Theory of Aerospace Propulsion

By Pasquale M. Sforza

Theory of Aerospace Propulsion, Second Edition, teaches engineering students how to utilize the fundamental principles of fluid mechanics and thermodynamics to analyze aircraft engines, understand the common gas turbine aircraft propulsion systems, be able to determine the applicability of each, perform system studies of aircraft engine systems for specified flight conditions and preliminary aerothermal design of turbomachinery components, and conceive, analyze, and optimize competing preliminary designs for conventional and unconventional missions. This updated edition has been fully revised, with new content, new examples and problems, and improved illustrations to better facilitate learning of key concepts.

• Includes broader coverage than that found in most other books, including coverage of propellers, nuclear rockets, and space propulsion to allows analysis and design of more types of propulsion systems
• Provides in-depth, quantitative treatments of the components of jet propulsion engines, including the tools for evaluation and component matching for optimal system performance
• Contains additional worked examples and progressively challenging end-of- chapter exercises that provide practice for analysis, preliminary design, and systems integration

• Language: English
• Publisher: Elsevier Science
• Release date: Aug 13, 2016
• ISBN: 9780128096017

Pasquale M. Sforza

Pasquale Sforza received his PhD from the Polytechnic Institute of Brooklyn in 1965. He has taught courses related to commercial airplane design at the Polytechnic Institute of Brooklyn and the University of Florida. His research interests include propulsion, gas dynamics, and air and space vehicle design. Dr. Sforza has also acted as Co-Editor of the Journal of Directed Energy and Book Review Editor for the AIAA Journal. His previous books include Theory of Aerospace Propulsion (Butterworth-Heinemann, 2011) and Commercial Airplane Design Principles, (Butterworth-Heinemann, 2014)

Book preview

9780128096017_FC

Theory of Aerospace Propulsion

Second Edition

Pasquale M. Sforza

University of Florida

Table of Contents

Cover image

Title page

Copyright

Preface to the Second Edition

Chapter 1: Propulsion Principles and Engine Classification

Abstract

1.1 Introduction to Aerospace Propulsion Engines

1.2 Conservation Equations

1.3 Flow Machines With No Heat Addition: Propellers, Fans, Compressors, and Turbines

1.4 Flow Machines With No Net Power Addition: Turbojets, Ramjets, Scramjets, and Pulsejets

1.5 Flow Machines With P = 0, Q = Constant and A0 = 0: The Rocket

1.6 The Special Case of Combined Heat and Power: The Turbofan

1.7 Aerospace Propulsion Fuels

1.8 Space Propulsion Engines

1.9 The Force Field for Airbreathing Engines

1.10 Summary

1.11 Useful Constants and Conversion Factors

1.12 Nomenclature

1.13 Exercises

Chapter 2: Quasi-One-Dimensional Flow Equations

Abstract

2.1 Introduction to the Flow Equations

2.2 Equation of State

2.3 Speed of Sound

2.4 Mach Number

2.5 Conservation of Mass

2.6 Conservation of Energy

2.7 Conservation of Species

2.8 Conservation of Momentum

2.9 The Impulse Function

2.10 The Stagnation Pressure

2.11 The Equations of Motion in Standard Form

2.12 Summary

2.13 Nomenclature

2.14 Exercises

Chapter 3: Idealized Cycle Analysis of Jet Propulsion Engines

Abstract

3.1 Introduction to Engine Cycle Analysis

3.2 General Jet Engine Cycle

3.3 Ideal Jet Engine Cycle Analysis

3.4 Ideal Turbojet in Maximum Power Takeoff

3.5 Ideal Turbojet in High Subsonic Cruise in the Stratosphere

3.6 Ideal Turbojet in Supersonic Cruise in the Stratosphere

3.7 Ideal Ramjet in High Supersonic Cruise in the Stratosphere

3.8 Ideal Turbofan in Maximum Power Takeoff

3.9 Ideal Turbofan in High Subsonic Cruise in the Stratosphere

3.10 Ideal Internal Turbofan in Supersonic Cruise in the Stratosphere

3.11 Ideal Scramjet in Hypersonic Cruise in the Stratosphere

3.12 Real Engine Operations

3.13 Summary

3.14 Nomenclature

3.15 Exercises

Chapter 4: Combustion Chambers for Airbreathing Engines

Abstract

4.1 Introduction to Combustion Chambers

4.2 Combustion Chamber Attributes

4.3 Modeling the Chemical Energy Release

4.4 Constant Area Combustors

4.5 Constant Pressure Combustors

4.6 Fuels for Airbreathing Engines

4.7 Combustor Efficiency

4.8 Combustor Configuration

4.9 Supersonic Combustion

4.10 Criteria for Equilibrium in Chemical Reactions

4.11 Calculation of Equilibrium Compositions

4.12 Adiabatic Flame Temperature

4.13 Summary

4.14 Nomenclature

4.15 Exercises

Chapter 5: Nozzles for Airbreathing Engines

Abstract

5.1 Introduction to Nozzles

5.2 Nozzle Characteristics and Simplifying Assumptions

5.3 Nozzle Flows With Simple Area Change

5.4 Mass Flow in an Isentropic Nozzle

5.5 Nozzle Operation

5.6 Normal Shock Inside the Nozzle

5.7 Two-Dimensional Considerations in Nozzle Flows

5.8 Conditions for Maximum Thrust

5.9 Afterburning for Increased Thrust

5.10 Nozzle Configurations

5.11 Nozzle Performance

5.12 Summary

5.13 Nomenclature

5.14 Exercises

Chapter 6: Inlets for Airbreathing Engines

Abstract

6.1 Introduction to Inlets

6.2 Inlet Operation

6.3 Inlet Mass Flow Performance

6.4 Inlet Pressure Performance

6.5 Inlets in Subsonic Flight

6.6 Normal Shock Inlets in Supersonic Flight

6.7 Internal Compression Inlets

6.8 Internal Compression Inlet Operation

6.9 Additive Drag

6.10 External Compression Inlets

6.11 Mixed Compression Inlets

6.12 Total Pressure Recovery With Friction and Shock Wave Losses

6.13 Hypersonic Flight Considerations

6.14 Summary

6.15 Nomenclature

6.16 Exercises

Chapter 7: Turbomachinery

Abstract

7.1 Introduction to Turbomachines for Propulsion

7.2 Thermodynamic Analysis of a Compressor and a Turbine

7.3 Energy Transfer Between a Fluid and a Rotor

7.4 The Centrifugal Compressor

7.5 Centrifugal Compressors, Radial Turbines, and Jet Engines

7.6 The Axial Flow Compressor

7.7 The Axial Flow Turbine

7.8 Axial Flow Compressor and Turbine Performance Maps

7.9 Three-Dimensional Considerations in Axial Flow Turbomachines

7.10 Summary

7.11 Nomenclature

7.12 Exercises

Chapter 8: Blade Element Theory for Axial Flow Turbomachines

Abstract

8.1 Introduction to Flows Through Blade Passages

8.2 Cascades

8.3 Straight Cascades

8.4 Elemental Blade Forces

8.5 Elemental Blade Power

8.6 Degree of Reaction and the Pressure Coefficient

8.7 Nondimensional Combined Velocity Diagram

8.8 Adiabatic Efficiency

8.9 Secondary Flow Losses in the Blade Passages

8.10 Compressor Blade Loading and Boundary Layer Separation

8.11 Characteristics of the Compressor Blade Pressure Field

8.12 Critical Mach Number and Compressibility Effects

8.13 Turbine Blade Heat Transfer

8.14 Summary

8.15 Nomenclature

8.16 Exercises

Chapter 9: Airbreathing Engine Performance and Component Integration

Abstract

9.1 Introduction to Airbreathing Engine Performance

9.2 Turbojet and Turbofan Engine Configurations

9.3 Operational Requirements

9.4 Compressor-Turbine Matching—Case 1: Nozzle Minimum Area and Combustor Exit Stagnation Temperature Specified

9.5 Compressor-Turbine Matching—Case 2: Mass Flow Rate and Engine Speed Specified

9.6 Inlet-Engine Matching

9.7 Thrust Monitoring and Control in Flight

9.8 Fuel Delivery Systems

9.9 Thrust Reversers

9.10 Estimating Thrust and Specific Fuel Consumption in Cruise

9.11 Engine Cost

9.12 Loads on Turbomachinery Components

9.13 Summary

9.14 Nomenclature

9.15 Exercises

Chapter 10: Propellers

Abstract

10.1 Introduction to Propellers

10.2 Classical Control Volume Analysis

10.3 Blade Element Analysis

10.4 Propeller Charts and Empirical Methods

10.5 The Variable Speed Propeller

10.6 Propeller Performance

10.7 Ducted Propellers

10.8 Turboprops

10.9 Geared Turbofans and Open Rotors

10.10 Summary

10.11 Nomenclature

10.12 Exercises

Chapter 11: Liquid Propellant Rocket Motors

Abstract

11.1 Introduction to Liquid Propellant Rocket Motors

11.2 Liquid Propellant Rocket Motor Nozzles

11.3 Specific Impulse

11.4 Liquid Propellants

11.5 Combustion Chambers for Liquid Propellant Rockets

11.6 Liquid Propellant Rocket Motor Operational Considerations

11.7 Characteristics of Real Liquid Propellant Rockets

11.8 Liquid Propellant Tanks and Feed Systems

11.9 Summary

11.10 Useful Constants, Definitions, and Conversion Factors

11.11 Nomenclature

11.12 Exercises

Chapter 12: Solid Propellant Rocket Motors

Abstract

12.1 Introduction to Solid Propellant Rocket Motors

12.2 Solid Propellant Rocket Description

12.3 Solid Propellant Grain Configurations

12.4 Burning Rate

12.5 Grain Design for Thrust-Time Tailoring

12.6 Combustion Chamber Pressure

12.7 Erosive Burning

12.8 Solid Propellant Rocket Motor Performance

12.9 Transient Operation of Solid Propellant Rocket Motors

12.10 Nozzle Heat Transfer

12.11 Solid Propellant Rocket Motor Sizing

12.12 Hybrid Rockets

12.13 Summary

12.14 Nomenclature

12.15 Exercises

Chapter 13: Space Propulsion

Abstract

13.1 Introduction to Space Propulsion

13.2 Space Propulsion Systems

13.3 Electric Propulsion Systems

13.4 Electrothermal Propulsion Devices

13.5 Electrostatic Propulsion Devices

13.6 Electromagnetic Propulsion Devices

13.7 Nuclear Propulsion Devices

13.8 Summary

13.9 Nomenclature

13.10 Exercises

Appendix A: Shock Waves, Expansions, Tables and Charts

A.1 Normal Shock Wave Relations

A.2 Oblique Shock Wave Relations

A.3 Prandtl-Meyer Expansion

A.4 Tables and Charts for Isentropic Compressible Gas Flows and Shock Waves in a Gas With γ = 1.4

A.5 Nomenclature

Appendix B: Properties of Hydrocarbon Fuel Combustion

B.1 Tables and Charts of Some Thermodynamic Properties

B.2 Nomenclature

Appendix C: Earth's Atmosphere

C.1 The Atmospheric Environment

C.2 The 1976 US Standard Atmosphere Model

C.3 Tables of Atmospheric Properties

C.4 Nomenclature

Appendix D: Boost Phase and Staging of Rockets

D.1 General Equations for Launch Vehicles

D.2 Simplified Boost Analysis With Constant Thrust and Zero Lift and Drag

D.3 Staging of Rockets

D.4 Single-Stage to Orbit (SSTO)

D.5 Two-Stage Vehicle to Orbit (TSTO)

D.6 Three-Stage Vehicle to Orbit

D.7 Staging Considerations

D.8 Nomenclature

Appendix E: Safety, Reliability, and Risk Assessment

E.1 System Safety and Reliability

E.2 Apportioning Mission Reliability

E.3 The Reliability Function

E.4 Failure Rate Models and Reliability Estimation

E.5 Apportionment Goals

E.6 Overview of Probabilistic Risk Assessment (PRA)

E.7 Launch Escape Systems and Crew Safety

E.8 Nomenclature

Appendix F: Aircraft Performance

F.1 The Range Equation

F.2 Take-Off Performance

F.3 Turboprop Powered Aircraft

F.4 The Air Data System

F.5 Nomenclature

Appendix G: Thermodynamic Properties of Selected Species

G.1 Tables of Thermodynamic Properties

G.2 Reference

G.3 Properties of Selected Species

Appendix H: Units and Conversion Factors

Index

Copyright

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Preface to the Second Edition

This edition reflects updates of and emendations to the first edition, which originally derived from notes developed and assembled over many years of teaching propulsion and high-speed airplane and spacecraft design courses at the Polytechnic Institute of Brooklyn (now part of New York University), as well as courses in propulsion and aerospace vehicle design at the University of Florida. The book remains aimed at presenting the theory and concepts of propulsion through a fundamental approach suitable for courses at the senior undergraduate and first year master’s level. The exercises are intended to promote an appreciation for applications of the theory to problems of practical interest.

General changes to the first edition include the addition of an introductory section at the start of each chapter as well as a summary section concluding each chapter. Many of the figures and graphs have been improved, additional example problems have been added, and the number of exercises increased. Errors appearing in the first edition have been tracked down and corrected.

Chapter 1, now titled Propulsion Principles and Engine Classification, is aimed at introducing the reader to the different types of jet propulsion engines and provides a quantitative foundation based on quasi-one-dimensional conservation equations. Airbreathing propulsion systems treated include propellers, turbojets, turbofans, pulsejets, ramjets, and scramjets. The section on the turbofan has been rewritten and sections on aerospace propulsion fuels and space propulsion engines have been added. The section dealing with the conditions for achieving maximum thrust has been expanded and moved to Chapter 5, which deals with nozzles.

Chapter 2 develops the quasi-one-dimensional equations that enjoy wide use in the design and analysis of various propulsion systems. The section on the conservation of chemical species equation was rewritten and the section on the equations of motion in standard form was expanded to include classical Fanno and Rayleigh flows.

Chapter 3 carries out an extensive set of analyses of the operation of a variety of airbreathing engines under conditions of ideal operation to maintain a focus on underlying concepts. Eight separate cases are studied: maximum power takeoff of turbojets and turbofans, high subsonic cruise of turbojets and turbofans, supersonic cruise of turbojets, turbofans, and ramjets, and hypersonic cruise of scramjets. These detailed analyses offer the equivalent of a set of sample exercises to aid the reader in understanding the ideal workings of the various engines and flight regimes. A section on the effect of the efficiencies of the various components of real engines is presented in a manner that should facilitate repeating the calculation of all the cases with reasonable concern for the losses common to practical operation.

Chapters 4, 5, and 6 each concentrate on one of the three engine components basic to jet propulsion principles: combustors, nozzles, and inlets. Fundamentals are discussed in some detail and application to actual hardware is shown. Chapter 4 deals with combustion chambers for airbreathing engines and treats constant area and constant pressure combustors and includes a new section on supersonic combustion. The calculation of chemical equilibrium composition and adiabatic flame temperature is described and a more comprehensive example problem for adiabatic flame temperature replaces the simpler one in the previous edition. Chapter 5 is now titled Nozzles for Airbreathing Engines denoting its particular emphasis. A section on conditions for maximum thrust has been added and includes the effects of stagnation temperature and back pressure on thrust. Chapter 6 is now called Inlets for Airbreathing Engines and includes a section on inlets in subsonic flight which discusses inlet lip design, inlet duct friction losses, and inlet boundary layer diverters. A section on total pressure recovery with friction and shock wave losses has been added.

Then Chapters 7 and 8 are devoted to the fundamentals of the turbomachinery required for operating airbreathing jet engines throughout the flight range up to and including supersonic speeds. Chapter 7 develops the gas dynamics and thermodynamics of turbomachinery needed for analyzing centrifugal flow compressors and axial flow compressors and turbines, including velocity diagrams and the development of performance maps from basic aerodynamic principles. Chapter 8 delves deeper into the details of flows within blade passages and discusses the important factors of boundary layer separation for compressors and heat transfer for turbines. A new section on calculating the optimum Mach number at the compressor face has been added.

Integrating the various components discussed in the previous five chapters into a working engine is the subject of Chapter 9. The description of different types of turbojet and turbofan engines now includes a discussion on the geared turbofan engine. The similarity variables important in the matching process are derived and detailed matching analyses for two basic design approaches are presented. Issues concerning inlet-engine matching, thrust monitoring and control in flight, and fuel delivery systems are discussed.

The last chapter concerned with aircraft flight operations is Chapter 10, which covers the operation of propellers and the application of the gas turbine engine to them. A section devoted to a simple analysis for determining static thrust and a section on geared turbofans and open rotor engines have been added along with an extensive example problem on turboprop performance.

Chapters 11 and 12 are concerned with liquid and solid propellant rocket engines, respectively, while Chapter 13 is devoted to space propulsion systems. Chapter 11 has been expanded and largely rewritten and a section on propellant density and specific impulse has been added. The section on liquid propellants now includes detailed assessments of the LH2-LOX, RP1-LOX, and the LCH4/LOX propellant combinations. The section on liquid propellant tank and feed system design has been updated and rewritten and now includes discussion of liquid propellant tank characteristics, analysis, and structural design, as well as liquid propellant feed systems and turbopump analysis and sizing considerations. Chapter 12 deals with solid propellant rocket motors and includes a new section on solid propellant rocket motor sizing and an associated worked example.

Chapter 13 covers the area of space propulsion with attention given to electric propulsion techniques that are of importance in satellite operations and space exploration and now incorporates nuclear propulsion and its possible role in interplanetary missions.

The eight appendices deal with important auxiliary information for the main text: Appendix A presents equations for the calculation of shock waves and expansions and includes tables and charts. Appendix B gives tables for the properties of hydrocarbon fuel combustion products and includes a narrative explaining the tables and their use. Appendix C gives a brief discussion of the physics of the earth’s atmosphere with expanded tables of atmospheric properties for greater utility. The material in Appendices D and E covers boost phase and staging of rockets and safety, reliability, and risk assessment using material from the author’s book Manned Spacecraft Design Principles (Elsevier, 2015). Appendix F deals with aircraft performance in takeoff and cruise using material from the author’s book Commercial Airplane Design Principles (Elsevier, 2014). Tables of thermodynamic properties of selected chemical species appropriate to propulsion applications are presented in Appendix G while H provides a listing of useful constants and conversion factors.

I would like to again acknowledge the inspiration provided by Professor Antonio Ferri, pioneer and champion of scramjet development, who taught propulsion courses I took as a graduate student at the Polytechnic Institute of Brooklyn many years ago. Appreciation is also due my close colleagues Professor Herbert Fox of the New York Institute of Technology and the late Professor Marian Visich of the State University of New York at Stony Brook for their long-term cooperation in, criticism of, and support for this book. Thanks are also due to a number of reviewers who have offered well-considered criticism of the first edition and provided useful suggestions and recommendations which I hope I have used wisely in preparing this edition.

Finally, and most importantly, I thank my wife, Anne, who continues to encourage, support, and assist me in these writing projects in spite of the time it takes me away from her company.

Pasquale M. Sforza, Professor of Aerospace Engineering, Emeritus

[e-mail]

Chapter 1

Propulsion Principles and Engine Classification

Abstract

The different types of aerospace propulsion engines are quantitatively described and the basic performance attributes of each by applying basic integral conservation equations. When work but no heat was added to the flow processed by the engine we had the case of the propeller, which turned out to be the most efficient propulsion device. Though efficient, the propeller was speed-limited because for a given power input, the thrust produced varied approximately inversely to the flight speed. When only heat is added but no net work is done on the fluid passing through engine, we have the case of the turbojet, ramjet, scramjet, or pulsejet engines, all of which are less efficient than the propeller, but capable of speeds much higher than those achievable with a propeller. A special case of adding only heat is the rocket, which takes in no ambient fluid but instead carries onboard all the propellants. The thrust of a rocket is independent of flight speed and because all the fluid it uses is carried onboard, the rocket is able to operate in the vacuum of space. By adding both work and heat to the fluid we arrive at the case of the turbofan which derives benefits of both the propeller and the turbojet. The ratio of the mass flow of the work-added part to the heat-added part is called the bypass ratio. The turbofan enjoys the lower specific fuel consumption provided by the fan section of the engine and the high-speed capability of the turbojet section of the engine. After illustrating the different engine types which use the principle of jet propulsion, attention was focused on the force field generated by such engines and the means by which the fundamental performance characteristics of the engines can be calculated.

Keywords

Propeller; Turbojet; Turbofan; Ramjet; Rocket; Specific fuel consumption; Specific impulse; Thrust

Chapter Outline

1.1 Introduction to Aerospace Propulsion Engines

1.2 Conservation Equations

1.2.1 Conservation of Mass

1.2.2 Conservation of Momentum

1.2.3 Conservation of Energy

1.3 Flow Machines with No Heat Addition: Propellers, Fans, Compressors, and Turbines

1.3.1 Zero Heat Addition with Ve > V0

1.3.2 Zero Heat Addition with Ve < V0

1.3.3 Zero Heat Addition with P = Constant > 0

1.3.4 Propulsive Efficiency

1.3.5 Example: Propeller Speed and Thrust

1.4 Flow Machines with No Net Power Addition: Turbojets, Ramjets, Scramjets, and Pulsejets

1.4.1 Heat Addition, Q > 0

1.4.2 Thrust Variation with Flight Speed

1.4.3 Overall Efficiency

1.4.4 Fuel Efficiency

1.4.5 Example: Turbojet Specific Fuel Consumption

1.5 Flow Machines with P = 0, Q = Constant and A0 = 0: The Rocket

1.5.1 Thrust Variation with Flight Speed

1.5.2 Propulsive Efficiency

1.5.3 Fuel Efficiency and Specific Impulse

1.6 The Special Case of Combined Heat and Power: The Turbofan

1.6.1 Very Small Bypass Ratio, β ≪ 1, the Turbojet

1.6.2 Small to Large Bypass Ratio, β ≤ 10, the Turbofan

1.6.3 Example: Turbofan Specific Fuel Consumption

1.6.4 Very Large Bypass Ratio, β ≫ 1, the Turboprop and the Open Rotor

1.7 Aerospace Propulsion Fuels

1.7.1 Jet Engine Fuels

1.7.2 Rocket Engine Fuels

1.7.3 Fuel Energy Content

1.8 Space Propulsion Engines

1.8.1 Heat Addition Using Nuclear or Electric Power

1.8.2 Electrostatic Acceleration

1.8.3 Electromagnetic Acceleration

1.9 The Force Field for Airbreathing Engines

1.9.1 Example: Jet Engine Performance

1.9.2 Example: Rocket Engine Performance

1.10 Summary

1.11 Useful Constants and Conversion Factors

1.12 Nomenclature

1.12.1 Subscripts

1.13 Exercises

Reference

1.1 Introduction to Aerospace Propulsion Engines

The operation of aerospace propulsion engines rests on the foundation of Newton's laws of motion. The second of these laws explains that the change in momentum of the fluid passing through an engine is equal to the force acting on the fluid. The third law states that the force acting on the fluid exerts a reaction, an equal and opposite force, on the boundaries separating the fluid and the engine. Indeed, such engines are often referred to as reaction motors.

In general, aerospace propulsion engines may be thought of as idealized flow machines in which fluid within the machine has work and/or heat added to it prior to its exit from the machine as a jet, thereby producing thrust according to the reaction principle described above. The fluid may enter the machine from the surroundings or may be carried entirely within the machine prior to being processed. The former engines are usually called airbreathing engines and the latter, rockets. Of primary interest is the magnitude of the thrust produced and the efficiency with which the heat and power is used in generating thrust. In this chapter we will classify the different types of aerospace propulsion engines and quantitatively describe the performance of each by applying the basic integral forms of the conservation equations of gas dynamics.

The most efficient flow machine is the propeller, in which work is done on the fluid passing through it, but no heat is added. The source of power is external to the propeller itself and may be an internal combustion engine, a gas turbine, or even an electric motor. In this chapter we show that though efficient, the propeller is speed-limited in the sense that for a given power input the thrust developed falls off with increasing speed.

Conversely, the turbojet, ramjet, scramjet, and pulsejet are flow machines in which heat is added to the airstream taken aboard, but no net work is done on it. The heat is added by burning fuel in that airstream. The thrust they develop has a relatively weak dependence on flight speed making them capable of good performance at much higher speeds than are achievable with propellers. However, they are less efficient than propellers and their ability to travel at higher, even supersonic, speeds is thus gained at the cost of greater fuel consumption. The rocket, a variant of this case, takes in no ambient fluid but instead carries onboard all the fluid to which heat is added, again by combustion, and subsequently ejected as a jet. The thrust of the rocket is independent of the flight speed but this advantage is paid for by even lower efficiency than the other jet engines. On the other hand, because all the necessary propellant is carried onboard the rocket is able to operate in the vacuum of space.

Rather than add power alone or heat alone to the working fluid we show that by adding a combination of power and heat we can derive some of the special benefits of both the propeller and the turbojet. This is the turbofan engine which is the common jet powerplant for both commercial and military aircrafts.

For space applications we examine means other than chemical combustion to energize an onboard propellant. Nuclear thermal propulsion (NTP) involves heating the onboard propellant by passing it through a nuclear fission reactor. Such rockets can provide high thrust with higher efficiency than a chemical rocket but pose radiation safety issues. Simple electrical resistance heating of a propellant is the basis of the resistojet while the arc jet uses an electric discharge to heat propellant. We also examine the highly efficient but low thrust ion rocket which generates thrust by electrostatic acceleration of ions and the magnetoplasmadynamic (MPD) rocket which uses electromagnetic acceleration to accelerate electric discharge-produced plasma. The ion and MPD rockets exhibit very high efficiency but are limited to relatively low thrust levels.

After illustrating the different engine types which use the principle of jet propulsion, attention is focused on details of the force field generated by such engines and the factors influencing the production of thrust.

1.2 Conservation Equations

A flow machine is one which ingests a stream of fluid, processes it internally in some fashion, and then ejects the processed fluid back into the ambient surroundings. An idealization of such a generalized flow machine is schematically depicted in Fig. 1.1.

f01-01-9780128093269

Fig. 1.1 Schematic diagram of idealized flow machine and associated streamtube control volume.

In order to develop the basic features of operation of the idealized flow machine without introducing unnecessary algebraic complexity, we make the following assumptions:

• The flow through the streamtube entering and leaving the machine is steady and quasi-one-dimensional. Mass cannot cross the streamtube surfaces.

• The entrance and exit stations shown are chosen sufficiently far from the flow machine entrance and exit such that the pressures at those stations are in equilibrium with their surroundings, that is, pe = p0.

• There is no heat transfer across the boundaries of the streamtube or the flow machine into the ambient surroundings.

• Frictional forces on the entering and leaving streamtube surfaces are negligible.

• Mass injected into or extracted from the fluid stream within the flow machine, if any, is negligible compared to the mass flow entering the flow machine.

With these restrictions in mind we may assess the consequences of applying the basic conservation principles to the streamtube control volume. A more detailed discussion and development of the conservation laws and associated thermodynamic principles is presented in Chapter 2. At this point we wish to use the simplest form of these laws to classify the wide variety of aerospace propulsion devices. However, there are some implications of the assumptions used which are important to keep in mind.

The assumption of steady flow implies that V0 is constant, that is, the idealized flow machine may be considered to be flying at the speed V0 through a stationary atmosphere with the ambient environmental values of pressure, density, and temperature denoted in Fig. 1.1 by p0, ρ0, and T0, respectively. Alternatively, we may consider our coordinate system to be fixed on the flow machine such that the atmosphere constitutes a free stream flow approaching at speed V0 with static conditions of pressure, density, and temperature denoted in Fig. 1.1 by p0, ρ0, and T0, respectively. This (Galilean) transformation of coordinates is possible because the motion is steady.

Another implication arising from the assumption that the flow machine is moving through the atmosphere at constant speed is that there must be no unbalanced force on the machine. Since there will be resistance to the motion arising from the aerodynamic drag D there must be another force acting which can balance it and so maintain the constant motion and that is the thrust F. The rate at which work must be done to maintain the motion is DV0 and because D = F the required power may also be written as FV0.

1.2.1 Conservation of Mass

Because mass can neither be created nor destroyed and any mass addition or subtraction within the flow machine is assumed negligible, the net change in the mass flow passing through the flow machine is zero. The assumption of quasi-one-dimensional flow and assigning a negative sign to flow into the control volume and a positive sign to flow leaving the control volume permits us to write this balance as

si1_e    (1.1)

This is equivalent to stating that the mass flow si2_e throughout the system. For density in kg/s, area in m², and velocity in m/s the mass flow has the units of kg/s.

1.2.2 Conservation of Momentum

Newton's second law of motion requires that the net change in momentum of the fluid passing through the streamtube is equal to the force acting on the fluid. Using the sign convention for mass flow given by Eq. (1.1) the change in the momentum of the fluid is given by

si3_e

Because the mass flow is constant this equation can be abbreviated to the following form:

si4_e    (1.2)

For mass flow in kg/s and velocity in m/s the force F has the unit kg m/s², which is called a Newton. The force acting on the fluid is denoted by F, and by Newton's third law of motion, the reaction on the control volume by the fluid is the force, −F. In general, the forces on the streamtube are negligible compared to those on the flow machine proper and are neglected. One practical case where this is not necessarily true is that of the so-called additive drag of inlets in supersonic flight, where the force on the entering streamtube surface may not be negligible. The issue of forces acting on the streamtube is discussed in more detail at the end of this chapter.

1.2.3 Conservation of Energy

The energy balance arising from the first law of thermodynamics for a flowing fluid requires that the net change in the total enthalpy of the flowing fluid is equal to the sum of the rate at which heat and work are added to the fluid, or

si5_e

   (1.3)

The quantities h, Q, and P denote enthalpy per unit mass in kJ/kg, heat addition per unit mass in kJ/kg, and power added in kW, respectively. Note that the kinetic energy per unit mass, V²/2 is expressed in m²/s² and therefore must be converted to the units of kJ/kg for use in Eq. (1.3). In the SI system of units 1 kJ = 10³ N m = 10³ kg m²/s² which results in 1 m²/s² = 10− 3 kJ/kg. Interpreting Eq. (1.3) in different limits of the variables permits us to illustrate the fundamental features of practical aerospace propulsion systems and components.

1.3 Flow Machines With No Heat Addition: Propellers, Fans, Compressors, and Turbines

Consider a typical airplane propeller driven by a motor, as shown schematically in Fig. 1.2. The motor here is an external source of power such as an internal combustion or gas turbine engine or an electric motor. This power source is used to spin the blades in a plane normal to the flight velocity V0. The rotating blades, which are basically wings, do work on the air, accelerating it in the downstream direction. The motor adds an insignificant amount of heat to the mass flow of air passing through the area swept out by the rotating blades. Heat generated by the motor itself would be confined to that transferred through the motor housing to the air in immediate contact with it or emanating from a relatively small exhaust pipe. Therefore it is reasonable to assume that for the air flow processed by a propeller Q = 0 so that

si6_e    (1.4)

f01-02-9780128093269

Fig. 1.2 General arrangement of a propeller driven by an external power source such as an internal combustion or gas turbine engine or an electric motor. Station 1 is in the plane of rotation of the propeller blades.

However, if no heat is added to the flowing fluid it is reasonable to expect that the temperature of the air would be essentially unchanged in passing through the rotating blades. Because the enthalpy is related to the temperature by the specific heat at constant pressure cp, the enthalpy difference he − h0 = cp(Te − T0) ~ 0 which, when using Eq. (1.2), transforms Eq. (1.4) into

si7_e

   (1.5)

Subsequently it will be useful to think of this power imparted to the flow to be the sum of a useful part that sustains the motion and another part that accounts for losses. Therefore we write

si8_e

Then the loss term is

si9_e

   (1.6)

From Eq. (1.5) we see that the power supplied to the fluid is approximately equal to the product of the force on the fluid and the average of the velocities entering and leaving the machine. In Chapter 10 we will prove that the average velocity is the velocity at the plane of rotation of the propeller (station 1 in Fig. 1.2) so that the mass flow

si10_e    (1.7)

1.3.1 Zero Heat Addition With Ve > V0

When the air is accelerated by the propeller from V0 in the undisturbed free stream to a higher value Ve far downstream from the propeller where pe recovers to p0, Eq. (1.2) shows that the force on the air F > 0. Furthermore, Eq. (1.5) shows that P > 0 and therefore work is done on the air by the propeller. In addition to the propeller that we see on general aviation and regional transport aircraft, this is also the case of the turbofan and the axial flow compressor, both found on commercial and military jets where the device does work on the fluid and produces a force on the fluid in the same sense as the entering velocity. Note that this means that the reaction force of the fluid on the machine is in the opposite sense, that is, a thrust is developed. Propellers and compressors will be discussed in detail in Chapters 10 and 7, respectively.

1.3.2 Zero Heat Addition With Ve < V0

If the flow machine decelerates the fluid from V0 in the undisturbed free stream to a lower value Ve far downstream from the device where pe recovers to p0, Eqs. (1.2), (1.5) show that the force on the fluid F < 0 and the power P < 0, so that work is done by the fluid. This is the case of the axial flow turbine which is used in jet engines to drive the compressor. This is also the case of wind turbine, where work is extracted from the wind and used to drive an electric generator. In these flow machines the fluid experiences a retarding force, that is, the force on the fluid is in the opposite sense to that of the incoming velocity. The reaction force on the machine is therefore in the same sense as the entering velocity and is therefore a drag force.

1.3.3 Zero Heat Addition With P = Constant > 0

For a constant value of power added to the fluid equation (1.5) shows that the thrust force developed drops off with flight speed:

si11_e    (1.8)

The variation of the thrust-to-power ratio of an aircraft propeller as a function of flight speed for various ratio of Ve/V0 is illustrated in Fig. 1.3.

f01-03-9780128093269

Fig. 1.3 The thrust-to-power ratio for a propeller as a function of practical flight speed V 0 with V 0 / V e as a parameter.

It is clear that the conversion of power applied to the fluid into thrust drops off rapidly with flight speed and that accelerating the flow to higher values of Ve yields diminishing returns in thrust for a fixed power input. From this observation and Eq. (1.2) we see that high propeller thrust is best accomplished by accelerating a large mass flow of air through a small velocity increase. It is further recognized from Fig. 1.3 that propeller thrust is highest at low speeds and this provides improved take-off acceleration and reduced runway lengths. Because of the rapid drop in F/P as a function of flight speed we also see that the maximum flight speed of a propeller-driven aircraft is limited by the power available.

1.3.4 Propulsive Efficiency

Remember that the flight speed V0 is constant and therefore the drag on the vehicle is equal to the thrust produced, D = F. The power required to keep the vehicle moving at constant speed V0 is DV0 = FV0. The power supplied to the fluid P = FVavg is not necessarily converted completely into useful thrust power FV0. Using Eq. (1.5) we define the propulsive efficiency ηp as the ratio of useful thrust power to total power imparted to the airstream:

si12_e

   (1.9)

This equation shows that the propulsive efficiency increases as the flight speed V0 approaches the jet exit velocity Ve, as shown in Fig. 1.4.

f01-04-9780128093269

Fig. 1.4 The efficiency of a propeller as a function of the ratio of the flight speed V 0 to the speed of the exit jet V e .

1.3.5 Example: Propeller Speed and Thrust

The North American P-51D Mustang (Fig. 1.5), one of the premier fighter aircraft in the Second World War, was powered by the Packard-built Merlin engine that produced a maximum shaft power of Ps = 1100 kW at sea level (z = 0) and Ps = 1000 kW at an altitude of z = 6000 m. The best propellers converted about 90% of that shaft power into power delivered to the air. The drag coefficient of the Mustang at high speed is approximately cD = 0.02 and the wing area S = 21.6 m². Then the total drag of the Mustang is D = cDqS = 0.432q, where q is the free stream dynamic pressure which is defined as si13_e . The dynamic pressure q is measured in N/m² (or Pa) in the SI system of units. In steady level flight the thrust F = D = 0.432q and the power transferred to the air is P = 0.9Ps. We wish to determine the speed achievable by the Mustang at 6000 m altitude and the corresponding propulsive efficiency.

f01-05-9780128093269

Fig. 1.5 A restored NACA P-51 Mustang in flight in Sep., 2000 (NASA).

At z = 6000 m the density ratio σ = ρ/ρsl = 0.5389 (see Appendix C) so the thrust is calculated to be

si14_e

measured in Newton. Using Eq. (1.5), where for consistency F must be expressed in kN, we find

si15_e

Solving for the flight velocity yields

si16_e    (1.10)

From Eq. (1.2), Eq. (1.7), and assuming incompressible flow (ρ0 = ρ1) we find

si17_e

We may rearrange this equation and use our result F = D = 0.432q to obtain

si18_e

The Mustang's propeller diameter is 3.4 m, so the area A1 = 9.07 m² and we solve the above equation to find Ve/V0 = 1.0236. Thus we see that the large mass flow of air passing through the propeller is accelerated only a very small amount to produce a substantial thrust. Using this ratio in Eq. (1.10) we may calculate the flight velocity to be V0 = 184.1 m/s (411 mph). Using Eq. (1.9) to calculate the propulsive efficiency we find

si19_e

Although the propulsive efficiency is high, Eq. (1.10) makes it quite clear that the flight speed of a propeller-driven aircraft is limited because si20_e so that increasing power provides continually diminishing returns, as can be seen from the derivative si21_e . Note that the mass conservation equation—Eq. (1.7)—with the incompressible flow assumption (ρ0 = ρ1 = ρe) shows that A0V0 = A1Vavg = AeVe. Thus the streamtube area A0 captured by the propeller is slightly larger than the propeller swept area A1. In the same fashion the downstream area Ae is slightly smaller than the propeller swept area A1.

1.4 Flow Machines With No Net Power Addition: Turbojets, Ramjets, Scramjets, and Pulsejets

Now we take the opposite tack and examine the effect of no net power being transferred into or out of the fluid passing through the flow machine. Then Eq. (1.3) with P = 0 reads as follows:

si22_e    (1.11)

Note that as in Eq. (1.5) we may write the kinetic energy term in Eq. (1.11) as

si23_e

   (1.12)

Substituting Eq. (1.12) into Eq. (1.11) and solving for the thrust yields

si24_e    (1.13)

1.4.1 Heat Addition, Q > 0

If sufficient heat is added to the fluid such that Q > 0 and that Q > (he − h0), Eq. (1.13) shows that F > 0 and therefore thrust is produced on the flow machine. This is the basis of operation of the simple jet engine. The general internal configuration of the practical jet engine is dependent upon the flight speed. The inlet captures the air to be processed by the flow machine. For flight in the range of 0 < M0 < 3 the jet engine requires a compressor to increase the pressure of the incoming air before fuel is added and burned, particularly in the low end of the speed range. The heated combustion products are then accelerated by a nozzle to produce thrust. The compressor must be driven by a shaft power source and this is most effectively supplied by coupling a gas turbine to it. The gas turbine extracts just enough power from the hot combustion gases to drive the compressor so that the net power into the fluid P = 0. Rotating compressors and turbines are generally called turbomachines. Such an arrangement is called a turbojet engine and is schematically illustrated in Fig. 1.6. The fifth basic assumption stated in Section 1.2 was that any mass injected into or extracted from the fluid stream within the flow machine is negligible compared to the mass flow entering the flow machine. In the case of fuel injection the mixture ratio of air to hydrocarbon fuel like kerosene for complete combustion is about 15. This results in temperatures much too high for practical engine structures so that excess air is added bringing the air-to-fuel ratio to about 50 (2% of the air flow) which certainly is in keeping with the idea of the added fuel mass flow being insignificant compared to the overall air flow. Interestingly, practical considerations require that about an equivalent mass flow of air be bled from the compressor to serve various pneumatic functions on an aircraft. Thus the net effect is that the mass flow through the engine is quite constant.

f01-06-9780128093269

Fig. 1.6 This schematic diagram of a typical turbojet engine shows the required turbomachinery components and a common station numbering scheme. The combustor burns the injected fuel supplying heat to the flow passing through the turbojet.

For supersonic flight in the Mach number range of 3 < M0 < 5 the ram pressure produced by the inlet in slowing down the incoming air to subsonic speeds obviates the need for the compressor and therefore its driving turbine. As a consequence, a practical jet engine for this flight regime is called a ramjet and its configuration is very simple, as shown in Fig. 1.7, where the turbomachinery no longer appears. This simplicity comes at a price however, because the ramjet cannot operate effectively at lower speeds. In particular, it generates no thrust at zero flight speed so that it cannot provide thrust for takeoff. The ramjet must be accelerated to near sonic or supersonic speeds by some other propulsive means before it can produce enough thrust to sustain flight of the vehicle it powers. It is therefore often used to power missiles launched from an aircraft in high-speed flight or by a rocket booster.

f01-07-9780128093269

Fig. 1.7 Schematic diagram of a typical ramjet engine. The high-speed ram compression obviates the need for the turbomachinery components. Only the combustor remains to burn the injected fuel supplying heat to the flow.

For hypersonic flight speeds, M0 > 5, the temperature increase accompanying the ram compression to subsonic speeds is so high that little or no additional heat can be added in the combustor by burning fuel. The only alternative is to use the inlet to slow the flow down from the flight speed to some lower supersonic Mach number, thereby not increasing the temperature too much. However, then fuel must be added to a supersonic stream, mixed, and combusted. Achieving this supersonic combustion is very difficult because the high speed in the combustor gives very little time for the mixing and combustion to take place. Such a supersonic combustion ramjet is popularly known as a scramjet. The general configuration is like that of the ramjet shown in Fig. 1.7.

The steady operation of the ramjet can only be maintained at speeds high enough to provide sufficient ram pressure in the combustor to ensure effective combustion. If instead a ramjet-like flow machine is operated in an unsteady mode, it is possible to produce thrust at zero forward speed. This is accomplished by the pulsejet, a propulsion device which is schematically illustrated in Fig. 1.8. The difference between the pulsejet and the ramjet is the addition of a grid of flapper valves between the inlet and the combustor.

f01-08-9780128093269

Fig. 1.8 Schematic diagram of a pulsejet engine. Time-varying back pressure due to combustion forces the flapper valves to cycle from an open to a shut position and back.

A pressure drop across the valve grid causes the valves to open as indicated in Fig. 1.8 permitting air to rush into the combustor. Injecting fuel at that point and igniting it causes a rapid rise in pressure in the combustor which forces the valves shut. The high-pressure combustion gases are ejected through an accelerating nozzle thereby producing thrust. The evacuation of the combustion gases causes a drop in pressure in the combustor sufficient to once again open the flapper valves allowing a fresh charge of air to enter. The entire process is repeated and another pulse of thrust is produced.

The frequency of pulsations is fixed by the natural frequency of the engine which in turn depends upon distance between the flapper valve grid and the nozzle exit. Like a resonating organ pipe, the shorter this length, the higher the frequency of the thrust pulsations. Examples of practical pulsejet engines range from the 3.35 m long engine V-1 buzz-bomb of the Second World War operating at about 50 Hz to model airplane engines on the order of a 0.5 m in length operating at around 350 Hz. Although the pulsejet has the advantage of being started under static conditions, the pressure losses in the flapper grid make it necessary to have a relatively large frontal area to permit the inlet to capture sufficient air flow for effective operation. The resulting ram drag penalties place a speed limitation on pulsejets well below that of the minimum operating speeds of the ramjet. Because the pressure pulsations are caused by detonations pulsejets are extremely noisy. Current research on pulsejets is focused on so-called pulse detonation engines designed to operate at much higher frequencies in order to achieve a smoother and more effective thrust delivery and permit operation at supersonic speed.

1.4.2 Thrust Variation With Flight Speed

Expanding Eq. (1.13) shows the thrust to be given by

si25_e    (1.14)

As the flight speed V0 increases, both the numerator and denominator increase so that thrust becomes approximately constant. Recall that in the case of the propeller equation (1.8) showed that F ~ 1/Vavg while Eq. (1.14) for a jet engine shows that F ~ V0/Vavg. This illustrates the basic advantage of the jet engine—its thrust is approximately independent of the flight speed and therefore it is not speed-limited, as is the propeller.

1.4.3 Overall Efficiency

The overall efficiency in the case of thrust produced by means of heat addition may be defined as the ratio of the thrust power required to maintain the motion to the rate of heat addition to the fluid and may be written as

si26_e    (1.15)

We may then define the thermal efficiency as

si27_e    (1.16)

Then, identifying V0/Vavg as the propulsive efficiency ηp defined in Eq. (1.8), the overall efficiency is expressed as the product of the propulsive and thermal efficiencies

si28_e    (1.17)

The thermal efficiency accounts for the fact that not all the heat added is converted to useable heat power, since some is rejected as increased internal energy in the exhaust gases. The propulsive efficiency accounts for the fact that the increased kinetic energy of the exhaust jet represents a loss in mechanical power to maintain flight at the speed V0. In a jet engine these two efficiencies generally drive in different directions, with the higher exhaust velocities sustainable at high thermal efficiency leading to lower propulsive efficiencies at a given flight speed. Whereas a propeller-driven aircraft will cruise at V0/Ve ~ 1, a subsonic turbojet-powered aircraft is more likely to cruise at V0/Ve ~ 0.6. From Fig. 1.4 we see that the propeller-driven aircraft has a propulsive efficiency ηp ~ 1, and the value for the turbojet-powered aircraft is ηp = 0.7.

1.4.4 Fuel Efficiency

The heat added to the flow is directly proportional to the rate of fuel consumption si29_e and to the specific energy content of the fuel Qf as expressed by

si30_e

The constant of proportionality is called the burner efficiency ηb so that the rate of heat addition through combustion becomes

si31_e    (1.18)

The burner efficiency ηb represents the ratio of the heat release actually transferred to the flowing combustion gases to the total heat release possible. Eq. (1.18) is based on the assumption that the rate of heat addition arises from the energy released in the chemical conversion of the fuel as characterized by Qf, which is called the heating value of the fuel per unit mass or the specific energy of the fuel. The characteristics of fuels used in aerospace propulsion applications are discussed in detail in Section 1.8.

A common measure of fuel efficiency for jet engines is the specific fuel consumption cj, which is defined in the English system of units as the ratio of the fuel mass flow to the thrust force produced and reported as pounds mass of fuel consumed per hour per pound force of thrust, lbm/lbf h. For convenience, and in applications where it should not cause confusion, we will simply denote pounds force as lb, that is, lb = lbf. In the SI system of units the specific fuel consumption cj is still the ratio of the mass flow of fuel consumed to the thrust force produced, but has the units of kg/h N. A nominal value for turbojets is cj = 1.0 lbm/h/lbf which, in SI units, is transformed to cj = 0.102 kg/h/N. It must be emphasized that care be taken to account properly for the units, particularly since both systems of units are in routine use in the aerospace industry.

Recall that the set of assumptions made at the outset included the requirement that the mass of fluid, in this case fuel, added to the general stream entering the flow machine is negligible, that is, si32_e As discussed previously and as will be shown in Chapter 4, practical fuel mass flows are much smaller than the air mass flow rate, so no loss in generality is incurred here by ignoring the added fuel mass flow. However, we can account for the fuel consumed by combining Eqs. (1.15), (1.18) to define the mass-based specific fuel consumption in the form shown below:

si33_e

   (1.19)

For a typical hydrocarbon jet fuel with Qf = 43,400 kJ/kg (see Table 1.1) and velocity measured in m/s, the specific fuel consumption in kilograms of fuel per hour (3600 s = 1 h) per Newton of thrust becomes

si34_e    (1.20)

Table 1.1

Properties of Some Aerospace Propulsion Fuels

a Chemical formulas for the Avgas, JP, and RP hydrocarbon mixtures are approximations.

b Freezing point.

c Boiling point at 1 atm pressure.

d Evaluated under the vapor pressure at 15°C.

e Evaluated at the b.p. and 1 atm pressure.

The units shown for cj, kg/N h, is essentially an acceleration (kg/N) divided by a time (h) and is therefore an inverse velocity. In the English system, the same hydrocarbon jet fuel has Qf = 18,660 Btu/lbm and with velocity measured in ft/s, we will denote the specific fuel consumption in pounds (mass) of fuel per hour per pound of thrust as follows:

si35_e    (1.21)

Burner efficiency is generally quite high, as will be seen in Chapter 4, and for the present purposes it may be taken as ηb = 0.95. However, the thermal efficiency bears a bit more consideration. Expanding Eq. (1.16) leads to the following representation of thermal efficiency:

si36_e    (1.22)

Under typical subsonic jet aircraft flight conditions in the stratosphere the atmospheric temperature T0 = 216 K, cp,0 = 1.04 kJ/kg K, and the fuel-to-air ratio si37_e . With a burner efficiency ηb = 0.95, and once again using Qf = 43.4 MJ/kg the thermal efficiency becomes

si38_e

The ratio of static to stagnation temperature in the exhaust, assuming that for the hot exhaust gases (see Table B.2) the ratio of specific heats γe ~ 4/3, is given by

si39_e    (1.23)

For a subsonic jet aircraft the exhaust Mach number Me ~ 1 so that a nominal value of the temperature ratio is Te/Tt,e = 0.857. A typical jet exhaust stagnation temperature (see Table 9.1) is Tt,e = 800 K making Te = 686 K with the corresponding specific heat value (see Table B.2) of cp,e = 1.1 kJ/kg K. Then the thermal efficiency is

si40_e

Therefore the thermal efficiency ηth = 35.7%. This leaves us with the determination of the propulsive efficiency ηp given in Eq. (1.9) which we may express in terms of Mach number M = V/a as follows:

si41_e

The sound speed a is discussed in detail in Chapter 2, but here we may use the usual definition a = (γRT)¹/² so that the propulsive efficiency becomes

si42_e    (1.24)

Using the nominal values selected in the discussion of thermal efficiency: γ0 = 7/5, T0 = 216 K, γe = 4/3, Te = 686 K, and Me = 1 we find

si43_e

Therefore the specific fuel consumption (kg/h N) for a nominal turbojet in subsonic flight in the stratosphere where a0 = 295 m/s becomes

si44_e

   (1.25)

This simple analysis for specific fuel consumption shows it to vary linearly with Mach number, as depicted in Fig. 1.9. The corresponding specific fuel consumption in lbm/h/lb is shown in Fig. 1.10.

f01-09-9780128093269

Fig. 1.9 Specific fuel consumption c j in kg/h/N as a function of flight speed for turbojets in cruise in the stratosphere and M  = 1 is indicated.

f01-10-9780128093269

Fig. 1.10 Specific fuel consumption c j in lbm/h/lb as a function of flight speed for turbojets in cruise in the stratosphere where M  = 1 is as indicated.

As shown by Eqs. (1.19), (1.20), high heating value, that is high Qf, fuels will reduce cj while reduced propulsive, thermal, and burner efficiencies will increase it. Combustor design can provide relatively high burner efficiencies with values of ηb > 90% readily attainable. The thermal efficiency, ηth, on the other hand, can vary widely because it depends directly on the temperature difference between the exhaust gas and the ambient temperature. On the other hand, Eq. (1.24) shows that the propulsive efficiency ηp depends on the square root of the ratio of the exhaust temperature to the ambient temperature and with good nozzle design is likely to be in the range of 50% < ηp < 70%. Assuming reasonable values like ηb = 0.95 and ηp = 0.6 in Eq. (1.20) leads us to the following estimate for specific fuel consumption (in kg/N h):

si45_e    (1.26)

A limitation of the turbojet engine is that only so much heat energy may be added to the flow in the combustor because of temperature limitations of the structural material, particularly the turbine blades, which are under the high stress of driving the compressor. Thus the stagnation temperature in the nozzle is limited, in turn limiting the magnitude of the velocity Ve, which, by Eq. (1.2), limits the thrust achievable. Aerodynamic drag exhibits a dramatic increase in the transonic region 0.80 < M < 1.2 and aircraft typically cannot produce sufficient thrust to achieve and sustain supersonic flight even in the low Mach number range below M = 2. To compensate for this lack of thrust needed to achieve supersonic flight, most aircrafts rely on adding more fuel directly into the nozzle downstream of the turbine and burning it to release more heat energy into the flow. This is possible because there is still sufficient oxygen in the flow to support further combustion. This simple arrangement which involves essentially no moving parts and adds little weight is called the jet engine afterburner and is shown schematically in Fig. 1.11. Clever aerodynamic and engine design permits aircraft like the F-22 Raptor to supercruise, that is, to fly continuously at low supersonic Mach number with no engine thrust enhancements.

f01-11-9780128093269

Fig. 1.11 Turbojet engine with an afterburner section introduced between the turbine and the nozzle.

In the afterburner, because more fuel is added to the same air flow passing through the engine, there is a substantial increase in the total fuel-to-air ratio being used by the engine, typically doubling to 0.04 from 0.02. In these mechanically simple afterburner units the burner efficiency achieved would likely be reduced to a value 90% to 60% of that achieved in the gas turbine combustor. In addition, the increased temperature in the nozzle tends to reduce the thermal efficiency to a value 50–30% of that achieved in the nonafterburning turbojet. The propulsive efficiency doesn't change appreciably so that the specific fuel consumption in supersonic afterburning mode may be as much as five times larger than in the subsonic turbojet. Careful design of the exhaust nozzle can improve the thermal efficiency substantially so that the increase in specific fuel consumption in afterburner mode may be reduced to about twice the value without afterburner.

1.4.5 Example: Turbojet Specific Fuel Consumption

The Pratt & Whitney J57 turbojet shown in Fig. 1.12 was the first jet engine to develop 45 kN (10,000 lb) thrust and powered the North American F-100 Super Sabre jet fighter to supersonic flight speed.

f01-12-9780128093269

Fig. 1.12 The Pratt & Whitney J57 turbojet was the first jet engine to develop 45 kN (10,000 lb) of thrust.

A later civilian version, the JTC3, processed a mass flow of 81.5 kg/s of air and produced 50 kN of thrust when powering Boeing 707 and Douglas DC-8 commercial airliners. Assume that a JTC3 engine is operated at sea level (T0 = 288 K) on a test stand (V0 = 0) and that it is burning Jet A fuel with an efficiency of ηb = 96%. Assuming that the engine is operating at full power and the temperature in the pressure-equilibrated jet is measured to be Te = 655 K, estimate the specific fuel consumption.

The thrust is given by

si46_e

   (1.27)

Considering that the fuel-to-air ratio f/a ≪ 1 we approximate the specific fuel consumption as

si47_e

   (1.28)

This solution depends upon the fuel-to-air ratio, f/a, and the pressure-equilibrated downstream velocity Ve, both of which are unknown at this point. However, we may rearrange Eq. (1.13) using Eq. (1.18) to write the thrust as follows:

si48_e

Solving for the specific fuel consumption yields

si49_e

The energy content of Jet A fuel is given in Table 1.1 as Qf = 43,400 kJ/kg. With Te = 665 K and T0 = 288 K we may use Table B.2 to estimate reasonable values for cp as follows: cp,e = 1.11 kJ/kg K and cp,0 = 1.03 kJ/kg K. Note that the specific heat is not sensitive to the pressure in the temperature range of interest. Then the specific fuel consumption is

si50_e

Converting the energy units in the denominator yields

si51_e

   (1.29)

Eliminating cj from Eqs. (1.28), (1.29) leads to the following result

si52_e    (1.30)

Using the given information that V0 = 0 in the thrust equation (1.27) we find

si53_e

   (1.31)

Substituting for Ve and V0 in Eq. (1.30) yields a fuel-to-air ratio f/a = 0.01495. Then the specific fuel consumption may be found from Eq. (1.28) to be cj = 2.438×10− 5 kg/s/N = 0.08779 kg/h/N. Conversion to English units yields the specific fuel