Powered Flight

David R. Greatrix

Powered Flight

The Engineering of Aerospace Propulsion

123

David R. GreatrixDepartment of Aerospace EngineeringRyerson UniversityVictoria Street 350Toronto, ON M5B 2K3Canada

ISBN 978-1-4471-2484-9 ISBN 978-1-4471-2485-6 (eBook)DOI 10.1007/978-1-4471-2485-6Springer London Heidelberg New York Dordrecht

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Preface

This book documents the engineering behind most of today’s aerospace propulsionsystems. Some of the material coverage in this book for a number of relevantpropulsion topics is more detailed, in the context of a university or technicalcollege course textbook at the introductory or advanced level, so that the readergets some appreciation of the mathematics (as well as the physics and history)involved in the science and engineering of propulsion. The wider coverage oftopics in this book gives more context to the reader, and allows one to observesimilarities and differences between systems, observations that otherwise wouldnot be possible in a book dedicated to only one, or a few, propulsion systems. Thebook has been divided into two main parts, mostly based on the historical sepa-ration of the two well-established general categories: airplane propulsion androcket propulsion. The roots of this book arise from several undergraduate andgraduate aerospace engineering courses taught by the author at Ryerson Universityover the years, including Aerospace Propulsion and Rocket Propulsion, but alsorelevant elements of courses like Flight Mechanics, Aircraft Performance and GasDynamics that help bring additional background.

In addition to teaching, the author conducts research in the areas of solid andhybrid rocket propulsion, and the flight dynamics of vehicles. It is hoped that thisresearch activity brings further background and insight in to this book. Forexample, in terms of historical context, the author can certainly understand thedemands faced, and the need for persistence required, by researchers past andpresent, who struggle forward in the face of underfunding and lack of interest fromthe surrounding technical community.

Prior to joining Ryerson University in 1993, the author worked in the Canadianaerospace industry for a number of years, as well as a few years at a Canadiangovernment laboratory. The author is presently an international member of theAIAA Solid Rockets Technical Committee, and over the years has been a frequentpresenter of technical papers at various propulsion-related conferences in NorthAmerica and Europe.

v

Acknowledgments

Assistance in the writing and publication of this book was provided by a number ofpeople, and I wish to express my gratitude for their help. I wish to express mythanks to the external reviewers who took the time to read the manuscript, andprovide constructive comments. I would like to acknowledge the various repre-sentatives of companies, technical/educational organizations, academic institutionsand government agencies who arranged for the usage of material helpful towardsunderstanding various topics under discussion in this literary effort. In terms ofbackground preparation for bringing forth the material presented in this book, I amindebted to my professional colleagues past and present, close and far away, and tomy undergraduate and graduate students past and present, who directly or indi-rectly have helped to raise my knowledge level from year to year to a point suchthat a comprehensive book on aerospace propulsion could ultimately be produced.In addition, some have laid the seeds of motivation that helped inspire the authorto initiate and proceed to complete this venture. Whether intentional or not, theyhave all contributed to the realization of this book, and these individuals have mythanks.

Colleagues who in different ways have been helpful in support of this literaryeffort include my current Ryerson University Propulsion Research Facility col-leagues, Jerry Karpynczyk and Peter Bradley, and from the past and presentAerospace Engineering program at Ryerson University, Profs. Cruchley, Downer,Mölder, McTavish, Liu, Kumar and Okouneva, and from the Mechanical Engi-neering program at Ryerson, Prof. Kawall. My thanks for the more recent assis-tance of Prof. Brian Cantwell of Stanford University, Prof. Frank Lu of theUniversity of Texas at Arlington, and Prof. William Roberts of North CarolinaState University, in graciously providing input to the present effort.

Professors who were instrumental in laying the earlier groundwork for theauthor’s subsequent efforts include Prof. Jeffrey Tinkler of the University ofManitoba, and Prof. James Gottlieb of the University of Toronto. The exchange ofeducational information over the years on gas turbine and other engines by mylong-time friend at GE Aviation, Greg Johnson, is acknowledged here with thanks.My father, John Greatrix, a long-time Royal Canadian Air Force fighter pilot and

vii

Transport Canada aviation inspector, has over the years provided useful infor-mation about aircraft, aircraft propulsion and aircraft flight operations, and I amhappy to acknowledge his underlying contribution to the present effort, and hislongstanding support. Finally, for her longstanding general support, I shouldmention Mary Carmel Roberta (Fraser) Greatrix… thanks, Mom.

viii Acknowledgments

Contents

1 Introduction to Aerospace Propulsion . . . . . . . . . . . . . . . . . . . . . 11.1 First Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 ‘‘Seemed Like a Good Idea at the Time’’ . . . . . . . . . . . . . . 31.3 From Design to Certification . . . . . . . . . . . . . . . . . . . . . . . 61.4 Integration of the Propulsion System

to the Flight Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Engine-Airframe Integration for Airplanes . . . . . . . . 101.4.2 Motor or Engine Integration into Rocket Vehicle . . . 131.4.3 Thruster-Spacecraft System Integration . . . . . . . . . . 17

1.5 Review of Gasdynamics and Thermodynamics . . . . . . . . . . . 191.6 Closing Comments for this Chapter . . . . . . . . . . . . . . . . . . 241.7 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.8 Solutions to Example Problems . . . . . . . . . . . . . . . . . . . . . 24References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Part I Airplane Propulsion

2 Introduction to Atmospheric Flight . . . . . . . . . . . . . . . . . . . . . . . 292.1 Introduction to Propulsion for Airplanes . . . . . . . . . . . . . . . 292.2 Example Mission Requirement: Range and Endurance

of Fixed-Wing Airplanes . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.1 Jet Aircraft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.2 Propellered Aircraft. . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Example Mission Requirement: Takeoffof Fixed-Wing Airplanes . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.1 Introduction to Takeoff Performance. . . . . . . . . . . . 382.3.2 Takeoff Analysis: Ground Roll. . . . . . . . . . . . . . . . 402.3.3 Takeoff Analysis: Climb Segment #1 . . . . . . . . . . . 432.3.4 Approximation Model for Ground Roll . . . . . . . . . . 45

ix

2.3.5 Approximation Model for Climb Segment #1. . . . . . 472.3.6 Approximated Overall Takeoff Distance . . . . . . . . . 49

2.4 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.5 Solution for Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3 The Propeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Actuator Disk Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 Momentum-Blade Element Theory . . . . . . . . . . . . . . . . . . . 693.4 Propeller Propulsive Efficiency . . . . . . . . . . . . . . . . . . . . . 763.5 Compressibility Tip Loss . . . . . . . . . . . . . . . . . . . . . . . . . . 773.6 Activity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.7 Blade Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.8 Note on Helicopter Rotors . . . . . . . . . . . . . . . . . . . . . . . . . 793.9 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.10 Solution for Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4 Internal Combustion Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Spark-Ignition Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.3 Compression-Ignition Engines . . . . . . . . . . . . . . . . . . . . . . 1064.4 Rotary Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.5 Turbosupercharging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.6 UAVs: A Realm for Innovation

and Alternative Technologies . . . . . . . . . . . . . . . . . . . . . . . 1144.7 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.8 Solutions to Example Problems . . . . . . . . . . . . . . . . . . . . . 119References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5 Pulsejet Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.2 Performance Considerations . . . . . . . . . . . . . . . . . . . . . . . . 1265.3 Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.4 Related Propulsion Technology . . . . . . . . . . . . . . . . . . . . . 136

5.4.1 Wave Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.4.2 Pulse Detonation Engine . . . . . . . . . . . . . . . . . . . . 138

5.5 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.6 Solutions to Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6 Gas Turbine Engines: Fundamentals . . . . . . . . . . . . . . . . . . . . . . 1476.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.2 An Introduction to Turbojet Engines . . . . . . . . . . . . . . . . . . 149

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6.3 Cycle Analysis of Turbojet Engine . . . . . . . . . . . . . . . . . . . 1526.4 Intake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.4.1 Subsonic Intake . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.4.2 Supersonic Intake . . . . . . . . . . . . . . . . . . . . . . . . . 1666.4.3 Supersonic Intake, Pitot Type. . . . . . . . . . . . . . . . . 1666.4.4 Supersonic Intake, Oblique Shock Type . . . . . . . . . 167

6.5 Centrifugal Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.6 Axial Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.7 Combustor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.8 Axial Turbine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.9 Afterburner for Jet Engines . . . . . . . . . . . . . . . . . . . . . . . . 2056.10 Exhaust Nozzle for Gas Turbine Engines. . . . . . . . . . . . . . . 2096.11 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2156.12 Solution to Example Problems . . . . . . . . . . . . . . . . . . . . . . 217References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

7 Turbofan Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.2 Cycle Analysis of Separate-Stream Turbofan Engine. . . . . . . 2387.3 Cycle Analysis of Mixed-Stream Turbofan Engine . . . . . . . . 2457.4 On-Design Versus Off-Design Cycle Analysis . . . . . . . . . . . 2477.5 Engine Health Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . 2487.6 Environmental Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

7.6.1 Environmental Issues: Noise . . . . . . . . . . . . . . . . . 2507.6.2 Environmental Issues: Air Pollution . . . . . . . . . . . . 2517.6.3 Environmental Issues: Dwindling Jet Fuel Supply. . . 252

7.7 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2547.8 Solutions to Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

8 Turboprop and Turboshaft Engines . . . . . . . . . . . . . . . . . . . . . . 2698.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2698.2 Cycle Analysis of Conventional

Free-Turbine Turboprop Engine . . . . . . . . . . . . . . . . . . . . . 2728.3 Installation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2778.4 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2788.5 Solutions to Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Part II Rocket Propulsion

9 Introduction to Space Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2939.1 Introduction to Propulsion for Rocket and Space Vehicles . . . 293

Contents xi

9.2 Mission Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2949.3 Launch Vehicle Trajectory to Orbit. . . . . . . . . . . . . . . . . . . 2989.4 Gasdynamics and Thermodynamics of Internal Flow. . . . . . . 3019.5 Rocket Nozzle Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3039.6 Combustion Considerations for Chemical Rockets . . . . . . . . 3159.7 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3179.8 Solutions to Example Problems . . . . . . . . . . . . . . . . . . . . . 318References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

10 Solid-Propellant Rocket Motors . . . . . . . . . . . . . . . . . . . . . . . . . 32310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32310.2 Performance Considerations . . . . . . . . . . . . . . . . . . . . . . . . 32410.3 Solid Propellant Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 33010.4 Solid-Propellant Burning Rate Models. . . . . . . . . . . . . . . . . 33210.5 Erosive Burning in SRMs . . . . . . . . . . . . . . . . . . . . . . . . . 33510.6 Acceleration Effects on Burning . . . . . . . . . . . . . . . . . . . . . 33910.7 Internal Ballistic Analyses for Steady

and Nonsteady Operation of SRMs . . . . . . . . . . . . . . . . . . . 34210.8 Finite-Difference Model for Quasi-Steady Internal

Ballistic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34610.9 Simpler Nonsteady Internal Ballistic Cases . . . . . . . . . . . . . 35010.10 Transient Burning of Solid Propellants . . . . . . . . . . . . . . . . 35110.11 Axial and Transverse Combustion Instability of SRMs . . . . . 35510.12 Structural Issues for SRMs. . . . . . . . . . . . . . . . . . . . . . . . . 35810.13 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36410.14 Solutions to Example Problems . . . . . . . . . . . . . . . . . . . . . 367References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

11 Liquid-Propellant Rocket Engines . . . . . . . . . . . . . . . . . . . . . . . . 38111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38111.2 Propellant Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38111.3 Propellant Storage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38411.4 Propellant Feed System . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

11.4.1 Pressure-Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38911.4.2 Turbopump-Feed . . . . . . . . . . . . . . . . . . . . . . . . . 390

11.5 Injectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39811.6 Thrust Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40111.7 Combustion Instability in LREs . . . . . . . . . . . . . . . . . . . . . 40511.8 Note on Thrust Vector Control . . . . . . . . . . . . . . . . . . . . . . 40611.9 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40911.10 Solutions to Example Problem . . . . . . . . . . . . . . . . . . . . . . 411References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

xii Contents

12 Hybrid Rocket Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41712.2 Performance Considerations . . . . . . . . . . . . . . . . . . . . . . . . 41912.3 Fuel Regression Rate in HREs . . . . . . . . . . . . . . . . . . . . . . 42112.4 Internal Ballistics of HREs . . . . . . . . . . . . . . . . . . . . . . . . 42312.5 Combustion Instability in HREs . . . . . . . . . . . . . . . . . . . . . 42712.6 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42812.7 Solutions to Example Problems . . . . . . . . . . . . . . . . . . . . . 429References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

13 Air-Breathing Rocket Engines. . . . . . . . . . . . . . . . . . . . . . . . . . . 43513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43513.2 Performance Considerations . . . . . . . . . . . . . . . . . . . . . . . . 43713.3 Combined-Cycle Engine Technology . . . . . . . . . . . . . . . . . 44013.4 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44813.5 Solutions to Example Problems . . . . . . . . . . . . . . . . . . . . . 449References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

14 Propulsion in Space: Beyond the Chemical Rocket . . . . . . . . . . . 45714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45714.2 Electric Propulsion for Spaceflight Applications . . . . . . . . . . 45714.3 Solar/Thermal Propulsion for Spaceflight. . . . . . . . . . . . . . . 46614.4 Nuclear/Thermal Propulsion for Spaceflight . . . . . . . . . . . . . 46814.5 Unconventional Rocket Engine Concepts. . . . . . . . . . . . . . . 47114.6 Non-Rocket Concepts for Advanced Aerospace

Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47514.7 Stratospheric Balloons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47814.8 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48014.9 Solutions to Example Problems . . . . . . . . . . . . . . . . . . . . . 482References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

15 Erratum to: Turboprop and Turboshaft Engines . . . . . . . . . . . . E1

16 Errata to: Powered Flight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . E3

Contents xiii

Appendix I: ICAO Standard Atmosphere. . . . . . . . . . . . . . . . . . . . . . 487

Appendix II: Modeling the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . 489

Appendix III: Unit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Appendix IV: Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

xiv Contents

Symbols

A Local core flow cross-sectional area, m2

Ac Mean chamber cross-sectional area, m2

Ae Nozzle exit plane cross-sectional area, m2

Aef Fan nozzle exit plane cross-sectional area, m2

Ainj Injector orifice cross-sectional area, m2

Aline Propellant line cross-sectional area, m2

Ap Port cross-sectional area, m2

At Nozzle throat cross-sectional area, m2

Aw Wetted area, m2

A1 Propeller disk area, m2

a Gas sound speed, m/s; also, coefficient, mass-flux-dependent burningrate, m/s-(kg/s m2)n

a‘ Longitudinal (or lateral) acceleration, m/s2

an Normal acceleration, m/s2

B Number of propeller blades; magnetic field strength, T (teslas)B Bypass ratio [TF]b Nonequilibrium two-phase sound speed, m/s; also, wing span, mC De St. Robert coefficient, m/s-Pan

CD Drag coefficientCDi Lift-induced drag coefficientCDo Zero-lift drag coefficientCd,inj Injector orifice flow discharge coefficientCF Thrust coefficient [rocket engine]CF,v Vacuum thrust coefficientCh Stanton numberCL Lift coefficient

xv

CL,me Lift coefficient for maximum enduranceCL,mr Lift coefficient for maximum air rangeCm Particle or droplet specific heat, J/(kg K)CP Propeller power coefficientCp Constant-pressure specific heat, gas phase, J/(kg K)CS Speed coefficient [propeller]Cs Specific heat, solid phase, J/(kg K)CT Propeller thrust coefficientCv Constant-volume specific heat, gas phase, J/(kg K)c Effective exhaust velocity, m/s; speed of light, m/s; airfoil chord, mc* Characteristic exhaust velocity, m/sD Aerodynamic drag, N; diameter, mDH Hydraulic diameter, mDN Bearing speed capability index, [(bearing bore) diameter, mm] 9 [speed

(of shaft rotation), rpm)]d Hydraulic diameter of core flow, m; propeller diameter, mde Nozzle exit diameter, mdm Particle mean diameter, mdp Propeller diameter, m; port diameter [SRM]dt Nozzle throat diameter, mE Total gas specific energy, J/kg; electric field strength, N/C or V/mEp Total particle specific energy, J/kge Oswald efficiency factores Surface ablation rate, m/sF Thrust, N; also, Fnet, Net thrustFgross Gross thrust, N (before effect of air-intake momentum drag included)Finstalled Installed thrust, N (net thrust produced from engine in place on aircraft)FQ Force contributing to torque on propeller or rotor shaft, NFnet Net thrust (including momentum drag effect), NFo Static thrust, Nf Darcy-Weisbach friction factor; frequency, Hz; fuel-to-air ratiof* Zero-transpiration friction factorfAB Fuel-to-gas ratio, afterburnerfact Actual fuel-air ratiofcyc Frequency of operational cycle, Hzflim Limit friction factor for negative erosive burningfstoich Stoichiometric fuel-air ratiof1L Fundamental [first] axial resonant [harmonic] acoustic frequency for

chamber of length L, Hzf1R Fundamental [first] radial resonant acoustic frequency for chamber of

radius R, Hz

xvi Symbols

f1T Fundamental [first] tangential resonant acoustic frequency for chamberof width D, Hz

G Axial mass flux, kg/(m2 s)Ga Accelerative mass flux, kg/(m2 s)g Gravitational acceleration, m/s2

go Reference [sea-level] gravitational acceleration, m/s2

H Headrise, m [turbopump]DHs Net surface heat release, J/kgh Effective convective heat transfer coefficient, W/(m2 K); altitude, m;

enthalpy, J/kg; wall thickness, mh* Zero-transpiration convective heat transfer coefficient, W/(m2 K)hAGL Altitude above ground level, mhc Convective heat transfer coefficient, W/(m2 K)hG Geometric altitude, mhp Pressure altitude, mhsc Screen [obstacle clearance] height, takeoff and landings, mhw, g Mean height of wing above ground, in ground roll, mDh Difference in pressure head, mI Electric current, A (amperes or C/s)Id Density specific impulse [average liquid or solid propellant specific

gravity 9 Isp], s; also, Iq [average propellant density 9 Isp], s�kg/mIsp Specific impulse, sItot Total impulse, N�sJ Propeller advance ratioj Electric current density, A/m2

K Lateral/longitudinal acceleration burning rate displacement orientationangle coefficient; lift-induced drag factor

Kb Limiting coefficient on transient burning rate, s-1

Kd Shear layer coefficient, m-1

k Gas thermal conductivity, W/(m K)kz Thermal conductivity, solid phase, W/(m K)L Aerodynamic lift, N; length, mL* Characteristic length, mLc Axial length of chamber, m‘c/p Effective length of PJ combustion tube [combustor + tailpipe], m‘f Length of HRE solid fuel grain, m‘p Length of SRM propellant grain, m; also Lp

Average molecular mass of gas, amu; also m

Mac Pitching moment about the aerodynamic centreMa Mach numberMa1 Mach number of flow upstream of oblique shock

Symbols xvii

Ma2 Mach number of flow downstream of oblique shockMa? Vehicle flight Mach numberMarel Relative flow Mach numberm Vehicle mass, kg; unit mass of gas, kg_ma Air mass flow [engine core intake], kg/s_mby Air mass flow through fan [TF, bypass flow], kg/s_mf Fuel mass flow, kg/s_mo Oxidizer mass flow, kg/smp Mean mass of particle or droplet, kg; mass of propellant, kgmpp Dry mass of EP powerplant, kgN Engine shaft speed, rpmNs Pump specific speed, rpm�(‘/min)1/2/m3/4

N1 Low-pressure shaft speed, rpm; also, NLP

N2 High-pressure shaft speed, rpm; also, NHP

Nu Nusselt numbern Propeller shaft rotational speed, revolutions/sec; burning rate exponent

[SRM, HRE]P Ideal power, WPinp Input electric power, WPjet Kinetic power of jet exhaust [EP], WPS Shaft power, WPSo Static shaft power, WPr Gas Prandtl numberp Local gas static pressure, Papc Chamber pressure, Pape Nozzle exit static pressure, Pap? Outside ambient air pressure, PaDpinj Injector pressure drop, PaQ Torque, N m; volumetric flow rate, m3/sQs Net near-surface heat release, J/kgq Dynamic pressure, Pa; electric charge magnitude, C (+/-, coulombs)qR Heat of reaction, J/kg, typically lower heating valueqrad Heat of radiation, J/kgR Specific gas constant, J/(kg K); propeller or rotor tip radius, m; air-

referenced range, m�Rc Degree of reaction, compressor�Rt Degree of reaction, turbine< Universal gas constant, J/(kg K)RE Mean earth radius, mRg Ground-referenced range (distance), mRed Local gas Reynolds number based on core hydraulic diameter

xviii Symbols

Rep Relative flow Reynolds number about particler Radial distance, m; oxidizer-to-fuel mixture ratio; compression ratiora Acceleration-dependent burning rate, m/srb Overall burning rate, m/srb,qs Quasi-steady burning rate, m/sre Erosive burning rate positive component, m/sro Base burning rate, m/srp Pressure-dependent burning rate, m/srst Stoichiometric mixture ratioru Velocity-dependent burning rate, m/srv Volumetric compression ratioS Wing reference area, m2; entropy, J/K; cross-sectional area of missile

body, m2

Ss Pump suction specific speed, rpm�(‘/min)1/2/m3/4

St Strouhal number [combustion]s Distance, m; entropy per unit mass, J/(kg K)sTO Takeoff distance, mT Local gas static temperature, KTas Auto-ignition temperature, KTds Decomposition gas temperature resulting from ablation, KTf Flame temperature, KTi Initial temperature, solid phase, KTo Stagnation temperature, KTp Particle temperature, KTs Burning surface temperature, KT? Outside ambient air temperature, K; local central core gas temperature,

Kt Time, s; airfoil thickness, mtB Burning [motor firing] time, stc Residence (stay) time for combustion, sDtc Effective combustion time period for one operational cycle, sDtcyc Time period of operational cycle, sU Local rotor tangential velocity, m/sUt Rotor tip speed, m/su Local axial gas velocity, m/s; internal energy, J/kgue Nozzle exit gas or exhaust jet velocity, m/s; also Ve

up Local axial particle velocity, m/su? Bulk axial gas velocity, m/sV Velocity, m/s; true airspeed, m/s; voltage, V (volts)DV Incremental vehicle velocity gain from motor thrust, m/sVE Overall resultant velocity, blade, m/s

Symbols xix

Ve,? Fully-expanded exhaust exit velocity, m/sVef,? Fully-expanded fan duct exhaust exit velocity, m/sVg Ground speed, m/sVR Resultant velocity [forward flight + rotation], blade, m/sVw Wind speed, m/sV? True airspeed of flight vehicle, m/s�V Volume of element, m3

�Vc Volume of chamber, m3

�Vp Volume of propellant, m3

�Vt Overall volume of tank, m3

�Vu Useable volume of tank, m3

D�Vmax Auxiliary-usage volume of propellant, m3

D�Vb Boil-off volume of propellant, m3

D�Vc Cooling bleed-off volume of propellant, m3

D�Vt Trapped (residual) volume of propellant, m3

D�Vu Ullage volume of propellant, m3

v Gas velocity, m/s; specific volume, m3/kgvf Normal gas flow velocity of flame, m/svinj Hydraulic injection velocity, m/sW Vehicle weight, NWc Work by compressor on air, JWE Empty vehicle weight, N; dry engine weight, NWF Fuel weight, NWO Initial vehicle weight, NWPL Payload weight, NWt Work by gas on turbine, Jw Induced velocity, m/s; swirl velocity, m/sx Axial distance, my Distance from propellant surface, ma Angle of attack, deg; Lenoir-Robillard model coefficientag Gas phase void fractionai Induced angle of attack, radap Particle mass loading fraction [SRM]as Thermal diffusivity, solid phase, m2/s; specific power, W/kg

[EP systems]b Propeller or rotor blade pitch angle, deg; angle of sideslip [aircraft],

deg; Lenoir-Robillard model coefficientbg Particle-to-gas mass flow ratiobref Reference propeller or rotor blade pitch angle at a specified radial

position, degC Dihedral angle (wing, blade), degc Ratio of specific heats of gas; flight path angle, degree or rad

xx Symbols

ca Ratio of specific heats of aird Atmospheric air pressure ratio wedge angle, degdmax Wedge angle at point of shock detachment, degdo Reference energy zone thickness, mdr Resultant energy zone thickness, me Effective propellant surface roughness height, m; emissivity coefficient

[radiation]; impeller slip factorec Combustion efficiency [HRE, LRE]f payload mass fraction [rocket vehicle]gb Burner [combustor] adiabatic component efficiencygc Compressor adiabatic component efficiencygd Diffuser [intake] adiabatic component efficiencygf Fan adiabatic component efficiencygfn Fan nozzle adiabatic component efficiencygm Turbomachinery adiabatic component efficiencygn Core nozzle adiabatic component efficiencygo Overall efficiencygp Jet propulsive efficiencygpr Propeller propulsive efficiencygt Turbine adiabatic component efficiency; thruster efficiency [EP systems]gth Thermal efficiencyh Atmospheric air temperature ratio pitch elevation angle, degree oblique

shock angle, deghR Runway upslope angle, deghr Resultant angle of stretched energy zone, degj Dilatation term, s-1

K Sweep angle (wing, blade), degk Nondimensional velocity ratio [propeller]; wavelength [EM], ml Absolute [dynamic] gas viscosity, kg/(m s); coefficient of rolling friction;

main rotor advance ratio [helicopter]t Kinematic gas viscosity, m2/sn Damping ratiopb Burner [combustor] stagnation pressure ratiopc Compressor stagnation pressure ratiopd Diffuser [intake] stagnation pressure ratiopf Fan stagnation pressure ratio [TF]q Local gas density, kg/m3

qm Density, particle or droplet material, kg/m3

qp Density, particles within gas flow volume, kg/m3

qq Net electric charge density, C/m3

qs Density, solid phase, kg/m3

r Atmospheric air density ratio; propeller or rotor solidity

Symbols xxi

rp Pressure-dependent burning-rate temperature sensitivity, K-1

ss Surface fluid shear stress, Pa/ Acceleration vector orientation angle, deg; ground effect factor; angle

of resultant velocity from plane of blade rotation, deg; equivalence ratio[combustion]; flow coefficient [turbine]; roll angle, deg [vehicle]

/d Acceleration vector displacement orientation angle, degw Rotor blade azimuth angle, deg [helicopter]; turbine stage loading

coefficientx Angular frequency, rad/sxn Natural frequency, rad/sxr Resonant frequency, rad/s

xxii Symbols