An Extension to Endoreversible Thermodynamics for Multi-Extensity Fluxes and Chemical Reaction Processes
Total Page：16
File Type：pdf, Size：1020Kb
An Extension to Endoreversible Thermodynamics for Multi-Extensity Fluxes and Chemical Reaction Processes Von der Fakult¨at fur¨ Naturwissenschaften der Technischen Universit¨at Chemnitz genehmigte Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt von Katharina Wagner, M. Sc. geboren am 25.10.1984 in Frankenberg eingereicht am 30. April 2014 Gutachter: Prof. Dr. Karl Heinz Hoffmann Prof. Dr. Bjarne Andresen Tag der Verteidigung: 20. Juni 2014 2 Bibliographische Beschreibung Wagner, Katharina Extensions to Endoreversible Thermodynamics for Multi-Extensity Fluxes and Chemical Reaction Processes Technische Universit¨at Chemnitz, Fakult¨at fur¨ Naturwissenschaften Dissertation (in englischer Sprache), 2014 102 Seiten, 28 Abbildungen, 82 Literaturzitate Referat In dieser Arbeit erweitere ich den Formalismus der endoreversiblen Thermodynamik, um Flusse¨ mit mehr als einer extensiven Gr¨oße sowie chemische Reaktionsprozesse modellieren zu k¨onnen. Mit Hilfe dieser Erweiterungen er¨offnen sich zahlreiche neue Anwendungsm¨oglichkeiten fur¨ endoreversible Modelle. Flusse¨ mit mehreren extensi- ven Gr¨oßen sind fur¨ die Betrachtung von Massestr¨omen ebenso n¨otig wie fur¨ Prozesse, bei denen sowohl Volumen als auch Entropie zwischen zwei Teilsystem ausgetauscht werden. Fur¨ sowohl reversibel wie auch irreversibel gefuhrte¨ chemische Reaktions- prozesse wird ein neues Teilsystem { der Reaktor\ { vorgestellt, welches sich ¨ahnlich " wie endoreversible Maschinen durch Bilanzgleichungen auszeichnet. Der Unterschied zu den Maschinen besteht in den Produktions- bzw. Vernichtungstermen in den Teil- chenzahlbilanzen sowie der m¨oglichen Entropieproduktion innerhalb des Reaktors. Beide Erweiterungen finden dann in einem endoreversiblen Modell einer Brennstoff- zelle Anwendung. Dabei werden Flusse¨ mehrerer gekoppelter Extensit¨aten fur¨ den Zustrom von Wasserstoff und Sauerstoff sowie fur¨ den Protonentransport durch die Elektrolytmembran ben¨otigt. Chemische Reaktionen treten in der Anode und Katho- de der Brennstoffzelle auf. Diese werden mit dem neu eingefuhrten¨ Teilsystem, dem Reaktor, eingebunden. Mit Hilfe des Modells werden dann Wirkungsgrad, Zellspan- nung und Leistung einer Brennstoffzelle unter Berucksichtigung¨ der Partialdrucke¨ der Substanzen, der Temperatur sowie der Dissipation beim Protonentransport be- rechnet. Dabei zeigt sich, dass experimentelle Daten fur¨ die Zellspannung sowohl qualitativ als auch n¨aherungsweise quantitativ durch das Modell abgebildet werden k¨onnen. Der Vorteil des endoreversiblen Modells liegt dabei in der M¨oglichkeit, mit nur einem Modell neben den genannten Kenngr¨oßen auch die abgegebene W¨arme sowie die Entropieproduktion zu quantifizieren und den einzelnen Teilprozessen zu- zuordnen. Schlagworte Endoreversible Thermodynamik, Nichtgleichgewichtsthermodynamik, Ideales Gas, van der Waals Gas, Brennstoffzelle, Modellierung, Kreisprozesse, Chemische Reak- tionen, Wirkungsgrad Abstract In this thesis extensions to the formalism of endoreversible thermodynamics for multi- extensity fluxes and chemical reactions are introduced. These extensions make it possible to model a great variety of systems which could not be investigated with standard endoreversible thermodynamics. Multi-extensity fluxes are important when studying processes with matter fluxes or processes in which volume and entropy are exchanged between subsystems. For including reversible as well as irreversible chemical reaction processes a new type of subsystems is introduced { the so called reactor. It is similar to endoreversible engines, because the fluxes connected to it are balanced. The difference appears in the balance equations for particle numbers, which contain production or destruction terms, and in the possible entropy production in the reactor. Both extensions are then applied to an endoreversible fuel cell model. The chemical reactions in the anode and cathode of the fuel cell are included with the newly introduced subsystem { the reactor. For the transport of the reactants and products as well as the proton transport through the electrolyte membrane, the multi-extensity fluxes are used. This fuel cell model is then used to calculate power output, efficiency and cell voltage of a fuel cell with irreversibilities in the proton and electron transport. It directly connects the pressure and temperature dependencies of the cell voltage with the dissipation due to membrane resistance. Additionally, beside the listed performance measures it is possible to quantify and localize the entropy production and dissipated heat with only this one model. Contents 1 Introduction 9 1.1 Document Structure . 11 2 Endoreversible Thermodynamics 13 2.1 General Formalism . 13 2.1.1 Subsystems . 14 Reservoirs . 15 Engines .............................. 15 2.1.2 Interactions . 16 2.2 Extension for Multi-Extensity Fluxes . 17 2.3 Ideal Gas . 21 2.3.1 Gibbs Equation and Principal Equation of State . 21 2.3.2 Infinite Reservoirs . 23 2.3.3 Finite Reservoirs . 24 Finite Reservoirs with Constant Pressure . 24 Finite Reservoirs with Constant Volume . 27 2.3.4 Reversible Engines . 30 Combining Gas Fluxes . 33 2.3.5 Interactions . 35 Interactions with Volume, Particle and Entropy Flux . 35 Interactions with Volume and Entropy Flux . 36 Interactions with Particle and Entropy Flux . 40 2.3.6 Example: Pressure Regulator . 42 2.4 Van der Waals Gas . 45 2.4.1 Gibbs Equation and Principal Equation of State . 48 2.4.2 Interactions . 49 Interactions with Volume and Entropy Flux . 49 2.4.3 Example: Pressure Regulator . 51 2.5 Summary ................................. 53 3 Chemical Reactions 55 3.1 Chemical Engines Without Reactions . 56 3.2 Chemical Engines with Reactions . 58 3.2.1 Reversible Reactions vs. Reversible Thermodynamic Processes 59 7 Contents 3.2.2 A New Subsystem { the Reactor . 60 3.2.3 Reversible Reaction Processes . 63 Reaction at Constant Volume . 64 Reaction at Constant Pressure . 65 3.2.4 Irreversible Reaction Processes . 67 Reaction at Constant Volume . 68 Reaction at Constant Pressure . 69 3.3 Summary ................................. 70 4 Fuel Cells 73 4.1 Structure of a PEM Fuel Cell . 73 4.2 Cell Voltage and Fuel Cell Efficiency . 74 4.3 Endoreversible Modeling of PEM Fuel Cells . 76 4.3.1 Reversible Fuel Cell Model . 76 4.3.2 Endoreversible Fuel Cell Model . 79 4.4 Summary ................................. 87 5 Conclusion 89 Nomenclature 91 Bibliography 93 8 1 Introduction Classical equilibrium thermodynamics has been developed in the 19th century. It offers a macroscopic theory for the description of thermodynamic systems in equi- librium. It concentrates on states of thermodynamic systems rather than processes. Thermodynamic processes are usually understood as a quasi-static sequences of equi- libria. Within the classical theory, processes are completely reversible, which for real systems would require infinitely slow execution. This leads to infinitely small rates, e.g. for energy conversion. The Carnot process [17] is an example for a completely reversible process. The Carnot efficiency ηC = 1 TL=TH gives the fraction of the heat, which is taken from a high temperature reservoir,− that is converted into work. The rest of the heat is released to a lower temperature reservoir. Since the Carnot process is completely reversible and there does not occur any dissipation, this is the highest efficiency reachable in a heat engine which operates between temperatures TL and TH. In real processes, however, finite rates are desired. Therefore, finite and thus ir- reversible fluxes are required. Over the years there have been various attempts in engineering and physics to include irreversibilities in thermodynamic descriptions, for instance under labels like finite-time thermodynamics, Onsager theory, extended thermodynamics and so on. A great number of publications was focused on irre- versibilities in heat engines (e.g. [6, 7]). In [57], Muller¨ et al. give an extensive overview of \Thermodynamics of irreversible processes { past and present", including the work of Cattaneo, Prigogine and Onsager. In 1979 one of the fields in finite-time thermodynamics was labeled \endoreversible" thermodynamics by Rubin [68, 69]. He defined an \endoreversible engine to be an engine such that during its operation its working fluid undergoes reversible transfor- mations." This definition includes all the reversible engines in classical equilibrium thermodynamics, such as the Carnot engine. \More importantly [. ], the subclass of engines which are coupled to the external world via irreversible processes" [68] is included in the definition of endoreversible engines. The most important and most investigated examples of such endoreversible engines are the Novikov heat en- gine [59, 60] and the Curzon-Ahlborn (CA) heat engine [25]. Fischer et al. [33] found that a simple Novikov model can accurately render an Otto engine if the parameters are adjusted appropriately. The CA engine has been discussed in great detail in literature. In its original form, 9 1 Introduction it consists of a Carnot heat engine that is coupled irreversibly to the high and low temperature reservoirs. The operating point of maximum efficiency for such an en- doreversible heat engine is not desirable, since the power output at this point goes to zero. However, the power over efficiency curve shows an operating point of maximum p power at the CA efficiency, ηCA = 1 TL=TH. This efficiency is a good approxima- tion of real power plant efficiencies (comparisons− can be found in [14] and [25]). Various