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Miloslav Pekař Ivan Samohýl The Thermodynamics of Linear Fluids and Fluid Mixtures The Thermodynamics of Linear Fluids and Fluid Mixtures Miloslav Pekaˇr Ivan Samohy´l • The Thermodynamics of Linear Fluids and Fluid Mixtures 123 Miloslav Pekarˇ IvanSamohy´l Faculty ofChemistry Instituteof Chemical Technology BrnoUniversityof Technology Prague 6 Brno Czech Republic Czech Republic ISBN 978-3-319-02513-1 ISBN 978-3-319-02514-8 (eBook) DOI 10.1007/978-3-319-02514-8 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013950364 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. 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While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface During the last 50 years our understanding of thermodynamics has achieved considerable progress, particularly in relation to irreversible or non-equilibrium phenomena. The principal approaches were reviewed in the book by [1] and are brieflysummarizedatthebeginningofChap.1.Animportantrolehasbeenplayed by phenomenological thermodynamics (or nonlinear thermomechanics) called rationalthermodynamics,asdevelopedbyTruesdellandothers.Ithasbeenshown that similarly as in other classical disciplines (e.g. mechanics or electromagne- tism),thethermodynamicsmaybedescribedthroughgeneralpostulatesvalidinall disciplines (e.g. First Law of thermodynamics (balance of energy), balance of mass, etc., or Second law of thermodynamics). Then special models of materials can be studied which are formulated by constitutive equations (e.g. uniform classical thermodynamics, mixtures with transport phenomena). As a result it has beennotnecessarytobeconfinedtoequilibriumphenomenaonly,andithasbeen possible also to describe the non-equilibrium phenomena, at least in principle. This book aims at providing consistent and integrated thermodynamic descriptionofchemicallyreactingsystemswhichareoftenencounteredinpractice based on methodology developed within the framework of continuum, rational thermodynamics.Becauseoftheextentandunderstandingofsuchabroadgoalwe limit the discussion to one phase, mostly fluid (gas or liquid) systems, pure sub- stanceor(evenchemicallyreacting)fluidmixtures,whichseemtocoverthemost important cases of applications. Although such modern thermodynamics is a mathematical theory established mainly by mathematicians, this book is focused onnon-mathematicians—physicists,chemistsandengineers.Theyareusuallyand typically studying systems where thermodynamic and transport phenomena and chemical reactions are running together (e.g. processes in industrial chemical reactors) using properties of these phenomena obtained from their separate and independent research. In other words, their approach is driven by (mostly tacit) assumption, that these separate knowledges are often also valid in such complex systems. For example, the oldest theory of non-equilibrium thermodynamics is based on a hypothesis called the ‘‘local equilibrium’’, i.e. the validity of classical thermodynamicrelationsinsystemswith(space)gradientsisassumedeventhough they were obtained in a uniform equilibrium system without such gradients. In this book we discuss models of traditional and industrially important situa- tions in chemically reacting mixtures, which confirm (in accordance with v vi Preface experience), the validity of such a hypothesis, together with its limits. All the methodology has clearly and explicitly given all starting axioms, all applied assumptions and simplifications; therefore the range of its validity as well as of derived models can be easily estimated and tested. Specifically, we show that rational thermodynamics applied to a model of ‘‘chemically reacting mixture of fluids with linear transport properties’’ (the linear fluids in brief) fulfils these ‘‘separated’’ properties, even in complex systems, where all these processes take place together—thermodynamic relations are the same as in classical equilibrium thermodynamics (i.e. the ‘‘local equilibrium’’ is proved to be valid)—classical linear transport laws are valid (e.g. Fick or Fourier laws)—classical chemical kinetics is valid (typically, the reaction rate depends (nonlinearly) on concentrations and temperature only), i.e. the mass action kinetics is proved as well. Thus, from the point of view of modern phenomenological thermodynamics, the current outputs of classical equilibrium thermodynamics (e.g. the description ofthermochemistryofmixtures)andthetasksofirreversiblethermodynamics,like thedescriptionoflineartransportphenomenaandnonlinearchemicalkinetics,are validmuchmoregenerally,e.g.evenwhenalltheseprocessesrunsimultaneously. As we noted above, these properties are not expected to be valid in any material models:insomemodelsthelocalequilibriummaynotbevalid,reactionratesmay depend not only on concentrations and temperature, etc. Webelievethatthephysicalcontentandinnerstructureofthistheoryisnotless interestingthanitsmathematicalformalismandthereforeinthisbookwestressthe physical meaning omitting mathematical technicalities wherever possible and try tobeconsistentandself-contained.Nevertheless,familiaritywithcalculus,vectors and tensors at introductory level, at least, is supposed; as suitable and concise study references the books by [2] or [3] can be recommended. Thebookisdivided intofour chapterscontaining27sections intotal.Thetext startswithgeneralconceptsanddevelopsandsimplifiesthemprogressivelytothe model of mixture of linear fluids for which explicit formulations can be derived and which is the model closest to common experience in chemistry and related fields. Chapter 1 states the general framework—besides others, it introduces generalvariables,explainstheconceptionofprimitivevariablesandstatesthetwo basicthermodynamiclawsinaverygeneralformastherelationshipsbetweenheat and work. In this chapter, we demonstrate that general formulation enables a construction (or the proof of existence) of quantities which are specific and basic for thermodynamics—internal energy, entropy (even non-equilibrium) and abso- lute temperature. Although in this generality the laws, variables or quantities are not directly used in the subsequent development Chap. 1 justifies their existence and applicability also in non-equilibrium states. Chapters2through4developmoderncontinuum(rational)thermodynamicsin its standard and most elaborated form. The most simple example or model— uniform systems (without space gradients of properties)—is discussed in Chap. 2 whichalsoservesasabasicandrelativelysimpleintroductiontothemethodology (Sects. 2.1 and 2.2 inthis chapter) which is not complicated by the description of spatial distribution. Four examples—models of uniform materials with increasing Preface vii complexityareusedtoexplainvariousaspectsofthemethodologyandtostressthe significance of constitutive equations and application of the Second Law in this approach.Section2.3givesbasicinformationaboutthelimitations ofthevalidity of material models (constitutive equations). Section 2.4 shows how chemical reactionsandtheirkineticsenterintothemethodologyandconstitutiveequations, whereasSect.2.5illustratesthedescriptionofequilibrium between phasesbyour approach. Chapter 2 thus gives an explanation of principles of how the non- equilibrium is treated, without complicating it by spatial description, and dem- onstrates how equilibrium is naturally incorporated as a final state of non-equilibrium development. Chapter 3 adds also the description of spatial distribution (gradients). Only single fluid is considered for the sake of simplicity and preparation of the basics for the subsequent treatment of mixtures. Mathematics necessary for the spatial description is introduced in Sect. 3.1. Section 3.2 in the same chapter stresses the importance of the referential frame (coordinate system) and its change in the mathematical description. Sections 3.3–3.6 shows the development of final materialmodel(of a fluid) within ourthermodynamic framework, consistent with general laws (balances) as well as with thermodynamic principles (the First and Second Laws and the principles of rational thermodynamics). The results of this development are simplified in Sect. 3.7 to the model of (single) fluid with linear transport properties. Sections 3.6 and 3.7 also show that the local equilibrium hypothesisisprovedforfluidmodels.ThelinearfluidmodelisusedinSect.3.8to demonstrate how the stability of equilibrium is analysed in our approach. The exposition culminates in Chap. 4 dealing with the mixture of fluids with linear transport properties and representing thus the most important part of this book. Section 4.1 explains the difference between the description of single-component and multi-component systems (mixtures). In Sects. 4.2–4.4 the basic principles and laws, presented in preceding chapters in the single-component version, are appropriately modified to mixtures and prepared to be used for the derivationofthermodynamicallyconsistentmodelsofamixture.Thepropertiesof mixturesaredescribedusingpartialquantitiessystematically.Specialattentionis paid to the accessibility of partial quantities from experiments—Sect. 4.4 also presents the special property related to this aspect, the mixture invariance. The derivationofconsistentmixturemodelisexemplifiedinSect.4.5inthemixtureof chemically reacting fluids with linear transport properties. In Sect. 4.6 the whole classical chemical thermodynamics is derived on the basis of this model and the validity of its equations also in the linear fluid mixture out of equilibrium is demonstrated. Of course, the local equilibrium is completely proved in this mixture model, again. Section 4.7 analyses the equilibrium in the mixture of linear fluidsindetailincludingitsstability.In Sect.4.8thelinearfluidmodelismodified to several yet simpler material models which reflect the systems analysed tradi- tionally in classical chemical thermodynamics. Among others, this enables to analyse the applicability of traditional instruments of chemical thermodynamics like activity or fugacity under non-equilibrium conditions. The consequences of the presented thermodynamic method on the rates of chemical reactions, i.e. on viii Preface chemical kinetics, are given in Sect. 4.9. Particularly, we show how the kinetic massactionlawnaturallyemergesfromthermodynamicconsiderationsandhowit can be generalised to non-ideal fluid mixtures. The last section, Sect. 4.10, elab- oratesonthetransportproperties(viscosityeffects,diffusion,heatconductionand corresponding cross effects) and transforms the transport equations derived in Sect. 4.5 to more practical forms. Several traditional models or phenomena then follow as natural consequences of presented thermodynamic methodology—e.g. Fick or Fourier laws, Sorret and Dufour effects, as well as various phenomenological coefficients which are in classical irreversible thermodynamics introduced a priori. Almosteverysectioncloseswithabriefsummarygivinganoutlineofthemost important information or equations and of what should be learned in the section. Thesummariesshouldserveasthestudyaidsandcanbereadalsobeforestudying the corresponding section. Some additional thermodynamic and particularly mathematical instruments are collected in Appendices 1–5. We want to express our gratitude to Prof. K. R. Rajagopal for the initiative to write this book and for stimulating discussions. Our thanks go to Drs. Miroslav Šilhavy´, Willy Pabst and Pavel Hrma. We thank also the representatives of the publisherforthepatienceandcaredevotedtothismanuscript.Theauthorswillbe grateful for any criticism concerning this book. August 2013 Miloslav Pekarˇ Ivan Samohy´l References 1. Lebon G., Jou D., Casas-Vázquez: Understanding Non-equilibrium Thermodynamics. Springer,Berlin-Heidelberg(2008) 2. Chadwick,P.:ContinuumMechanics.GeorgeAllenandUnwin,London(1976) 3. Aris, R.: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall, EnglewoodCliffs1962,reprintedbyDoverPubl.,NewYork(1989) Contents 1 Thermodynamics and Its Concepts in Nonequilibrium. . . . . . . . . . 1 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 General Concepts and Framework, Thermodynamic Systems, Processes, and the Universe. . . . . . . . . . . . . . . . . . . 8 1.3 The First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . 14 1.4 The Second Law of Thermodynamics. . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 Thermodynamics of Uniform Systems. . . . . . . . . . . . . . . . . . . . . . 35 2.1 Energy Balance, Entropy Inequality and Constitutive Principles, and Equations in Uniform Systems. . . . . . . . . . . . . 35 2.2 Constitutive Equations of Uniform Fluids. . . . . . . . . . . . . . . . 41 2.3 Level of Description and Internal Variables . . . . . . . . . . . . . . 51 2.4 Uniform Reacting Mixture in Closed System . . . . . . . . . . . . . 53 2.5 Phase Equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3 Continuum Thermodynamics of Single Fluid . . . . . . . . . . . . . . . . 67 3.1 Kinematics of Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Change of Frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3 Balances of Mass, Momentum, and Moment of Momentum . . . 86 3.4 Energy Balance and Entropy Inequality . . . . . . . . . . . . . . . . . 94 3.5 Constitutive Principles and Constitutive Equations for the Single Substance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.6 Principle of Admissibility—Constitutive Equations of Single Material. Fluid with Viscosity and Heat Conduction . . . . . . . . 104 3.7 Fluid with Linear Transport Properties. . . . . . . . . . . . . . . . . . 117 3.8 Equilibrium Processes in Linear Fluid . . . . . . . . . . . . . . . . . . 123 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 ix x Contents 4 Continuum Thermodynamics of Mixture of Linear Fluids. . . . . . . 143 4.1 Principles of Mixture Theory. . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2 Balances of Mass and Stoichiometry of Chemical Reactions. . . 147 4.3 Balances of Momentum and Moment of Momentum in Reacting Mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.4 Balance of Energy and Entropy Inequality in Reacting Mixture: Mixture Invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.5 Chemically Reacting Mixture of Fluids with Linear Transport Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.6 Thermodynamic Relationships in the Linear Fluid Mixture. . . . 184 4.7 Equilibrium in the Linear Fluid Mixture. . . . . . . . . . . . . . . . . 208 4.8 Special Cases of Linear Fluid Mixtures. Chemical Potentials and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.9 Chemical Reactions and their Kinetics . . . . . . . . . . . . . . . . . . 247 4.10 Transport Phenomena in the Linear Fluid Mixture. . . . . . . . . . 257 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Appendix A.1 Empirical Temperature, Ideal Gas, and Carnot Cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Appendix A.2 Representations of Linear Isotropic Functions. . . . . 283 Appendix A.3 Concave Functions. . . . . . . . . . . . . . . . . . . . . . . . . 293 Appendix A.4 Nonorthogonal Bases . . . . . . . . . . . . . . . . . . . . . . . 295 Appendix A.5 Inequalities, Theorem of I-Shih Liu . . . . . . . . . . . . 296