Lecture Notes on Intermediate Thermodynamics
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LECTURE NOTES ON INTERMEDIATE THERMODYNAMICS Joseph M. Powers Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, Indiana 46556-5637 USA updated 28 January 2019, 3:07pm 2 CC BY-NC-ND. 28 January 2019, J. M. Powers. Contents Preface 7 1 Review 9 2 Cycle analysis 17 2.1 Carnot....................................... 17 2.1.1 Analysis for a calorically perfect ideal gas . 17 2.1.2 Exergy................................... 24 2.2 Rankine ...................................... 28 2.2.1 Classical.................................. 28 2.2.2 Reheat................................... 33 2.2.3 Regeneration ............................... 33 2.2.4 Losses ................................... 37 2.2.5 Cogeneration ............................... 37 2.3 Airstandardcycles ................................ 37 2.4 Brayton ...................................... 38 2.4.1 Classical.................................. 38 2.4.2 Regeneration ............................... 43 2.4.3 EricssonCycle............................... 46 2.4.4 Jetpropulsion............................... 48 2.5 Reciprocating engine power cycles . 53 2.6 Otto ........................................ 55 2.7 Diesel ....................................... 59 2.8 Stirling....................................... 64 2.9 Refrigeration ................................... 65 2.9.1 Vapor-compression ............................ 65 2.9.2 Airstandard................................ 69 3 Gas mixtures 71 3.1 Somegeneralissues................................ 71 3.2 Idealandnon-idealmixtures. 75 3.3 Idealmixturesofidealgases . 75 3.3.1 Daltonmodel ............................... 76 3 4 CONTENTS 3.3.1.1 Binary mixtures . 79 3.3.1.2 Entropy of mixing . 82 3.3.1.3 Mixturesofconstantmassfraction . 85 3.3.2 Summary of properties for the Dalton mixture model . 86 3.3.3 Amagatmodel .............................. 92 3.4 Gas-vapormixtures................................ 92 3.4.1 Firstlaw.................................. 95 3.4.2 Adiabaticsaturation ........................... 97 3.4.3 Wet-bulbanddry-bulbtemperatures . 99 3.4.4 Psychrometricchart ........................... 99 4 Mathematical foundations of thermodynamics 101 4.1 Exact differentials and state functions . 101 4.2 Twoindependentvariables............................ 106 4.3 Legendretransformations. 109 4.4 Heatcapacity ................................... 114 4.5 VanderWaalsgas ................................ 118 4.6 Redlich-Kwonggas ................................ 124 4.7 Compressibility and generalized charts . 125 4.8 Mixtures with variable composition . 126 4.9 Partialmolarproperties ............................. 129 4.9.1 Homogeneousfunctions. 129 4.9.2 Gibbsfreeenergy ............................. 129 4.9.3 Otherproperties ............................. 130 4.9.4 Relation between mixture and partial molar properties . 132 4.10 Irreversible entropy production in a closed system . ...... 133 4.11 Equilibrium in a two-component system . 136 4.11.1 Phase equilibrium . 136 4.11.2 Chemical equilibrium: introduction . 139 4.11.2.1 Isothermal, isochoric system . 140 4.11.2.2 Isothermal, isobaric system . 145 4.11.3 Equilibrium condition . 147 5 Thermochemistry of a single reaction 149 5.1 Molecularmass .................................. 149 5.2 Stoichiometry ................................... 151 5.2.1 Generaldevelopment . .. .. 151 5.2.2 Fuel-airmixtures ............................. 159 5.3 First law analysis of reacting systems . 161 5.3.1 Enthalpyofformation .......................... 161 5.3.2 Enthalpy and internal energy of combustion . 163 5.3.3 Adiabatic flame temperature in isochoric stoichiometric systems . 164 CC BY-NC-ND. 28 January 2019, J. M. Powers. CONTENTS 5 5.3.3.1 Undiluted, cold mixture . 164 5.3.3.2 Dilute, cold mixture . 165 5.3.3.3 Dilute, preheated mixture . 167 5.3.3.4 Dilute, preheated mixture with minor species . 168 5.4 Chemical equilibrium . 169 5.5 Chemical kinetics of a single isothermal reaction . 175 5.5.1 Isochoricsystems ............................. 175 5.5.2 Isobaricsystems.............................. 183 5.6 Some conservation and evolution equations . 189 5.6.1 Total mass conservation: isochoric reaction . 189 5.6.2 Element mass conservation: isochoric reaction . 190 5.6.3 Energy conservation: adiabatic, isochoric reaction . 191 5.6.4 Energy conservation: adiabatic, isobaric reaction . 193 5.6.5 Entropy evolution: Clausius-Duhem relation . 196 5.7 Simple one-step kinetics . 199 6 Thermochemistry of multiple reactions 203 6.1 Summary of multiple reaction extensions . 203 6.2 Equilibrium conditions . 210 6.2.1 Minimization of G via Lagrange multipliers . 210 6.2.2 Equilibration of all reactions . 216 6.3 Concise reaction rate law formulations . 218 6.3.1 Reaction dominant: J (N L).................... 219 ≥ − 6.3.2 Species dominant: J < (N L)..................... 220 − 6.4 Adiabatic, isochoric kinetics . 220 7 Kinetics in some more detail 223 7.1 Isothermal, isochoric kinetics . 226 7.1.1 O O dissociation............................ 226 − 2 7.1.1.1 Pair of irreversible reactions . 226 7.1.1.1.1 Mathematical model . 226 7.1.1.1.2 Example calculation . 232 7.1.1.1.2.1 Species concentration versus time . 233 7.1.1.1.2.2 Pressure versus time . 234 7.1.1.1.2.3 Dynamical system form . 236 7.1.1.1.3 Effectoftemperature. 240 7.1.1.2 Single reversible reaction . 241 7.1.1.2.1 Mathematical model . 241 7.1.1.2.1.1 Kinetics . 241 7.1.1.2.1.2 Thermodynamics . 243 7.1.1.2.2 Example calculation . 244 7.1.2 Zel’dovich mechanism of NO production ................ 247 CC BY-NC-ND. 28 January 2019, J. M. Powers. 6 CONTENTS 7.1.2.1 Mathematicalmodel . 247 7.1.2.1.1 Standardmodelform. 248 7.1.2.1.2 Reducedform..................... 249 7.1.2.1.3 Example calculation . 252 7.1.2.2 Stiffness, time scales, and numerics . 258 7.1.2.2.1 Effectoftemperature. 258 7.1.2.2.2 Effect of initial pressure . 261 7.1.2.2.3 Stiffness and numerics . 263 7.2 Adiabatic, isochoric kinetics . 264 7.2.1 Thermalexplosiontheory . 265 7.2.1.1 One-step reversible kinetics . 265 7.2.1.2 First law of thermodynamics . 266 7.2.1.3 Dimensionless form . 269 7.2.1.4 Example calculation . 270 7.2.1.5 High activation energy asymptotics . 272 7.2.2 Detailed H O N kinetics ..................... 276 2 − 2 − 2 Bibliography 281 CC BY-NC-ND. 28 January 2019, J. M. Powers. Preface These are lecture notes for AME 50531 Intermediate Thermodynamics (AME 54531 for students in our London Programme), the second of two undergraduate courses in thermody- namics taught in the Department of Aerospace and Mechanical Engineering at the Univer- sity of Notre Dame. Most of the students in this course are juniors or seniors majoring in aerospace or mechanical engineering. The objective of the course is to survey both practical and theoretical problems in classical thermodynamics. The notes draw heavily on the text specified for the course, Borgnakke and Sonntag’s (BS) Fundamentals of Thermodynamics, Eighth Edition, John Wiley, New York, 2013, es- pecially Chapters 8-14. In general the nomenclature of BS is used, and much of the notes follow a similar structure as that text. In addition, Abbott and van Ness’s Thermodynamics, McGraw-Hill, New York, 1972, has been used to guide some of the mathematical develop- ments. Many example problems have been directly taken from BS and other texts; specific citations are given where they arise. Many of the unique aspects of these notes, which focus on thermodynamics of reactive gases with detailed finite rate kinetics, have arisen due to the author’s involvement in research supported by the National Science Foundation under Grant No. CBET-0650843. The author is grateful for the support. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. These notes emphasize both problem-solving as well as some rigorous undergraduate-level development of the underlying classical theory. It should also be remembered that practice is essential to the learning process; the student would do well to apply the techniques presented here by working as many problems as possible. The notes, along with information on the course, can be found on the world wide web at https://www3.nd.edu/ powers/ame.50531. At this stage, anyone is free to make copies for their own use. I would∼ be happy to hear from you about suggestions for improvement. Joseph M. Powers powers@nd.edu https://www3.nd.edu/ powers ∼ Notre Dame, Indiana; USA CC BY: $\ = 28 January 2019 The content of this book is licensed under Creative Commons Attribution-Noncommercial-No Derivative Works 3.0. 7 8 CONTENTS CC BY-NC-ND. 28 January 2019, J. M. Powers. Chapter 1 Review Review BS, Chapters 1-7. Here is a brief review of some standard concepts from a first course in thermodynamics. property: characterizes the thermodynamics state of the system • – extensive: proportional to system’s extent, upper case variables V : total volume, SI-based units are m3, ∗ U: total internal energy, SI-based units are kJ, ∗ H: total enthalpy, SI-based units are kJ, ∗ S: total entropy, SI-based units are kJ/K. ∗ – intensive: independent of system’s extent, lower case variables (exceptions are temperature T and pressure P , which are intensive). Intensive properties can be on a mass basis or a molar basis: v: specific volume, SI-based units are m3/kg, ∗ u: specific internal energy, SI-based units are kJ/kg, ∗ h: specific enthalpy, SI-based units are kJ/kg, ∗ s: specific entropy, SI-based