Non-equilibrium Thermodynamics

The importance of thermodynamics, particularly its Second Principle, to all branches of science in which systems with very large numbers of particles are involved cannot be overstated. This book offers a panoramic view of non-equilibrium thermodynamics. Perhaps the two most attractive aspects of thermodynamic equilibrium are its stability and its independence from the specifics of the particular system involved. Does an equivalent exist for non-equilibrium thermodynamics? Many researchers have tried to describe such stability in the same way that the Second Principle describes the stability of thermodynamic equilibrium - and failed. Most of them invoked either entropy, or its production rate, or some modified version of it. In their efforts, however, those researchers have found a lot of useful stability criteria for far-from-equilibrium states. These criteria usually take the form of variational principles, in terms of the minimization or maximization of some quantity. The aim of this book is to discuss these variational principles by highlighting the role of macroscopic quantities. This book is aimed at a wider audience than those most often exposed to the criteria described, i.e., undergraduates in STEM, as well as the usual interested and invested professionals. Preface 7 Contents 9 List of Main Variables and Acronyms 14 Latin 14 Greek 14 Acronyms 14 1 Looking for the Holy Grail? 21 2 Thermodynamic Equilibrium 27 2.1 Some Fundamental Concepts 27 2.2 A Minimum Amount of Work 28 2.3 Thermodynamic Potentials 29 2.4 The Impact of Magnetic Field 30 2.5 A Symmetry 32 3 Local Thermodynamic Equilibrium 33 3.1 Le Châtelier's Principle 33 3.2 Local Thermodynamic Equilibrium and Le Châtelier's Principle 36 3.3 Some Consequences of Local Thermodynamic Equilibrium 39 3.4 The Role of Gravity 41 3.4.1 Collapse 41 3.4.2 Constant Gravitational Field 45 3.5 Continuous Versus Discontinuous Systems 45 3.6 General Evolution Criterion 46 4 Linear Non-equilibrium Thermodynamics 49 4.1 Discontinuous Systems 49 4.1.1 What is the Linear Non-equilibrium Thermodynamics 49 4.1.2 Onsager's Symmetry 51 4.1.3 Rayleigh's Dissipation Function 52 4.1.4 Minimum Entropy Production in Discontinuous Systems 54 4.1.5 The Least Dissipation Principle 55 4.1.6 The Balance of Entropy in a Copper Wire 56 4.1.7 Wiedemann-Franz' Law 59 4.1.8 Seebeck Effect 60 4.1.9 Peltier Effect 61 4.1.10 Thomson Effect 62 4.1.11 Kelvin's Thermocouple Equations 63 4.1.12 Knudsen Versus Pascal 63 4.2 Entropy Balance in Fluids 66 4.2.1 Dissipationless Fluids 66 4.2.2 Viscous Fluids 68 4.2.3 Joule-Thomson Throttled Expansion 69 4.2.4 Fluids with Electromagnetic Fields 70 4.2.5 Fluids with Many Non-reacting Species 72 4.2.6 Fluids with Many Species Reacting with Each Other 72 4.2.7 Fluids with Gravity 73 4.2.8 Local form of the Entropy Balance 74 4.2.9 Global form of the Entropy Balance: Back to the Copper Wire 75 4.3 Continuous Systems 76 4.3.1 A Slight Abuse of Notation 76 4.3.2 Thermodynamic Forces and Fluxes in Continuous Systems 76 4.3.3 Entropy Production Due to Diffusion 78 4.3.4 Saxen's Laws 79 4.3.5 Fick's Law 80 4.3.6 Soret Effect and Dufour Effect 83 4.3.7 Entropy Production Due to Reactions Among Species 84 4.3.8 Coupling of Diffusion and Reactions 86 4.3.9 Stability Versus the Coupling of Diffusion and Reactions 88 4.3.10 Minimum Entropy Production in Continuous Systems 88 5 Beyond Linear Non-equilibrium Thermodynamics 92 5.1 Gage et al.'s Theorem 92 5.2 Heat Conduction 95 5.2.1 Fourier's Law 95 5.2.2 Stability Versus Fourier's Law 97 5.3 Minimum Entropy Production 97 5.3.1 Joule Heating: Kirchhoff's Principle 97 5.3.2 Electric Arc 99 5.3.3 A Tale of Two Resistors 99 5.3.4 Back to Ohm 102 5.3.5 An Auxiliary Relationship 102 5.3.6 What if Joule Heating is Negligible? 103 5.3.7 Viscosity: Korteweg–Helmholtz' Principle 106 5.3.8 Maximum Economy: Yardangs, Rivers and the Human Blood 109 5.3.9 Porous Media 109 5.3.10 Stability Versus Kirchhoff's and Korteweg–Helmholtz' Principles 112 5.3.11 Convection at Moderate ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper R a) /StPNE pdfmark [/StBMC pdfmark Ra ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 115 5.3.12 Turbulent Flow Between Fixed Parallel Surfaces 116 5.4 Bejan's `Constructal Law' 117 5.5 Zipf's Principle of Least Effort 119 5.5.1 Of Words and Bells 119 5.5.2 City Air Makes You Free 123 5.5.3 Pareto 128 5.5.4 A Tale of Two Cities 128 5.5.5 Travels with Entropy 130 5.6 Maximum Entropy Production 132 5.6.1 Muffled Intuitions 132 5.6.2 Maximum Versus Minimum 135 5.6.3 A Thought Experiment 137 5.6.4 Again, the Copper Wire 141 5.6.5 Two Remarkable Exceptions 142 5.6.6 Heat Conduction in Gases 143 5.6.7 Convection at Large ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper R a) /StPNE pdfmark [/StBMC pdfmark Ra ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 143 5.6.8 The H-Mode 144 5.6.9 Shock Waves 145 5.6.10 Dunes 146 5.6.11 Detonation Versus Shock Waves 146 5.6.12 Solids 147 5.6.13 Earth's Oceans and Atmosphere 148 5.7 Lotka and Odum's Maximum Power Principle 151 5.8 Oscillating Relaxed States 154 5.8.1 Rules of Selection 154 5.8.2 Biwa et al.'s Experiment 155 5.8.3 Meija et al.'s Experiment 156 5.8.4 Hong et al.'s Experiment 158 5.8.5 Flame Quenching 158 5.8.6 Holyst et al.'s Simulations 159 5.8.7 Rauschenbach's Hypothesis 159 5.8.8 Rayleigh's Criterion of Thermoacoustics 160 5.8.9 Rijke's Tube 162 5.8.10 Sondhauss' Tube 166 5.8.11 Welander's Loop 167 5.8.12 Eddington and the Cepheids 170 6 A Room, a Heater and a Window 176 6.1 When Principles Collide 176 6.1.1 The Problem 176 6.1.2 Insufficient Approaches 177 6.1.3 Excess Entropy Production and Dissipative Structures 177 6.1.4 Selective Decay 180 6.1.5 Maximal Entropy 182 6.1.6 Extended Irreversible Thermodynamics 183 6.1.7 Steepest Ascent 183 6.1.8 Second Entropy 184 6.1.9 Information Thermodynamics and MaxEnt 184 6.1.10 Orthogonality Principle 187 6.1.11 Quasi-Thermodynamic Approach 188 6.1.12 Gouy-Stodola's Theorem and Entropy Generation 189 6.1.13 Much Ado for Nothing? 190 6.2 One Principle to Bind Them All? 191 6.2.1 A 1st Necessary Condition for Stability 191 6.2.2 Convection at Moderate ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper R a) /StPNE pdfmark [/StBMC pdfmarkRaps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark, Retrieved 193 6.2.3 Detonation Waves, Retrieved 194 6.2.4 Two Applications of Bejan's Constructal Law 195 6.2.5 Kohler's Principle 197 6.2.6 Entropy Production in a Radiation Field 198 6.2.7 Uniform Temperature: A Reciprocal Problem,... 199 6.2.8 ...Joule Heating,... 200 6.2.9 ...Viscous Heating,... 201 6.2.10 ... And Porous Media 202 6.2.11 A 2nd Necessary Condition for Stability 204 6.2.12 Entropy Production of a Radiating Body 205 6.2.13 No Heater, Two Windows 206 6.2.14 Resistors, Again 208 6.2.15 A 2nd Necessary Condition for Stability—General Form 209 6.2.16 Heat Conduction in Gases, Retrieved 210 6.2.17 Shock Waves and Dunes, Again 211 6.2.18 Back to Entropy Generation 212 6.2.19 Convection, Again 214 6.2.20 Crystal Growth, Retrieved 215 6.2.21 Liesegang and Gelation 215 7 The Garden of Forking Paths 220 Appendix 232 A.1 Proof of the General Evolution Criterion 232 A. 2 Euler-Lagrange Equations 233 A. 3 Lagrange Multipliers 234 A. 4 Proof of Gage et al.'s Theorem 235 Index 238