Compressible Flow with Application to Shocks and Propulsion is part of the series "Mathematics and Physics for Science and Technology", which combines rigorous mathematics with general physical principles to model practical engineering systems with a detailed derivation and interpretation of results. Volume V presents the mathematical theory of partial differential equations and methods of solution satisfying initial and boundary conditions, and includes applications to: acoustic, elastic, water, electromagnetic and other waves; the diffusion of heat, mass and electricity; and their interactions. This is the second book of the volume.
The first book of volume V starts with the classification of partial differential equations and proceeds with similarity methods that apply in general to linear equations with constant coefficients and all derivatives of the same order, such as the Laplace and Biharmonic equations, without and with forcing. The similarity solutions are also applied to Burger's non-linear diffusion equation. First-order linear and quasi-linear partial differential equations with variable coefficients are considered, with application to the representation of conservative/non-conservative, solenoidal/rotational and Beltrami/helical vector fields by one, two or three scalar and/or one vector potential in relation with exact, inexact and non-integrable differentials. The latter appear in the first and second principles of thermodynamics that specify the constitutive and diffusive properties of matter as concerns thermal, mechanical, elastic, flow, electrical, magnetic and chemical phenomena and their interactions.
The book is intended for graduate students and engineers working with mathematical models and can be applied to problems in mechanical, aerospace, electrical and other branches of engineering dealing with advanced technology, and also in the physical sciences and applied mathematics.
Author(s): Luis Manuel Braga da Costa Campos, Luís António Raio Vilela
Series: Mathematics and Physics for Science and Technology
Publisher: CRC Press
Year: 2022
Language: English
Pages: 293
City: Boca Raton
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Table of Contents
List of Notes, Tables, and Diagrams
Preface
Organization and Presentation of the Subject Matter
About the Authors
Acknowledgements
Physical Quantities
Chapter 2: Thermodynamics and Irreversibility
2.5 Equation of State and Thermodynamic Cycles
2.5.1 Work for Distinct Pressure – Volume Relations
2.5.2 Equation of State ( Clapeyron 1834) for an Ideal Gas
2.5.3 Functions of State for an Ideal Gas
2.5.4 Equation of State for a Perfect Gas
2.5.5 Functions of State and Adiabatic Conditions
2.5.6 Thermodynamic Coefficients for an Ideal Gas
2.5.7 Thermodynamic Derivatives for Perfect Gases
2.5.8 Equation of State of First (Second) Order in Entropy (Mass Density)
2.5.9 Tait (1888) Equation of State for Water
2.5.10 Adiabatic Process for a Perfect Gas
2.5.11 Isochoric Process for an Ideal Gas
2.5.12 Isobaric Process (Charles 1787, Gay-Lussac 1802, Dalton 1802)
2.5.13 Isothermal Process (Boyle 1662, Mariotte 1876)
2.5.14 Polytropic Relation and Thermodynamic Cycles
2.5.15 Heat and Work in a Thermodynamic Cycle
2.5.16 Effeciency of the Carnot (1878) Cycle for Any Substance
2.5.17 The Four-Stroke Piston Engine (Otto 1878; Diesel 1913)
2.5.18 Three Thermodynamic Diagrams for the Carnot Cycle
2.5.19 Carnot Cycle for a Perfect Gas
2.5.20 Power of the Carnot Cycle for a Perfect Gas
2.5.21 Work to Transfer Heat from a Cold to a Hot Body
2.5.22 Reversed Engine Cycles: Heat Pump and Refrigerator
2.5.23 Atkinson (1882) Cycle and Variable Valve Timing
2.5.24 Comparison of Efficiencies of Atkinson and Carnot Cycles
2.5.25 Stirling (1916) Air-Independent Engine
2.5.26 Heat and Work in the Stirling Cycle
2.5.27 Power and Efficiency of the Stirling Cycle
2.5.28 More Work in the Carnot than in the Stirling Cycle
2.5.29 Conditions at the Four Corners of the Thermodynamic Diagram
2.6 Adiabatic Compressible Fluid Flow
2.6.1 Equation of Continuity or Mass Conservation
2.6.2 Equation of Momentum for an Inviscid Fluid (Euler 1755)
2.6.3 Equation of State for an Isentropic Flow
2.6.4 Equation of Energy for a Non-Dissipative Fluid
2.6.5 Conservative Equations for Non–Dissipative or Isentropic Flows
2.6.6 Conservation of Vorticity in a Homentropic Flow
2.6.7 Convection Derivative (Lie 1888) of the Density, Velocity and Vorciticy
2.6.8 Conservation of Circulation along Streamlines (Kelvin 1869)
2.6.9 Potential Flow of a Barotropic Fluid (Bernoulli 1738)
2.6.10 Irrotational Incompressible Unsteady Flow
2.6.11 Irrotational Homentropic Unsteady Flow
2.6.12 Enthalpy for Incompressible, Barotropic, Polytropic and Adiabatic Flows
2.6.13 Irrotational Homentropic Flow of a Perfect Gas
2.6.14 Relation between Critical and Stagnation Variables
2.6.15 Local, Stagnation and Critical Mach Numbers
2.6.16 Sound Speed, Temperature, Mass Density and Pressure
2.6.17 Incompressible, Subsonic, Transsonic, Supersonic and Hypersonic Flows
2.6.18 Orbital/Escape Speeds and Ionization/Dissociation
2.6.19 Sound, Heat, Ionization and Dissociation Barriers
2.6.20 Piston-Engined Propeller-Driven Versus Jet-Powered Aircraft
2.6.21 Turbojet, Turboprop, Turboshaft, Turbofan and Propfan
2.6.22 Afterburning, Pulse Jets, Ramjets, Rockets and Ionic Propulsion
2.6.23 Turboramjet, Ramrocket, Turboelectric and Variable-Cycle Engines
2.6.24 Thermodynamic Cycle of the Jet Engine (Barber 1791, Brayton 1930)
2.6.25 Comparison of the Carnot, Stirling and Barber-Brayton Cycles
2.6.26 Four Stages of the Barber-Brayton Cycle
2.6.27 Heat, Work and Efficiency of the Barber-Brayton Cycle
2.6.28 Maximum Work, Power and Thrust of a Turbojet Engine
2.6.29 International Standard Atmosphere (ISA) at Sea Level
2.6.30 Inlet and Exhaust of a Turbojet Engine
2.6.31 Pressure, Density and Temperature in a Turbojet Cycle
2.6.32 Heat Rate and Power Per Unit Mass
2.6.33 Work Per Unit Mass and Power
2.6.34 Exhaust Speed and Thrust of a Turbojet Engine
2.6.35 Rotational, Non-Isentropic, Unsteady Flow
2.7 Vortex Sheet and the Normal Shock (Rankine 1870; Huginot 1887)
2.7.1 Matching Conditions across Flow Discontinuities
2.7.2 Tangential Discontinuity between Two Jets
2.7.3 Compression across a Normal Shock Front
2.7.4 Deviation of Velocity towards an Oblique Shock Front
2.7.5 Conserved and Discontinuous Quantities across a Normal Shock
2.7.6 Entropy Production in a Weak Shock
2.7.7 Compatibility with the Second Principle of Thermodynamics
2.7.8 Adiabatic Versus Shock Pressure – Volume Relations
2.7.9 Thermodynamic Conditions for a Strong Shock
2.7.10 Pressure and Temperature Ratios for an Ideal Gas
2.7.11 Geometric/Arithmetic Mean of Velocities
2.7.12 Speed of Advance of the Shock Front
2.7.13 Arithmetic Larger Than Geometric Average
2.7.14 Comparison of the Adiabatic and Shock Loci
2.7.15 Downstream in Terms of Upstream Shock Conditions
2.7.16 Variations across Shock in Terms of Pressure Ratios
2.7.17 Shock Relations in Terms of Upstream Mach Number
2.7.18 Relation between Upstream and Downstream Mach Numbers
2.7.19 Upstream Supersonic (Subsonic) Stagnation Mach Number
2.7.20 Subsonic Downstream Local, Stagnation and Critical Mach Numbers
2.7.21 Upstream Flow before the Shock
2.7.22 Downstream Flow after the Shock
2.7.23 Differences and Ratios across the Shock Front
2.7.24 Entropy Production in a Strong (Weak) Shock
2.7.25 Conservation or Variation of Stagnation Flow Variables
2.8 Oblique Shock (Busemann 1931) and Adiabatic Turn (Prandtl, Meyer 1908)
2.8.1 Mach (1883) Cone for a Supersonic Velocity
2.8.2 Continuous Tangential Velocity for an Oblique Shock
2.8.3 Jump of Normal Velocities across an Oblique Shock
2.8.4 Supersonic Flow Past Convex and Concave Corners
2.8.5 Incident Velocity within the Mach Cone
2.8.6 Oblique Shock Polar (Busemann 1931) and Folium (Descartes 1638)
2.8.7 Normal Shock and Sound Wave
2.8.8 Weak/Strong Oblique Shocks and Maximum Deflection
2.8.9 Pressure, Mass Density, Velocity and Entropy Jumps
2.8.10 Deflection of the Velocity across an Oblique Shock
2.8.11 Relation between Upstream and Downstream Mach Numbers
2.8.12 Relation between the Angles of Incidence and Deflection
2.8.13 Shock with Large Upstream Mach Number
2.8.14 Circular Polar for a High Mach Number Shock
2.8.15 Relations Across a High Mach Number Shock
2.8.16 Oblique Shock with Small Deflection of Velocity
2.8.17 Incidence Angle Close to the Mach Angle
2.8.18 Relations between Flow Variables and Deflection Angle
2.8.19 Gradual Rotation of a Front along a Smooth Concave Wall
2.8.20 Supersonic Expansion Fan in a Convex Corner (Prandtl – Meyer 1908)
2.8.21 Direct and Inverse Problems of Adiabatic Turn
2.8.22 Non-Linear Second-Order Approximations for Weak Shocks
2.8.23 Speed of Advance of the Shock relative to the Sound Speed
2.8.24 Oblique Shock with Two Upstream Incidence Angles
2.8.25 Upstream Flow Variables and Angle of Deflection
2.8.26 Downstream Angle of Incidence and Flow Variables
2.8.27 Changes across the Oblique Shock Front
2.8.28 Entropy Production and Supersonic Shock Front
2.8.29 Conserved or Decreased Stagnation Flow Variables
2.9 The “Choked” or “Shocked” Nozzle
2.9.1 Unsteady Quasi-One-Dimensional Flow
2.9.2 Steady Flow in a Duct of Varying Cross-Section
2.9.3 Incompressible Flow in a Convergent or Divergent Nozzle
2.9.4 Compressible Flow in a Throated Nozzle (Körting 1878; Laval 1888)
2.9.5 Sonic Condition in a Convergent – Divergent Nozzle
2.9.6 Maximum Volume Flux for Incompressible Nozzle Flow
2.9.7 Maximum Mass Flow Rate for Compressible Nozzle Flow
2.9.8 Nozzle Flow Variables in Terms of Stagnation Values
2.9.9 Nozzle Flow Variables in Terms of Critical Values
2.9.10 Mass Flow Rate and Variables in a “Choked” Nozzle
2.9.11 Perfectly Expanded and Under(Over)Expanded Flows
2.9.12 Location of the Normal Shock in an Overexpanded Nozzle
2.9.13 Shock Location via Pressure or Mass Density
2.9.14 Hydrostatic Equilibrium of Incompressible/Compressible Fluids
2.9.15 Isothermal/Polytropic Atmosphere of a Perfect Gas
2.9.16 Sea Level, Troposphere, Tropopause and Stratosphere
2.9.17 Pressure, Mass Density, Temperature and Sound Speed
2.9.18 Six Flow Regimes in a Throated Nozzle
2.9.19 Inlet and Exhaust of a Jet Engine at Sea Level
2.9.20 Conditions at the Throat of a Convergent – Divergent Nozzle
2.9.21 Approach to Land and Climb after Take-Off
2.9.22 Low(High)-Speed Flight at Sea Level (The Tropopause)
2.9.23 Nozzle Inlet Conditions for Cruise Flight at the Tropopause
2.9.24 Choked Throated Nozzle at the Cruise Condition
2.9.25 Thrust and Power for a Perfect Expansion
2.9.26 Specific Impulse and Thrust of a Rocket Engine
2.9.27 Minimum Data to Describe the Operation of a Rocket
2.9.28 Critical Mach Number at the Nozzle Exit
2.9.29 Nozzle Exit Flow in Vacuo
2.9.30 Total Thrust and Nozzle Exit Area
2.9.31 Physical Conditions in the Combustion Chamber
2.9.32 Flow Conditions at the Throat of the Nozzle
2.9.33 Altitude for Perfectly Expanded Nozzle Flow
2.9.34 Exit Conditions for Overexpanded Nozzle at Atmospheric Pressure
2.9.35 Thrust at Lift-Off for Overexpansion at Sea Level
2.9.36 Inexistence of a Normal Shock Inside or at the Nozzle Exit
2.9.37 Strong Oblique Shock at the Nozzle Exit
2.9.38 Flow Conditions Upstream of the Oblique Shock
2.9.39 Flow Variables Downstream of the Oblique Shock
Note 2.1 The First (Second) Principles of Thermodynamics and Reversible (Irreversible) Processes
Note 2.2 Thickness, Dissipation and Internal Structure Of Shock Waves
Note 2.3 Equations of Non-Dissipative and Dissipative Fluid Mechanics
Note 2.4 Equations of Viscous Single-Phase Flow
Note 2.5 Equation of Energy for a Dissipative Fluid
Note 2.6 Equation of Energy with Viscosity and Heat Conduction
Note 2.7 Equations of Fluid Mechanics for a Perfect Gas
Note 2.8 Adiabatic and Isothermal Sound Speeds
Note 2.9 Conservative Form for a Viscous, Thermally Conducting Flow
Note 2.10 Heat, Mass and Energy Fluxes
Note 2.11 Viscous, Thermally Conducting Two-Phase Flow
Note 2.12 Energy Density and Convective and Diffusive Fluxes
Note 2.13 Entropy Production by Coupled Heat and Mass Diffusion
Note 2.14 Thermal Conductivity, Mass Diffusivity and Cross-Coupling Coefficients
Note 2.15 Velocity, Temperature, Pressure, Mass Density and Mass Fraction
Note 2.16 Steady One-Dimensional Flow with Triple Diffusion
Note 2.17 Linearized Flow of Perfect Gas with Viscous, Thermal and Mass Diffusion
Note 2.18 Temperature Due to Coupled Thermal and Mass Diffusion
Note 2.19 Coupling of Temperature and Mass Fraction Due to Double Diffusion
Note 2.20 Velocity, Mass Density and Pressure in a Flow with Triple Diffusion
Note 2.21 Six Physical Diffusion Mechanisms
2.10 Conclusion
Bibliography
1. Thermodynamics
2. Heat
References
Index
