction; and higher Mach number, altitude, thrust and powerplant efficiencies. 

Therefore, the performance of the irreversible Brayton cycle will be assessed by 

defining new and currently available optimization functions and applied specifically 

to aircraft gas turbine / jet engines. 

1.1 Purpose of Thesis 

The increased awareness that energy is a limited resource has triggered researchers to 

develop new techniques to enhance the utilization of energy conversion devices. In 



4 

addition, discharges and radiative forcing (RF) associated to the aeronautical sector 

along with their impacts upon atmospheric and environmental pollution are a 

significant concern; with aviation ranking amongst the ten largest emitters. The 

primary emitters of greenhouse gases that supports radiative forcing include: carbon 

dioxide CO2, carbon monoxide CO, water H2O, nitrous oxide NOX, sulphur oxides 

SOX and volatile organic compounds VOCs.  

Therefore, performance assessment pertaining to aeronautical thrust generating power 

plants must be investigated with regards to ecological and sustainable conditions also 

taking into account fuel and power consumption. Thus, various optimization criterion 

functions can be utilized as tools by the aeronautical sector to create ‘next generation 

power plants’ that are advantageous from both an economic and environmental 

standpoint. 

The first law of thermodynamics deals with the quantity of energy where energy 

cannot be created or destroyed but can only be converted from one form to another. 

The second law deals with the quality of energy and its degradation during a process, 

entropy generation and the lost opportunities to do work. In compliance with the 

second law of thermodynamics performance optimization functions are defined in 

terms of: exergy (availability); reversible work (maximum useful work that can be 

obtained as a system undergoes a process between two specified states) and 

irreversible work (wasted work potential during a process as a result of irreversibilities 

known as exergy destruction or lost work) [7-10]. 

While numerous scientists have examined the performance of internal combustion 

engines and industrial based air breathing engines, this work is an addendum of the 

currently defined optimization functions such as maximum power (MP), maximum 

power density (MPD), ecological coefficient of performance (ECOP) and ecological 

function (ECOL) for aeroplane jet engines. Power density abides to the designated 

proportion concerning power to maximum specific volume of a powerplant 

configuration. Whilst, ECOP represents a proportion of power generated to loss rate 

of availability and ECOL as power generated less the loss rate of opportunity. In 

addition, newly defined parameters are introduced as power loss parameter (PLOS), 

effective power loss parameter (EPLOS) and Carnot-Brayton shape factor (CBSF) for 

a better assessment of the performance and power losses throughout the operation of 



5 

the engine cycle. Further more, the optimization functions, of MP, MPD, ECOP and 

ECOL are compared with respect to each other at their optimal operation conditions. 

In pursuance of expanding the traditional irreversible Brayton configuration 

ecompasing aeronatical powerplants implementable for aeroplanes, more advanced 

investigations must be conducted to achieve: augmented propulsion efficiency and 

proportions of power generation directed towards engine mass, volume, and inlet area. 

The purpose is to attain a increased power generation to powerplant dimension (mass) 

via a much required thermodynamically efficient demeanor considering actual gas 

turbine powerplants where maximum ECOP, ECOL, power density and power 

operations are utilized for the foundation and resolution with regards to optimum 

working circumstances and precursory design limitations for actual air breathing 

powerplants at flight conditions. 

This research will study the impact of the prescribed optimization criterions targeted 

towards the aviation industry as a function of changes in compressor pressure ratio 𝜃𝑐, 

compressor and turbine efficiencies (𝜂𝑐 and 𝜂𝑡 respectively), cycle temperature ratio / 

maximum cycle temperature, altitude along with flight Mach number 𝑀∞ where 

applicable with respect to the jet engine being considered. Consequently, the classical 

irreversible Brayton configuration applicable to airbreathing powerplants is extended 

and includes the impacts of all the engine components (from free straem to inlet to 

outlet) as members of the thermodynamic cycle representation. 

1.2 Literature Review 

Angulo-Brown [11] proposed an ecological (ECOL) optimization criterion for finite-

time Carnot heat engines as, 𝐸𝐶𝑂𝐿 = 𝑊̇ − 𝑇0𝑆𝑔̇, where 𝑊̇ is the power output, 𝑇0𝑆𝑔̇ 

is the loss rate of availability. However, the ecological coefficient of performance 

(ECOP) became a performance criterion for diverse energy transformation power 

plants for the environment [12] and defined by proportions of power generated to the 

loss rate of attainability 𝐸𝐶𝑂𝑃 = 𝑊̇/𝑇0𝑆𝑔̇.  

Further researchers made explorations utilizing ECOP and ECOL for the investigation 

of both Carnot and Brayton thermodynamic configurations to enhance performance 

along with reducing green house gases. Cheng and Chen [13] utilized ECOL 

applicable to Carnot heat engines, and augmented regarding to the cycle temperature 



6 

and heat conductance fraction. Furthermore, Cheng and Chen [14] performed an 

analogous examination for an irreversible Brayton heat engine while perfecting for 

thermal conductance along with adiabatic temperature fraction. Whereas Blank [15] 

studied a rerciprocal sink outlet temperature to attain the most optimal power of an 

openended irreversible Brayton and Joule-Brayton heat engine configuration. Üst et 

al. [16] utilized ECOP for an irreversible Carnot heat engine with losses as a 

consequence of the heat exchange over finite temperature variations, heat leak and 

internal irreversibilities. Likewise, Üst et al. [17] directed an investigation for an 

irreversible Bryton heat engine and extrapolated correlative assessments for thermal 

efficiencies and power outputs. Furthermore, Üst et al. [18] employed the irreversible 

regenerative Brayton heat engine configuration and demonstrated benefits of entropy 

generation rate, thermal efficiency and investment expenditure. Olivera et al. [19] 

created a representation of an irreversible Brayton configuration to explored optimal 

quantities of temperature and pressure ratios of the cycle for the initial stages of 

compression and expansion. In contrast, Haseli [20] carried out an investigation for a 

trilogy of arrangements and correlated operational conditions at maximum thermal 

efficiency and work generated and minimum entropy output for the openended power 

Brayton configurations. Contrarily, Gonca [21] conducted a review for a three-shaft 

aeroderivative powerplant with a focus on energetic, exergetic and ecological abstracts 

such as power, power density, second-law efficiency (exergy efficiency), exergy 

destruction, ecological coefficient of performance (ECOP) and effective ecological 

power density (EFECPOD). The impact of engine design criterions (namely: gas 

generator speed, air mass flow rate, equivalence ratio, pressure ratio, etc.) on the 

performance attributes were evaluated by engaging temperature-reliant specific heats 

and heat transformation deficits. Although Balli [22] investigated a two-shaft high 

bypass fraction turbofan engine and expressed approximated results for exergy 

efficiency, waste exergy fraction, eco-friendly and economical impact element. 

Durmusoglu and Üst, on the other hand, [23] conducted a thermo-ecological 

augmentation of an irreversible regenerative close-ended Brayton configuration using 

a design function derives as the fraction of net power generated to total value change 

(F); where the price change incorporates variables of fuel, expenditure, eco-friendly 

and functioning and preservation financial estimates. Furthermorer, Sadatsakkak et al. 

[24] evaluated an irreversible regenerative close-ended Brayton configuration 

imparting three design theorems: power generation, thermo-ecological requirements 



7 

and an economical equality. Moreover, Açıkkalp [25] examined ecological thermo-

environmental formulations for the evaluation of performance for an irreversible solid-

oxide fuel cell Brayton heat engine in addition to power, exergy destruction, energy 

and exergy efficiencies. Finally, Gonca [26] investigates the impact of engine design 

criterions employing performance parameteres including effective power (EP), 

effective power density (EPD), ecological coefficient of performance (ECOP), exergy 

destruction (EXED) along with exergy efficiency (EXEF) of a regenerative aero-

derivative engine. 

Najjar and Balawneh [34] examined the changes of compressor pressure ratio and 

turbine inlet temperature and its impact pertaining to specific thrust and specific fuel 

consumption (SFC). Najjar and AbuEisheh [35] adapted an exergy assessment towards 

the evaluation of specific thrust and SFC. Also, Najjar and el-Sharif [36] utilized 

thermodynamic parameters and optimization gradient process based methods for 

reductions on SFC of a turbofan engine. In addition, Najjar and Balawneh [37] 

examined compressor and optimum compressor ratio as well as the turbine inlet 

temperature (TIT) to evaluate the impact on SFC and specific thrust. De Sa and Al 

Zubaidy [38] evaluated the impact of atmospheric temperature on engine performance. 

Badami et al. [39] conducted an empirical and algebraic investigation for a miniature 

model turbojet engine and evaluated its true performance. Salpingidou et al 0 

investigaed the thermodynamic configuration of a taditional restorative aero-

derivative powrplant, by positioning a heat exchanger downstream of the power 

turbine, along with drawing comparisons between two nonconventional recuperative 

aero-derivative powerplant configurations. Wang et al. [41] carried out performance 

evaluations on a three spool engine. Guha [42][43] studied diverse variables and 

applied augmentation methods for the ascertion of highest thermal efficiency and 

specific thrust pertaining to turbofan powerplants. Patel et al. [44] considered 

efficiency, thrust and fuel consumption augmentation pertaining to turbojet 

powerplants. Kodal [45] implemented the maximum power density technique for 

irreversible amalgamated Carnot configurations. Chen et al. [46][47] conducted 

maximum power density assessments for Brayton configurations, where the 

intercooling pressure ratio is augmented for normalized power density; inaddition to 

an Atkinson powerplant configuration whilst the impact of temperature ratio was 

analyzed. Gonca et al. [48][49] evaluated the power density of a Miller configuration 



8 

and then conducted an analogous investigation for a Dual-Atkinson configuration 

beneficial towards optimization of internal combustion powerplant performances. In 

addition, Al-Sharki et al. [50] investigated the Miller configuration efficiency utilizing 

the maximum power density approach. Cheng et al. 0 have also inquiered into Brayton 

configurations intended for high power producing hypersonic ‘next generation type’ 

aircraft/spacecraft vehicle. 

Contrastingly, researchers have utilized evaluations via ECOP and ECOL for both 

Carnot and Brayton thermodynamic configurations aiming at enhanced performance 

and reduced green house gases. Notwithstanding, ECOP and ECOL are techniques that 

can be applied by the aeronautical sector for the conceptualization of ‘new generation 

engines’ that are ecologically and environmentally advantageous [31-46]. Moreover 

[52], investigators have studied performance preconditions and their effect on gas 

turbine cycle performance and module conception. Zenkner et al. [68] examined the 

maximum fuel expenditure and accessible installation area towards powerplant inlet 

augmentation. J. J. Otter et al. [69] evaluated aerodynamic installation consequences 

as well as effects on central nozzle dimension. Chen et al. [70] studied divergences of 

engine modules and their consequence on the performance of an adaptive engine cycle. 

In terms of the ramjet, tubojet with afterburner and turbine based combined cycle 

(TBCC) engines, researchers have used the application of exergy exploration on 

various aspects in accordance to task requirements. Şöhret et al. [73] applied an exergy 

efficiency analysis for a ramjet engine using hydrogen fuel on a component (inlet, 

combustion zone and nozzle) and overall engine level. Latypov [76] conducted an 

exergy investigation based on various energy supplies to the air flow of the ramjet 

duct. Latypov [77] also assessed the specific impulse and thrust-economic 

characteristics of the ramjet using exergy analysis. Ayaz et al. [78] used exergy 

analysis on a generic ramjet engine under three different Mach regiems. Moorhouse 

[79] expanded the exergy method to the design of a complete aircraft vehicle based on 

mission requirements including component level evaluation. Moorhouse et al. [80] 

further expands his study to the applicaion of hypersonic vehicle design as an energy 

problem. Moorhouse et al. [81] also applied the exergy consept to the hypersonic inlet 

flow problem to determine the optimal shock-on-lip position for off-nominal flight 

conditon. Marlet et al. [82] also made use of exergy evaluation for a combined ramjet 

and turbojet engine during transient maneuvers as well as the wake region of the 



9 

turbojet engine. Ispir et al. [83] used an exergy simulation based platform for the 

thermodynamic cycle and performance optimization of the STRATOFLY MR3 

aircraft vehicle in DMR mode, ATR combustor, regenerator, nozzle, turbomachinery 

components and air turbo rocket bypass line. Ehtaei et al. [84] utilized an exergy 

approach for a turbojet engine with an afterburner (J85-GE-21) on a component level 

where the highest exergy efficiency was observed for the compressor and nozzle. Roth 

et al. [85] considered the loss management method for the analysis and quntification 

of technology impact of the F-5E/J85-GE-21 engine/airframe combintions and its 

relation to vehicle mass properties (weight). Camberos et al. [86] have published a 

book specifically describing the advantage of exergy analysis in the field of 

astronautics and aeronautics for various types of propulsion systems and even applying 

the concepts of exergy to airfoil drag evaluation. Hayes et al. [87] showed that exergy 

can be adopted to various aspects in aerospace including design, performance and 

thermodynamic analysis of commercial aerospace systems, propulsion systems, 

aerodynamic and structural optimization, multi-disciplinary optimization based on the 

Breguet equation and mapping exergy over a variable flight envelope. Riggins et al. 

[88] also makes use of the laws of thermodynamics for the evaluation of a hyperspace 

vehicle applicble to both ramjets or scramjets using individual stream tubes as 

components within the overall fluid control volume. Balli 0 conducted a study of 

exergy destruction rates within engine components which were split into 

endogenous/exogenous and avoidable/unavoidable parts on a military turbojet engine 

with afterburner. Balli [90] then used the J85 turbojet engine with afterburner to assess 

the performance, exergetic, exergoeconomic, sustainability and environmental 

damage cost at Idle (ID), Intermediate (INT), Military (MIL) and Afterburner (AB) 

operation modes. Balli [91] further considered the afterburning effect on energetic and 

exergetic performance of an experimental Turbojet Engine (TJE) and to determine 

thermodynamic inefficiencies at military (MIL) and afterburner (AB) operation 

modes. Akkaya et al. [92] defined an exergetic performance coefficient (EPC) to assess 

a fuel cell power generation system (fuel cell stack, afterburner, fuel and air 

compressors, and heat exchangers) fed by hydrogen. Yüksel et al. [93] evaluated the 

exergetic analyses at Military (MIL) and Afterburner (AB) process modes of the (J85-

GE-5H) military turbojet engine using kerosene (JP-8) and hydrogen (H2) fuels. Balli 

et al. [94] conducted a performance assessment for both MIL and AB operation modes; 

and while under afterburner operation, examined energetic and exergetic performances 



10 

and the effects on the environmental, ecological and sustainability metrics of the 

engine. Akkaya et al. [95] utilize an exergetic performance coefficient (EPC) for a gas 

turbine to investigate design parameters including fuel utilization, current density, 

recuperator effectiveness, compressor pressure ratio and pinch point temperature, to 

achieving higher exergy output with lower exergy loss in the system. Bastani et al. 

[96] applied exergy analysis and showed that the greatest exergy loss is in the 

afterburner due to its high irreversibility; therefore, the optimization of afterburner has 

an important role in reducing the exergy loss of total turbojet engine cycle. Yüksel et 

al. [97] conducted an exergy-based economic and sustainability analysis for a (J85-

GE-5H) military turbojet engine (TJE) using kerosene and H2 fuel under MIL and AB 

regeims where higher exergy destruction occurred in the afterburner exhaust duct 

(ABED) and combustion chamber (CC) which led to higher exergy destruction costs. 

Niknamian [98] exergy analysis on J85-GE-21 turbojet engine and system 

optimization based on PSO (Particle Swarm Optimization) methods which showed 

that highest and lowest exergy efficiency of the engine components corresponded to 

the diffuser and compressor respectively. Sürer et al. [99] performd a critical mini 

review exergy analyses of jet engines which concluded that if there is no afterburner, 

the combustion chamber has the greatest exergy destruction and thus minimum exergy 

efficiency due to its highly thermodynamically irreversible process; whereas the 

presence of an afterburner constitutes the biggest exergy destruction and smallest 

exergy efficiency. Dong et al. [100] revealed that the exergy analysis method can be 

used as a direct indicationof the weaknesses of an entire energy system, reveal the 

interactions among system components and estimate the realistic work potential of 

different subsystems; it also provides a significant guidance for the improvement of 

engine performance, reduction of fuel consumption and optimization of engine 

combustion. Noori et al. [101] made use of four objective functions (𝐹𝑠, TSFC, 𝜂𝑡ℎ and 

𝜂𝑝) for the optimization of an ideal turbojet engine with afterburner. Nasab et al. [102] 

conducted an exergy analysis for the J85-GE-21 turbojet engine with afterburner 

where the highest exergy efficiency was demonstrated by the diffuser and the lowest 

belonged to the compressor.  

It is clear that the use of exergy as an analysis tool provides an advantage in the 

evaluation and optimization of aircraft gas turbine propulsion systems. Identification 

of the level of exergy destruction can be made on a component level and subsequently 



11 

exploit optimization functions for the improvement of TSFC, ST, 𝜂𝑡ℎ and 𝜂𝑝 thus 

reducing the ecological impact of aircraft based engines on the environment.  

1.3 Conjecture 

A comparative performance analysis for various optimization criterion functions is to 

be carried out for an irreversible Brayton cycle applicable to aircraft jet engines: 

Turbojet (Without an Afterburner), Turbojet (With an Afterburner), Ramjet and 

Turboramjet. Where the turboramjet engine cycle is identified as Turbine Based 

Combined Cycle Engines (TBCC). Such hybrid cycle engines can be applied to 

UAV’s, UCAV’s and powering future hypersonic flight vehichles. 

The main focus of the comparative performance analysis for the various optimization 

criterion functions will be applied to the aircrat turbojet cycle without an afterburner; 

whereas the tubojet with an afterburner, ramjet and turboramjet engines will be 

assessed on the basic principle of maximum power generation capability.  

It is aspired that this investigation would bestow beneficial comprehension in the 

precursory configuration of airborn based gas turbibe engines (Turbojet: No 

Afterburner, Turbojet: With Afterburner, Ramjet and Turbo-Ramjet) and construct a 

platform towards a journey in promoting adaptive powerplant modules and 

configurations for the inception of truly intelligent powerplants; a gas turbine with the 

capability of evaluating its present operating condition and work towards a logical 

appreciable and favorable power interim (ECOL or ECOP or MP or MPD) such that 

the developer’s and engine’s expected performance capability is realized. 

The program utilized to investigate the correlative performances of an irreversible 

Brayton configuration relevant for aircraft jet engine cycles is the scholistic platform 

of MATLAB 2018b supplied by the MathWorks corporation (See APPENDIX D). 

 



12 



13 

2. THEORETICAL MODEL OF TURBOJET: NO AFTERBURNER 

2.1 Objective Formulations and Established Variables 

The irreversible Brayton cycle and T-s diagram characterization of the turbojet cycle 

configuration are shown in Figure 2.1. The elementary states of the Brayton cycle are: 

air compression accomplished via a multistage compressor; squeezed fluid is further 

combined with fuel to be ignited at static pressure constraints inside a combustion 

chamber; lastly, high temperature gases leaving the combustor are dialated across a 

turbine and nozzle to generate work. The representation of the turbojet engine applied 

for investigation operates between a high temperature heat source, 𝑇𝐻, and low 

temperature heat sink, 𝑇𝐿. The rate of heat transmitted from the heat source to the jet 

engine and rate of heat casted to the heat sink from the jet engine are 𝑄̇𝐻 and 𝑄̇𝐿 

respectively. 

 

 

 
(a) (b) 

Figure 2.1: Engine arrangement (a) and T-s schematic representation of a turbojet cycle (b). 

 

The rate of total heat trasmitted 𝑄̇𝐻𝑇 and rejection 𝑄̇𝐿𝑇 are defined as: 

𝑄̇𝐻𝑇 =  𝑄̇𝐻 +  𝑄̇𝐿𝐾        (2.1) 

𝑄̇𝐿𝑇 =  𝑄̇𝐿 +  𝑄̇𝐿𝐾        (2.2) 

where 𝑄̇𝐿𝐾 is the rate of heat seepage coming out of a high and flowing towards a low 

heat reservoir and can be asserted as: 



14 

𝑄̇𝐿𝐾 =  𝑚̇𝑎𝜉𝐶𝑝𝑎(𝑇𝐻 − 𝑇𝐿)       (2.3) 

where 𝜉 is a percent of internal conductance and 𝑚̇𝑎 as air mass flow rate to the 

powerplant. In agreement with the first law of thermodynamics, the power extracted 

from the turbojet powerplant configuration is: 

𝑊̇ =  𝑄̇𝐻𝑇 −  𝑄̇𝐿𝑇 =  𝑄̇𝐻 −  𝑄̇𝐿 .      (2.4) 

The total heat trasmission rates coming out of the high temperature source is expressed 

as: 

𝑄̇𝐻𝑇 =  𝑚̇𝑓𝑄𝑅𝜂𝑏        (2.5) 

where 𝑄𝑅 is the amount of heat discharged from the fuel per unit mass, the mass flow 

rate of fuel is 𝑚̇𝑓 and the combustion efficiency is 𝜂𝑏. 

A compression ratio parameter, 𝜃𝑐 is delineated as: θc =  (𝑃02 𝑃01⁄ )(γ−1) 𝛾⁄ ; as well as 

a cycle temperature ratio (𝛼) designation of: 𝛼 = 𝑇03 𝑇𝑎⁄ . 

Implementing an energy balance between the inlet and outlet of the combustor the fuel 

to air ratio, f  resolves as: 

𝑓 =  
CptT03− CpaT02

QRηb−CptT03
        (2.6) 

The expression of the turbojet variables including: thrust (F), specific thrust (FS), thrust 

specific fuel consumption (TSFC), air specific impulse (Ia), thermal (th), and 

propulsive  (p) efficiencies, power density (𝑊𝑑
̇ ), rate of entropy generation (𝑆̇𝐺), 

ECOL and ECOP have been prescribed in accordance to the engine configuration 

imparted in Figure 2.1. Furthermore, variations of altitude were evaluated utilizing the 

methodology provided by Airbus [30]. 

Moreover, the dimensional deviation of respective powerplant constituents and their 

collaboration to the overall performance was analyzed subject to maximum power, 

power density, ECOP and ECOL constraints. Dimensional deviations were expressed 

with reference to variations of specific volume (∆𝜈) and dimensionless pressure (Ṕ) 

and temperature (Ť). Furthermore, the span of the combustion chamber was assessed 

for optimum working constraints utilizing the equations imparted by Mattingly [73]. 

𝐿 = 𝑐𝑃𝑡2
−𝑟/√𝑇𝑡3        (2.7) 



15 

Equation 2.7 above was calibrated using a 𝑃𝑡2 = 2.5 and 𝑇𝑡3 = 6 and 𝐿 = 0.526 𝑚 

provided by Farokhi [74] such that the constant 𝑐 was resolved. 

The thermal efficiency for the turbojet powerplant cycle is written as: 

𝜂𝑡ℎ =  
𝑊̇

𝑄̇𝐻𝑇
=  

𝑊̇

𝑚̇𝑓𝑄𝑅𝜂𝑏
 .       (2.8) 

The thrust formula is acquired by implementing the integral form of the momentum 

equality by properly designating a control volume across the engine [23]. 

F =  ṁ𝑎[(1 + f)C5 − Ca] + 𝐴5(𝑝5 − 𝑝𝑎)     (2.9) 

where C5 = exit speed, Ca = free stream speed, A5 = nozzle exit cross sectional area 

and f = proportion of fuel to air mass flow rate. Accepting that perfect expansion occurs 

acorss the nozzle and utilizing a per unit weight foundation, specific thrust may be 

defined in the form below: 

F𝑆 = (1 + f)C5 − Ca.        (2.10) 

The thrust specific fuel consumption is directly enumrated as: 

TSFC =
𝑓

𝐹𝑆
         (2.11) 

and the air specific impulse becomes: 

Ia =
𝐹𝑆

𝑔
 .         (2.12) 

The power density equals frations of power to the maximum specific volumes of the 

engine configuration and designated as: 

𝑊𝑑
̇ =  

𝑊̇

𝑣5
.         (2.13) 

The propulsive efficiency is delineated as: 

𝜂𝑝 =  
𝐹 𝐶𝑎

𝑊̇
 .         (2.14) 

The destruction of exergy for a powerplant equals the reversible power less the actual 

power generated and formulated such that: 

𝑋̇𝐷𝐸𝑆 =  𝑊̇𝑟𝑒𝑣 − 𝑊̇ .        (2.15) 

The power manifested from a Carnot heat powerplant is known as the reversible 

power: 



16 

𝑊̇𝑟𝑒𝑣 =  𝑚̇𝑓𝑄𝑅𝜂𝐶𝑎𝑟𝑛𝑜𝑡       (2.16) 

where 𝜂𝐶𝑎𝑟𝑛𝑜𝑡 equals a Carnot efficiency. The formulation of a traditional Brayton 

configuration can be obtained from thermodynamic textbooks such as [28]. The 

fraction of destroyed exergy to the reversible power is considered as the power loss 

(PLOS) parameter and given as: 

𝑃𝐿𝑂𝑆 =  
𝑋̇𝐷𝐸𝑆

𝑊̇𝑟𝑒𝑣
 .        (2.17) 

The fraction of ideal minus real power from the Brayton configuration to the reversible 

power is designated as the effective power loss (EPLOS) and written as: 

𝐸𝑃𝐿𝑂𝑆 =  
𝑊̇𝐵𝑟𝑎𝑦− 𝑊̇

𝑊̇𝑟𝑒𝑣
 .        (2.18) 

The rate of entropy generation of the turbojet powerplant configuration is prescribed 

in the form of: 

𝑆̇𝐺 =  
𝑄̇𝐿𝑇

𝑇𝐿
−  

𝑄̇𝐻𝑇

𝑇𝐻
        (2.19) 

The ratio of power output to the loss rate of availability is known as the ECOP 

objective function and written as: 

𝐸𝐶𝑂𝑃 =  
𝑊̇

𝑇0𝑆̇𝐺
         (2.20) 

The ecological equality is delineated as: 

𝐸𝐶𝑂𝐿 = 𝑊̇ − 𝑇0𝑆̇𝐺 .        (2.21) 

A portion of the total energy released from fuel combustion is inaccessible energy 

while the residual energy is considered to be transformable reversible power; that is, 

the power produced fron the ideal Carnot cycle. However, Brayton cycles are he basis 

of turbojet engine operation. As a consequence of the powerplant configuration and 

operational variations amongst the representations of the Brayton and Carnot 

configurations, performance assessments regarding ideal Carnot configurations cannot 

be an exemplary evaluation. Furthermore, the Carnot and Brayton cycles are not 

interchangeable and cannot expect that the power generated by either cycle to be 

interchangeable with one other; each cycle is unique from the other and must be 

assessed with their own respective operational states and cycle classification(s). 

Therefore, the performance of a real physical airbreathing engines must be investigated 



17 

and compared to its ideal case: i.e. an ideal Brayton cycle vs. aircraft engine Brayton 

cycle with losses. In addition, discussions of performance enhancement investigations 

with rgards to the Carnot cycle are profoundly misleading. The distinctions amongst 

PLOS and EPLOS depict the power that cannot be produced due to the representative 

configuration of the powerplant: Carnot vs. Brayton. Thus, the variable, Carnot 

Brayton Shape Factor (CBSF), which identifies non-reproducible power between the 

cycle configurations of Carnot and Brayton and specified as: 

𝐶𝐵𝑆𝐹 = 𝑃𝐿𝑂𝑆 − 𝐸𝑃𝐿𝑂𝑆 =
𝑊̇𝑟𝑒𝑣−𝑊̇𝐵𝑟𝑎𝑦

𝑊̇𝑟𝑒𝑣
.     (2.22) 

Based on the second law of thermodynamics, EPLOS is a performance analytical 

parameter which may be utilized in depicting impacts of internal irreversibilities on 

power production deficits from an ideal to an actual Brayton configuration. Whereas, 

PLOS indicates the overall power reduction from the Carnot cycle’s reversible power 

output. Therefore, in terms of power output, PLOS and EPLOS can be used as a means 

of correlations between two discretionary operational (design) constraints on a 

hundred percent scale, in addition to assessing the significant objective theorems 

collectively. Moreover, CBSF shows nor-reproducible power as a consequence of the 

preference of thermodynamic powerplant (the extent of how mcuh the power 

generated y the Brayton or any other cycle configuration - Otto, Rankine, Diesel, 

Stirling, Ericsson, etc.- approaches that of the ideal Carnot cycle). Therefore, the 

cooperative utilization of the prescribed objective functions of PLOS, EPLOS and 

CBSF, endows the performance evaluation of the irreversible Brayton configuration 

utilizing numerous working conditions. 

2.2 Mass Flow, Engine Speed and Shaft Force Models and Altitude 

The mass flow rate of air for the turbojet engine cycle is defined utilizing free stream 

standard atmospheric static criteria and given as: 

𝑚̇0 =  
𝑃0𝐴0𝑀0

√𝑇0
√

𝛾

𝑅
        (2.23) 

Where 𝐴0 is the upstream capture area from the engine inlet at altitude along with free 

stream flight Mach number and written as: 

(
𝐴0

𝐴1
)

𝑀𝐴𝑋
=  

1

𝑀0
[

2

𝛾+1
(1 +

𝛾−1

2
𝑀0

2)]

𝛾+1

2(𝛾−1)
      (2.24) 



18 

For supersonic conditions the presumtion is that 𝐴0 ≈  𝐴1. The corrected gas generator 

speed (RPM) for the compressor or turbine depends on the tip speed Mach number of 

the rotor and written as: 

𝑁

√𝜃
=  

60

𝜋𝑑
𝑀𝑡𝑖𝑝√

𝛾𝑅𝑇𝑟𝑒𝑓

1+
𝛾−1

2
𝑀2

       (2.25) 

Where the compressor or turbine diameter is 𝑑 and the temperature normalization 

parameter 𝜃 is defined as 𝑇𝑡/𝑇𝑟𝑒𝑓. Stagnation conditions for the turbine inlet  are 

presumed, therfore the Mach number of the flow entering the turbine can be considered 

as negligible; therefore, the Mach number at the tip of the turbine rotor is assumed to 

be sonic and under these constraints the preceding formula is truncated to: 

𝑁 =  
60

𝜋𝑑
𝑀𝑡𝑖𝑝√𝛾𝑅𝑇4        (2.26) 

The proportion of the turbine work to the exit velocity of the turbine is defined as the 

shaft force of the turbojet powerplant and written as: 

𝐹𝑠ℎ𝑎𝑓𝑡 =  𝑊̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 𝐶4⁄ .       (2.27) 

Reference [29] can be consulted for the complete comprehensive formulations. 

 

Alterations in altitude (ISA conditions) were examined for the turbojet cycle utilizing 

the temperature and pressure formulations provided by Airbus [30]. Beneath the 

tropopause (i.e. ≤ 11000 meters) the temperature and pressure equations are defined 

as: 

𝑇 = 𝑇0 − 6.5 
ℎ(𝑚)

1000
         (2.28) 

𝑤ℎ𝑒𝑟𝑒 𝑇0 = 150𝐶 𝑓𝑜𝑟 ℎ ≤ 11000 𝑚 

𝑃 = 𝑃0 (1 − 0.0065
ℎ

𝑇0
)

5.2561

       (2.29) 

𝑤ℎ𝑒𝑟𝑒 𝑇0 (𝐾), ℎ(𝑚) 𝑎𝑛𝑑 𝑃(ℎ𝑃𝑎) 

Beyond the tropopause the temperature and pressure formulations provided in the form 

of: 

𝑇 = −56.50𝐶 𝑓𝑜𝑟 ℎ > 11000 𝑚 

𝑃 = 𝑃11𝑒
−

𝑔

𝑅𝑇11
(ℎ−ℎ11)

        (2.30) 



19 

𝑤ℎ𝑒𝑟𝑒 𝑃11 = 226.32 ℎ𝑃𝑎, 𝑇11 = 216.65 𝐾 𝑎𝑛𝑑 ℎ11 = 11000 𝑚 

2.3 Algebraic Optimization Methodology 

The correlative performance assessments implemented for irreversible Brayton 

turbojet powerplant configuration utilizes the scholastic rendition of MATLAB 2018b 

software dispensed by the MathWorks corporation. 

Various algorithm approaches are created to accomodate the requisites of numerous 

types of augmentation chalenges. The primary classification of design perspectives 

are: classical optimization techniques, linear programming, nonlinear programming, 

geometric programming, dynamic programming, integer programming, stochastic 

programming, evolutionary algorithms, etc. The methodology of obtaining the 

criterions that provide the minimum or maximum value of an equation, where the 

functions constitute the endevour needed or the sought after interest is known as 

optimization. This research utilizes model optimization to augment (maximize or 

minimize) an objective equality without breaching capital restrictions; this is known 

as arithmetic data processing [31][32]. Thus, the compressor pressure ratio parameter 

(𝜃𝑐) being varied in order to resolve the maximum ECOP, ECOL, MP and MPD and 

minimum PLOS for changes of elevation and free stream flight Mach number 𝑀∞; 

taking the equivalent mathematical process as the differential of 𝜃𝑐. Moreover, the 

optimum operating conditions of maximum TSFC, 𝐼𝑎, 𝑓, EPLOS, thrust, and 𝜂𝑡ℎ. was 

resolved algebraically (Figure 2.15 to Figure 2.23). Assessments and correlations were 

subsequently conducted. 

 

  



20 

2.4 Outcomes and Considerations 

The evaluation of the environmental and cost effective consequences of the 

irreversible Brayton turbojet engine cycle at flight conditions, requiers a sequence of 

algebraic computations carried out consturcted upon numerical conceptualizations 

prescribed in the preceding subdivision. The maximums of power and power density, 

ECOP and ECOL principle expressions were examined and investigated based on 

optimal circumstances. The operational criterias (altitude and 𝑀∞) at each optimum 

circumstance, were corelated to one another subject to PLOS, EPLOS and CBSF 

specifications (Figure 2.15 to Figure 2.25). Performance assessmnets of fundamental 

turbojet engine variables along with their diversification with regards to constituent 

efficiencies, altitude and heat leakage were also examined. Table 2.1 constitutes the 

delegated variable inputs applied throughout this study; any deviation from the 

prescribed values are disclosed in the individual graphs. Normally, 𝑇𝐿 may be distinct 

from 𝑇𝑎, however, in this work 𝑇𝐿 ≅ 𝑇𝑎. Essential formulae for the turbojet engine 

configuration state point computation may be obtained in Saravanamutto et al [27].  

 

Table 2.1: Delegated variable inputs. 

Inputs 

TL = 200 K j = 0.95 g = 1.333 

TH = 2200 K m = 0.99 R = 287 J/kg K 

Ta = 223.3 K 

Pa = 26.5 kPa 
b = 0.98 Di = 0.8 m 

ALT = 10000 m Cpa = 1.005 kJ/kg K QR = 43000 kJ/kg 

 = 0.01 Cpg = 1. 148 kJ/kg K c = 0.87 

i = 0.93 a = 1.4 t = 0.9 

M = 0.5 Pb = 0.96 T03 = 1200 K 

 

Changes of the optimization functions, including PLOS and the thermal efficiency, 

with regards to power are depicted in Figure 2.2. For a compressor efficiency (𝜂𝑐) of 

95%, the maximum thermal efficiency gives the maximum ECOP and minimum PLOS 

at a turojet power generation marginally higher than 12000 kW. Conversely, at 

maximum power density, the power discharge from a given turbojet configuration 



21 

approaches 14000 kW for a comparmize on moderate reductions of thermal efficiency, 

ECOP and PLOS. 

Given Figure 2.3, the changes in the normalized construct of power, power density, 

ECOP, specific volume difference, 𝜈𝑑 =  (𝜈5 − 𝜈𝑎) 𝜈𝑎⁄ ) and ECOL collectively with 

the cycle thermal efficiency are given apropos the compression ratio parameter (𝜃𝑐). 

An increase in the compression ratio parameter results in an overall specific volume 

difference reduction, therefore, stipulating a reduction in engine size. Figure 2.3 shows 

distinctively that the optimal compressor ratio parameter at maximum ECOP is higher 

than that at maximum power density; in addition, the optimal 𝜃𝑐 at maximum power 

density is greater than that at maximum power conditions. Therefore, the overall 

outcome is that: 𝜃𝑐 𝐸𝐶𝑂𝑃 > 𝜃𝑐 𝑀𝑃𝐷 > 𝜃𝑐 𝑀𝑃. It is observed from Figure 2.3, that 

maximum ECOP operating conditions indemnifies favorable thermal efficiency and 

reduced powerplant dimensions at higher compression ratios. Correlations amongst 

the maximums of ECOP, power density and power circumstances were interpolated 

using Figure 2.3. The engine size in terms of normalized specific volume, power and 

thermal efficiency variations with regards to the maximums of ECOP power density 

and power conditions may be interpreted by sketching relevant contours that 

deliberately intersect the admissible lines on the graph. Bear in mind, that ECOL lies 

is on the secondary y-axis (right) and negative in quantity. While, the designated 

operating conditions are still physically applicable to the case(s) in question, it is 

important to address that ECOL can carry both positive or negative values owing to 

the intrinsic characteristic of its formulation. 

 

 
𝑊𝑜𝑟𝑘 (𝑘𝑊) 

 



22 

Figure 2.2: Various optimization functions vs. power at 𝜼𝒕 = 𝜼𝒄 = 𝟎. 𝟗𝟓. 

 

𝑊̅̇ 
 

 

𝑊̇𝑑
̅̅ ̅̅  

 

 

𝜂𝑡ℎ 

 

 

𝑣𝑑̅̅ ̅ 

 

 

𝐸𝐶𝑂𝑃̅̅ ̅̅ ̅̅ ̅̅  

  

𝐸
𝐶

𝑂
𝐿

 

𝜃𝑐  

Figure 2.3: Non-dimentionalized power, power density, specific volume variance, ECOP, 

ECOL and 𝜼𝒕𝒉 for variations of compressor pressure ratio parameter, 𝜽𝒄. 

 

The irreversible Brayton turbojet engine cycle was analyzed and compared with 

regards to the Carnot configuration utilizing the second law of thermodynamics. In 

order to provide a set graphical rendition of the different types of power generations 

or deficits with regards to the assessment equalities, Figure 2.4 is given. The solutions 

in Figure 2.4b were reconstructed to a 100% scale with regards to 𝑊̇𝑟𝑒𝑣 to be 

recognized and correlated with the Carnot configuration. The greatest achievable fuel 

heat discharge rate is denoted by 𝑄̇𝑓𝑢𝑒𝑙. If 𝑄̇𝑓𝑢𝑒𝑙 could be wholely translated into 

power, the topmost curve in Figure 2.4b is attained. Notwithsatnding, the second law 

of thermodynamics restricts that the most comprehensible reversible power that can 

be attained from the Carnot configuration (maximum power) can only be 𝑊̇𝑟𝑒𝑣. The 

heat leak rate, 𝑊̇𝑙𝑘, for the specified Brayton configuration will inflict additional power 

deficits. The achievable power from an ideal Brayton configuration is given as 𝑊̇𝐵𝑟𝑎𝑦; 

in addition the power decrease from 𝑊̇𝑟𝑒𝑣 to 𝑊̇𝐵𝑟𝑎𝑦 is a direct cause of the 

thermodynamic cycle configuration preference. Commensing with the ideal to the real 

Brayton configuration, even more power vanishes from cumulative irreversibilities of 

the respective engine modules (diffuser, compressor, burner, turbine and nozzle) that 

contribute to the powerplant configuration. Thus, the power generated from the actual 

Brayton configuration is prescribed by 𝑊̇. Moreover, the derivable effective power for 

 



23 

a real Brayton configuration is assessed with regards to the attainable specific thrust 

(thrust force multiplied by the flight speed). Therefore, ultimately attaining the 

effective applicable power of a real Brayton configuration which is prescribed as 

𝑊̇𝑢𝑠𝑒𝑓𝑢𝑙. Therefore, out-of the initial starting position of pure carburent heat discharge 

rate the culmination of power deficits are a consequence of powerplant differences and 

inevitably irreversibilities corresponding to physical life powerplants. Even at 

minimum PLOS, approximately 60 percent power reduction is observed; which indeed 

is an elevated quantity of power deficit. Notwithstanding, PLOS may be utilized as an 

approximated performance criterion amonst the ideal Carnot and real Brayton 

configuration, EPLOS, on the other hand, provides an improved functional correlation 

amongst the ideal and real Brayton configuration. For minimum PLOS operation, 

EPLOS is approximately 25 percent, which is precisely associated to intrinsic 

irreversibilities. The contrasts amongst PLOS and EPLOS is CBSF, this is roughly 35 

percent of the total obtainable power and is forfeited due to the thermodynamic 

powerplant configuration of Brayton vs. Carnot. Thus, an effectively advocation for a 

collection of suitable projected powerplant performances and evaluate the effect 

amongst similar powerplant configurations classification via applying the newly 

asscribed optimization functions. 

 

𝑃
𝑜

𝑤
𝑒𝑟

 (
𝑘

𝑊
) 

 

𝑃
𝐿

𝑂
𝑆

 𝑎
𝑛

𝑑
 𝐸

𝑃
𝐿

𝑂
𝑆

 

 
𝜃𝑐 

(a) 

       𝜃𝑐 

       (b) 

Figure 2.4: Manifestations of power losses (a) and their non-dimentionalized forms (b) for 

variations of 𝜽𝒄 

 

 

 



24 

Furthermore, evaluating PLOS for variations of NG and shaft force (Figure 2.5a and 

Figure 2.5b), at reduced NG speeds increased shaft force outputs may be obtained as 

the compressor and turbine efficiency of the powerplant increases. Changes of power 

with regards to PLOS and EPLOS are given in Figure 2.6a and Figure 2.6b for different 

quantities of 𝜂𝑐. Where for maximum power conditions the practical operation range 

for the compression parameter is 𝜃𝑐 = 2 ± 0.4 with an exergy destruction range of 60 

to 90 percent seen from PLOS values. This portrays the amount of destruction against 

the environment, even when improved performance is obtained from a power 

perspective. Moreover, as 𝜂𝑐 nears unity, values of EPLOS at MP conditions decreases 

on the order of 15 percent. Also, EPLOS increases as 𝜃𝑐 increases, in the absence of 

an extrema circumstance. Therefore, it bcomes discernible that regardless if minimum 

PLOS or maximum power generation is targeted, there is an invariable limitation of 

rquired quantity desired.  

Maximum power generated by the cycle transpires at quantities of 𝜃𝑐 = 2.0 ± 0.4 

(Figure 2.7) for varying 𝑀∞ and with a reasonable accomodation on PLOS and 

EPLOS. Minimum PLOS developed for quantities of 𝜃𝑐 in the range of 2.5 and 3, 

which demands increased compression ratios and therefore substantially larger 

compressors. Nonetheless, for ecofriendly deliberation, a minimum PLOS criteria is 

recommended. On the other hand, examining Figure 2.7b, EPLOS quantities are 

reduced at maximum power than at minimum PLOS constraints; suggesting engine 

constituent efficiencies are a source of higher power deficits at minimum PLOS 

operations. 

A consolidation with regards to minimum TSFC and maximum 𝐼𝑎 can be attained at 

minimum PLOS quantities (Figure 2.8). Maximum 𝐼𝑎 quantities traspire when 𝜃𝑐 is 

lower than 2.5; on the other hand, minimum TSFC quantities are obtained when 𝜃𝑐 is 

larger from 3. Where 𝜃𝑐 = 2.5 𝑎𝑛𝑑 3 grants a compressor pressure ratios (𝑃02 𝑃01⁄ ) 

of 25 and 47 respectively. Although, for 𝜃𝑐 ≥ 3 a designer must contemplate material 

impediments in terms of temperature particularly for combustion chambers and the 

turbine blades. 

When the compressor ratio parameter is 𝜃𝑐  ≤ 1.75 (Figure 2.9), and 𝑀∞ is ~1.4 and 

above, PLOS increases, 𝜂𝑝 is still favourable and thrust generated increases. However, 

for 𝜃𝑐 > 1.75 and 𝑀∞ is ~1.4 and below, the reduction in PLOS and increase in thrust 



25 

output arrives for a compromise on 𝜂𝑝. For both thrust and 𝜂𝑝, at 𝑀∞ above 1.4, the 

quantities of PLOS experience a rapid increase; although 𝜂𝑝 is still advantageous, the 

𝜃𝑐 of the cycle must be reduced to attain favourable thrust outputs. Given a prescribed 

compressor and a targeted 𝜃𝑐 design point, the Mach number transforms to an 

operational condition of preference; however, flight operation is favoured within the 

minimum PLOS range where the destruction of exergy is reduced and thus less 

detrimental to the ecology. Consequently, an operational zone chosen in the region of 

1.4 <𝑀∞<1.6 then ecological liabilities (lower PLOS) as well as having the capability 

for higher thrust generation can be fulfilled. The steep increase in PLOS is ascribed to 

the sharp reduction of power generated at increased values of 𝜃𝑐, which subsequently 

truncates the mathematical statement of PLOS to the ratio of 𝑊̇𝑟𝑒𝑣 to 𝑊̇𝑟𝑒𝑣; thus, under 

these conditions PLOS nears unity. In addition, EPLOS encounters this similar 

situation, where a decrease in 𝑊̇ the mathematical definition reduces to the proportion 

of ideal Brayton to ideal Carnot power generation; thus, EPLOS also nears unity.  

 

𝑃
𝐿

𝑂
𝑆

 

 

𝑃
𝐿

𝑂
𝑆

 

 

𝑁𝐺 (𝑅𝑃𝑀) 

(a) 

 𝑆ℎ𝑎𝑓𝑡 𝐹𝑜𝑟𝑐𝑒 (𝑘𝑁) 
(b) 

Figure 2.5: Contours of PLOS for variations of (a) NG and (b) Shaft Force for distinct 

quantities of 𝜼𝒄 = 𝜼𝒕 

 

 

 

  

 
 



26 

 

𝑃
𝑜

𝑤
𝑒𝑟

 (
𝑘

𝑊
) 

 
𝑃

𝐿
𝑂

𝑆
 

𝑃
𝑜

𝑤
𝑒𝑟

 (
𝑘

𝑊
) 

 

𝐸
𝑃

𝐿
𝑂

𝑆
 

𝜃𝑐 

(a) 

𝜃𝑐 

(b) 

Figure 2.6: Contours of power and PLOS (a) and EPLOS for variations of 𝜽𝒄 for dsitinct 

quantities of 𝜼𝒄 = 𝜼𝒕
1  

 

𝑃
𝑜

𝑤
𝑒𝑟

 (
𝑘

𝑊
) 

 

𝑃
𝐿

𝑂
𝑆

 

𝑃
𝑜

𝑤
𝑒𝑟

 (
𝑘

𝑊
) 

 

𝐸
𝑃

𝐿
𝑂

𝑆
 

𝜃𝑐 
(a) 

𝜃𝑐 

(b) 

Figure 2.7: Contours of power and PLOS (a) EPLOS for variaitons of 𝜽𝒄 for distinct 

quantities of 𝑴∞ 

 

                                                 

 
1: Figure 2.6 to Figure 2.9: Dashed lines lie on the secondary y-axis (right) and represent values of 

PLOS and EPLOS only. 

 

 

 
 



27 

𝐼𝑎 

 

𝑃
𝐿

𝑂
𝑆

 

𝑇
𝑆

𝐹
𝐶

 

 

𝑃
𝐿

𝑂
𝑆

 

𝜃𝑐 
(a) 

𝜃𝑐 

(b) 

Figure 2.8: Contours of 𝑰𝒂and PLOS (a) and TSFC (b) 𝜽𝒄 for distinct quantities 𝑴∞ 

 

𝑇
ℎ

𝑟𝑢
𝑠𝑡

 (
𝑘

𝑁
) 

 

𝑃
𝐿

𝑂
𝑆

 

𝜂
𝑝
 

 

𝑃
𝐿

𝑂
𝑆

 

𝑀∞ 

(a) 

𝑀∞ 
(b) 

Figure 2.9: Contours of thrust and PLOS (a) and 𝜼𝒑 (b) for variaitons of 𝑴∞ for distinct 

quantities of 𝜽𝒄 

 

For a PLOS assessment for variations of power and for changes in compressor 

efficiency (Figure 2.10), power deficit decreases and the power output increases as 𝜂𝑐 

increases. It is worthy of noting that for a range of 𝜂𝑐 from 70 to 100 percent, the power 

generated by the powerplant increases two-fold. For an increasing inlet flight Mach 

number, the required power generation from the powerplant must increase (Figure 

2.11). While the power increases at increasing Mach number, quantities of minimum 

PLOS are not seen to vary for the selected operational condition under fixed 

irreversibilities. Moreover, Figure 2.12 illustrates that while the internal conductance 

 

 

 

 



28 

𝜉 for heat leakage drops PLOS also reduces; therefore, the working fluid is efficient in 

transforming the heat produced by the powerplant into accessible power at a minimum 

power deficit. Likewise, for engine operation at higher altitudes (Figure 2.13) both 

PLOS and the power generated by the turbojet decreases. Furthermore, for variations 

in altitude, the minimum quantities of PLOS for variations of power alludes to a linear 

association amongst the variables. Also, for 𝑀∞ below sonic operation (Figure 2.14), 

there is a distictive benifit for the optimums of 𝑇𝑆𝐹𝐶∗, 𝐼𝑎
∗ and 𝑓∗ at maximum power 

and minimum PLOS of the powerplant. 

𝑃
𝐿

𝑂
𝑆

 

 
𝑃𝑜𝑤𝑒𝑟 (𝑘𝑊) 

Figure 2.10: Contours of PLOS for variaitons of power for distinct quantities of 𝜼𝒄 = 𝜼𝒕 
 

𝑃
𝐿

𝑂
𝑆

 

 
𝑃𝑜𝑤𝑒𝑟 (𝑘𝑊) 

Figure 2.11: Contours of PLOS for variaitons of power for distinct quantities of 𝑴∞ 
 

 

 



29 

𝑃
𝐿

𝑂
𝑆

 

 
𝑃𝑜𝑤𝑒𝑟 (𝑘𝑊) 

Figure 2.12: Contours of PLOS for variaitons of power for distinct quantities of heat leakage 

rates. 
 

𝑃
𝐿

𝑂
𝑆

 

 
𝑃𝑜𝑤𝑒𝑟 (𝑘𝑊) 

Figure 2.13: Contours of PLOS for variations of power for distinct quantities of altitudes 
 

𝑇
𝑆

𝐹
𝐶

∗
 

 

𝑓∗ 

 

𝑀∞ 

(a) 

 𝑀∞ 
(b) 

 

 

 

 

 



30 

𝐼𝑎
∗ 

 

𝑀∞ 
(c) 

Figure 2.14: Contours of 𝑻𝑺𝑭𝑪∗ (a), 𝒇∗ (b) and 𝑰𝒂
∗   (c) for variations of 𝑴∞ at maximum 

power and minimum PLOS conditions 

 

In Figure 2.15 through Figure 2.23, the optimum operational criterias of maximum 

power, maximum power density, ECOP and ECOL are evaluated amongst one another 

for selected principle performance variables as well as PLOS, EPLOS and CBSF. 

Figure 2.15 shows the contours of optimum EPLOS quantities with regards to altitude 

and flight Mach number. Maximum power operations indicate the minimal EPLOS 

quantities with regards to the remaining optimization theorems, in addition to 

stipulating a higher resiliance to internal irreversibilities. On the other hand, the 

maximum power conditions decreased thermal efficiencies and increased PLOS 

quantities (Figure 2.16 and Figure 2.17). Thus, maximum power operations are more 

detrimental for the environment, even if they exhibit the most reduced internal 

irreversibility impacts between the examined optimization theorems. Therefore, 

maximum power operation increased specific impulse and thrust (Figure 2.18 and 

Figure 2.19). In addition, maximum power operation have higher TSFC quantities 

(Figure 2.21) while producing the largest amount of power (Figure 2.22) for a 

moderate decrease in compression ratios (Figure 2.20) which results in reduced 

compressor sizes. The non-producible power owing to the configuration of the Carnot 

and Brayton configuration is portrayed in Figure 2.23; and is a comparitive assessment 

of the ideal Brayton configuration to the 100% scale of the Carnot configuration. 

Figure 2.23 indicates that for higher altitudes a reduced amount of power is forfeited, 

thus, the power generated by the Brayton cycle nears that of the Carnot cycle. Whereas, 

 



31 

𝑀∞ is seen to have a insignificant impact with regards to power generation; as a result, 

the forward speed of the aircraft is inconsequential. Depending on the design 

application, various testimonies can be obtained; operation at the maximum ECOL 

criteria renders higher EPLOS and lower PLOS quantities between the remaining 

optimization statements. Despite the impacts of the internal irreversibilities being 

moderately increased, the maximum ECOL criteria confers a more beneficial 

economical and environmental outcome.  

The evaluation of Figure 2.17 reveals that for maximum power and ECOL operations, 

lower PLOS values are achieved at elevated altitudes and develope more beneficial 

circumstances for ecological deliberations; on the other hand 𝑀∞ is seen to have 

marginal impact. However, if it is fundamental to have a larger power generation the 

turbojet must function at reduced altitudes and increased 𝑀∞ when maximum power 

and ECOL operation are used, this is shown in Figure 2.22. Thus, it is up to the 

engineer to determine the design state of preference. Maximum power and ECOL 

operation can be more ecologically favourable (greater altitudes for reduced power 

generation) and alternatively present undesireable conditions (reduced altitudes higher 

𝑀∞ and increased power generation); or perhaps a consolidation amongst the 

operational altitudes and power generations. 

 

 

Figure 2.15: Optimum EPLOS contours for prescribed ALT and 𝑴∞. 

 

0,17

0,19

0,21

0,23

0,25

0,27

0,29

0,31

0,33

0,35

0 5 10 15

E
P

L
O

S

Altitude (km)

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

0,16

0,18

0,20

0,22

0,24

0,26

0,28

0,2 0,7 1,2

E
P

L
O

S

Mach Number

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density



32 

 

Figure 2.16: Optimum 𝜼𝒕𝒉 contours for prescribed ALT and 𝑴∞. 

 

 

Figure 2.17: Optimum PLOS contours for prescribed ALT and 𝑴∞. 

 

 

Figure 2.18: Optimum 𝑰𝒂 contours for prescribed ALT and 𝑴∞. 

 

0,30

0,32

0,34

0,36

0,38

0,40

0,42

0,44

0 5 10 15

T
h

e
r
m

a
l 

E
ff

ic
ie

n
c
y

Altitude (km)

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

0,38

0,39

0,40

0,41

0,42

0,43

0,2 0,7 1,2

T
h

er
m

a
l 

E
ff

ic
ie

n
cy

Mach Number

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

0,54

0,56

0,58

0,60

0,62

0,64

0,66

0,68

0 5 10 15

P
L

O
S

Altitude (km)

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

0,53

0,54

0,55

0,56

0,57

0,58

0,59

0,2 0,7 1,2

P
L

O
S

Mach Number

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

48

53

58

63

68

73

0 5 10 15

S
p

e
c
if

ic
 I

m
p

u
ls

e

Altitude (km)

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

50

55

60

65

70

75

80

0,2 0,7 1,2

S
p

ec
if

ic
 I

m
p

u
ls

e

Mach Number

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density



33 

 

Figure 2.19: Optimum Thrust contours for prescribed ALT and 𝑴∞ 

 

 

Figure 2.20: Optimum 𝜽𝒄 contours for prescribed ALT and 𝑴∞. 

 

 

Figure 2.21: Optimum TSFC contours for prescribed ALT and 𝑴∞. 

 

18

28

38

48

58

68

0 5 10 15

T
H

R
U

S
T

 (
k

N
)

Altitude (km)

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

25

27

29

31

33

35

37

39

41

0,2 0,7 1,2

T
H

R
U

S
T

 (
k

N
)

Mach Number

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

1,8

1,9

2,0

2,1

2,2

2,3

2,4

2,5

2,6

2,7

2,8

2,9

3,0

0 5 10 15

𝜃
𝑐

Altitude (km)

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

1,8

1,9
2,0

2,1
2,2

2,3

2,4
2,5

2,6
2,7

2,8

2,9
3,0

3,1

0,2 0,7 1,2

𝜃
𝑐

Mach Number

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

0,000024

0,000026

0,000028

0,000030

0,000032

0,000034

0,000036

0 5 10 15

T
S

F
C

Altitude (km)

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

0,000022

0,000024

0,000026

0,000028

0,000030

0,000032

0,000034

0,000036

0,000038

0,2 0,7 1,2

T
S

F
C

Mach Number

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density



34 

 

Figure 2.22: Optimum power contours for prescribed ALT and 𝑴∞. 

 

 

Figure 2.23: Optimum CBSF contours for prescribed ALT and 𝑴∞. 

 

The performance assessment, at flight conditions, of the irreversible Brayton turbojet 

engine was carried out utilizing power and power density approach in addition to 

ECOP and ECOL, selected variables were held constant while others varied. The 

ambient condition prescribed for the powerplant were at an initial: altitude of 10 km, 

inlet temperature of 𝑇𝑎 = 223.3 𝐾, inlet pressure of 𝑃𝑎 = 26.5 𝑘𝑃𝑎 and an inlet speed 

of sound of 𝑎 = 299.5 𝑚/𝑠. The given predetermined variables were: 𝑀∞ = 0.5, 𝜂𝑐 =

0.87, 𝜂𝑡 = 0.90, 𝜂𝑖 = 0.93, 𝜂𝑗 = 0.95, 𝜂𝑚 = 0.99, 𝜂𝑏 = 0.98, 𝑄𝑅 = 43000 𝑘𝐽/𝑘𝑔 

and a pressure drop from inlet to outlet of the combustor given as 4% (i.e. ΔPb = 0.04). 

Variations of certain criteria such as 𝑀∞, are prescribed in the applicable graphic 

legends. 

8800

10800

12800

14800

16800

18800

20800

22800

24800

26800

28800

30800

0 5 10 15

P
o

w
e
r
 (

k
W

)

Altitude (km)

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

10800

12800

14800

16800

18800

20800

22800

24800

0,2 0,7 1,2

P
o

w
er

 (
k

W
)

Mach Number

MAX ECOL

MAX ECOP

MAX Power

MAX Power Density

0,26

0,28

0,30

0,32

0,34

0,36

0,38

0,40

0,42

0,44

0,46

0,48

0 5 10 15

C
B

S
F

Altitude (km)

MAX ECOL
MAX ECOP
MAX Power
MAX Power Density

0,26

0,28

0,30

0,32

0,34

0,36

0,38

0,40

0,42

0,2 0,7 1,2

C
B

S
F

Mach Number

MAX ECOL
MAX ECOP
MAX Power
MAX Power Density



35 

The contours of air specific impulse, 𝐼𝑎 and thermal efficiency, 𝜂𝑡ℎ wit