Table Of ContentCompressible Flow with
Applications to Engines,
Shocks and Nozzles
Mathematics and Physics for Science and Technology
Series Editor: L.M.B.C. Campos
former Director of the Center for Aeronautical
and Space Science and Technology
Lisbon University
Volumes in the series:
TOPIC A – THEORY OF FUNCTIONS AND POTENTIAL PROBLEMS
Volume I (Book 1) – Complex Analysis with Applications to Flows and Fields
L.M.B.C. Campos
Volume II (Book 2) – Elementary Transcendentals with Applications to Solids
and Fluids
L.M.B.C. Campos
Volume III (Book 3) – Generalized Calculus with Applications to Matter and
Forces
L.M.B.C. Campos
TOPIC B – BOUNDARY AND INITIAL-VALUE PROBLEMS
Volume IV – Ordinary Differential Equations with Applications to Trajectories
and Oscillations
L.M.B.C. Campos
Book 4 – Linear Differential Equations and Oscillators
L.M.B.C. Campos
Book 5 – Non-Linear Differential Equations and Dynamical Systems
L.M.B.C. Campos
Book 6 – Higher-Order Differential Equations and Elasticity
L.M.B.C. Campos
Book 7 – Simultaneous Differential Equations and Multi-Dimensional
Vibrations
L.M.B.C. Campos
Book 8 – Singular Differential Equations and Special Functions
L.M.B.C. Campos
Book 9 – Classification and Examples of Differential Equations and their
Applications
L.M.B.C. Campos
Volume V – Partial Differential Equations with Applications to Waves and
Diffusion
L.M.B.C. Campos & L.A.R. Vilela
Book 10 – Vector Fields with Applications to Thermodynamics and
Irreversibility
L.M.B.C. Campos & L.A.R. Vilela
Book 11 – Compressible Flow with Applications to Engines, Shocks and Nozzles
L.M.B.C. Campos & L.A.R. Vilela
For more information about this series, please visit: http://www.crcpress.com/
Mathematics-and-Physics-for-Science-and-Technology/book-series/CRCMATPHYSCI
Compressible Flow with
Applications to Engines,
Shocks and Nozzles
Mathematics and Physics
for Science and Technology
Luis Manuel Braga da Costa Campos
and
Luís António Raio Vilela
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2023 Luis Manuel Braga da Costa Campos and Luís António Raio Vilela
CRC Press is an imprint of Taylor & Francis Group, LLC
The right of Luis Manuel Braga da Costa Campos and Luís António Raio Vilela to be identified as
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Library of Congress Cataloging‑in‑Publication Data
Names: Campos, Luis Manuel Braga da Costa, author. | Vilela, Luís António
Raio, author.
Title: Partial differentials with applications to thermodynamics and compressible flow / Luis Manuel
Braga da Costa Campos and Luís António Raio Vilela.
Description: First edition. | Boca Raton, FL : CRC Press, 2021. | Series:
Mathematics and physics for science and technology | Includes
bibliographical references and index. | Summary: “This book is part of the series “Mathematics and
Physics Applied to Science and Technology.” It combines rigorous mathematics with general physical
principles to model practical engineering systems with a detailed derivation and interpretation of
results”-- Provided by publisher.
Identifiers: LCCN 2021005670 (print) | LCCN 2021005671 (ebook) | ISBN
9781032029870 (v. 1 ; hardback) | ISBN 9781032030838 (v. 1 ; paperback)
| ISBN 9781032030838 (v. 2 ; hardback) | ISBN 9781032030814 (v. 2 ; paperback) | ISBN
9781003186595 (v. 1 ; ebook) | ISBN 9781003186588 (v. 2 ; ebook)
Subjects: LCSH: Thermodynamics. | Fluid dynamics. | Gases,
Compressed--Mathematical models. | Differential equations, Partial.
Classification: LCC QC168 .C29 2021 (print) | LCC QC168 (ebook) | DDC
515/.353--dc23
LC record available at https://lccn.loc.gov/2021005670
LC ebook record available at https://lccn.loc.gov/2021005671
ISBN: 978-1-032-02989-4 (hbk)
ISBN: 978-1-032-03081-4 (pbk)
ISBN: 978-1-003-18658-8 (ebk)
DOI: 10.1201/9781003186588
Typeset in Times
by SPi Technologies India Pvt Ltd (Straive)
to Lev Davidovich Landau
Contents
List of Notes, Tables, and Diagrams .......................................................................xiii
Preface ......................................................................................................................xv
About the Authors ..................................................................................................xvii
Acknowledgements .................................................................................................xix
Physical Quantities ..................................................................................................xxi
Chapter 2 Thermodynamics and Irreversibility ..................................................233
2.5 Equation of State and Thermodynamic Cycles .......................233
2.5.1 Work for Distinct Pressure – Volume Relations ........233
2.5.2 Equation of State (Clapeyron 1834) for an
Ideal Gas....................................................................236
2.5.3 Functions of State for an Ideal Gas ...........................238
2.5.4 Equation of State for a Perfect Gas ...........................239
2.5.5 Functions of State and Adiabatic Conditions ............241
2.5.6 Thermodynamic Coefficients for an Ideal Gas .........245
2.5.7 Thermodynamic Derivatives for Perfect Gases .........247
2.5.8 Equation of State of First (Second) Order in
Entropy (Mass Density) ............................................248
2.5.9 Tait (1888) Equation of State for Water ....................249
2.5.10 Adiabatic Process for a Perfect Gas ..........................251
2.5.11 Isochoric Process for an Ideal Gas ............................252
2.5.12 Isobaric Process (Charles 1787, Gay-Lussac
1802, Dalton 1802)....................................................252
2.5.13 Isothermal Process (Boyle 1662, Mariotte 1876) .....253
2.5.14 Polytropic Relation and Thermodynamic Cycles .....253
2.5.15 Heat and Work in a Thermodynamic Cycle ..............254
2.5.16 Efficiency of the Carnot (1878) Cycle for Any
Substance...................................................................255
2.5.17 The Four-Stroke Piston Engine (Otto 1878;
Diesel 1913) ..............................................................258
2.5.18 Three Thermodynamic Diagrams for the Carnot
Cycle .........................................................................260
2.5.19 Carnot Cycle for a Perfect Gas..................................260
2.5.20 Power of the Carnot Cycle for a Perfect Gas ............262
2.5.21 Work to Transfer Heat from a Cold to a Hot Body .....264
2.5.22 Reversed Engine Cycles: Heat Pump and
Refrigerator ...............................................................265
2.5.23 Atkinson (1882) Cycle and Variable Valve Timing ...266
2.5.24 Comparison of Efficiencies of Atkinson and
Carnot Cycles ............................................................270
2.5.25 Stirling (1916) Air-Independent Engine ....................271
vii
viii Contents
2.5.26 Heat and Work in the Stirling Cycle .........................274
2.5.27 Power and Efficiency of the Stirling Cycle ...............274
2.5.28 More Work in the Carnot than in the Stirling
Cycle .........................................................................276
2.5.29 Conditions at the Four Corners of the
Thermodynamic Diagram .........................................276
2.6 Adiabatic Compressible Fluid Flow ........................................279
2.6.1 Equation of Continuity or Mass Conservation ..........280
2.6.2 Equation of Momentum for an Inviscid Fluid
(Euler 1755) ..............................................................281
2.6.3 Equation of State for an Isentropic Flow ..................282
2.6.4 Equation of Energy for a Non-Dissipative Fluid.......283
2.6.5 Conservative Equations for Non-Dissipative or
Isentropic Flows ........................................................285
2.6.6 Conservation of Vorticity in a Homentropic
Flow ...........................................................................287
2.6.7 Convection Derivative (Lie 1888) of the Density,
Velocity and Vorticity ................................................289
2.6.8 Conservation of Circulation along Streamlines
(Kelvin 1869) ............................................................290
2.6.9 Potential Flow of a Barotropic Fluid (Bernoulli
1738) .........................................................................291
2.6.10 Irrotational Incompressible Unsteady Flow ..............292
2.6.11 Irrotational Homentropic Unsteady Flow .................293
2.6.12 Enthalpy for Incompressible, Barotropic,
Polytropic and Adiabatic Flows ................................293
2.6.13 Irrotational Homentropic Flow of a Perfect Gas .......295
2.6.14 Relation between Critical and Stagnation
Variables ....................................................................296
2.6.15 Local, Stagnation and Critical Mach Numbers .........297
2.6.16 Sound Speed, Temperature, Mass Density and
Pressure .....................................................................298
2.6.17 Incompressible, Subsonic, Transsonic,
Supersonic and Hypersonic Flows ............................300
2.6.18 Orbital/Escape Speeds and
Ionization/Dissociation ............................................302
2.6.19 Sound, Heat, Ionization and Dissociation
Barriers ......................................................................303
2.6.20 Piston-Engined Propeller-Driven versus Jet-
Powered Aircraft .......................................................304
2.6.21 Turbojet, Turboprop, Turboshaft, Turbofan and
Propfan ......................................................................306
2.6.22 Afterburning, Pulse Jets, Ramjets, Rockets and
Ionic Propulsion ........................................................309
2.6.23 Turboramjet, Ramrocket, Turboelectric and
Variable-Cycle Engines .............................................312
Contents ix
2.6.24 Thermodynamic Cycle of the Jet Engine (Barber
1791, Brayton 1930)..................................................316
2.6.25 Comparison of the Carnot, Stirling and Barber-
Brayton Cycles ..........................................................318
2.6.26 Four Stages of the Barber-Brayton Cycle .................320
2.6.27 Heat, Work and Efficiency of the Barber-Brayton
Cycle .........................................................................321
2.6.28 Maximum Work, Power and Thrust of a Turbojet
Engine .......................................................................322
2.6.29 International Standard Atmosphere (ISA) at Sea
Level ..........................................................................323
2.6.30 Inlet and Exhaust of a Turbojet Engine .....................325
2.6.31 Pressure, Density and Temperature in a Turbojet
Cycle .........................................................................326
2.6.32 Heat Rate and Power per Unit Mass .........................328
2.6.33 Work per Unit Mass and Power ................................328
2.6.34 Exhaust Speed and Thrust of a Turbojet Engine .......330
2.6.35 Rotational, Non-Isentropic, Unsteady Flow ..............331
2.7 Vortex Sheet and the Normal Shock (Rankine 1870;
Huginot 1887) .........................................................................332
2.7.1 Matching Conditions across Flow
Discontinuities...........................................................333
2.7.2 Tangential Discontinuity between Two Jets ..............334
2.7.3 Compression across a Normal Shock Front ..............336
2.7.4 Deviation of Velocity towards an Oblique
Shock Front ...............................................................337
2.7.5 Conserved and Discontinuous Quantities across a
Normal Shock............................................................338
2.7.6 Entropy Production in a Weak Shock ........................340
2.7.7 Compatibility with the Second Principle of
Thermodynamics .......................................................341
2.7.8 Adiabatic versus Shock Pressure – Volume
Relations ....................................................................342
2.7.9 Thermodynamic Conditions for a Strong Shock .......344
2.7.10 Pressure and Temperature Ratios for an Ideal Gas ...346
2.7.11 Geometric/Arithmetic Mean of Velocities ................347
2.7.12 Speed of Advance of the Shock Front .......................348
2.7.13 Arithmetic Larger Than Geometric Average .............349
2.7.14 Comparison of the Adiabatic and Shock Loci...........351
2.7.15 Downstream in Terms of Upstream Shock
Conditions .................................................................353
2.7.16 Variations across Shock in Terms of Pressure
Ratios.........................................................................354
2.7.17 Shock Relations in Terms of Upstream Mach
Number ......................................................................356
2.7.18 Relation between Upstream and Downstream
Mach Numbers ..........................................................357
