Class Note for ECE 449 at UA 2
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Date Created: 02/06/15
A I MIME 449549 Cuntinnuns 35M whalingl Convective Mass Flows III 0 In this lecture we shall concem ourselves once more with convective mass and heat ows as we still have not gained a comprehensive understanding of the physics behind such phenomena 0 We shall start by looking once more at the capacitive eld 0 We shall then study the internal energy of matter 0 Finally we shall look at general energy transport phenomena which by now include mass ows as an integral aspect of general energy ows November 3 2003 start Presentation AL I 449549 Gluntinnuus System whaling l Table of Contents Capacitive Fields 0 Internal energy of matter Busbonds and bus junctions Heat conduction Volume work 0 General mass transport Multi phase systems Evaporation and condensation Thermodynamics of mixtures Multielement systems November 3 2003 start Presentation A I MIME 449549 Cuntinnuns 35M mmm Capacitive Fields III 0 Let us brie y consider the following electrical circuit i1 i3 Cz i2 l l Cl Cz C3 I I 11 13 12 13 2 Mail 1114413 quot215 i3 C1 C3 u l vi 1 0 04gt 1 5quot 0x 0 11 1 i3 3 November 3 2003 start Presentation I 449549 Gluntinnuus System whaling Capacitive Fields IV quot ir uruq 11311 i 2 0 CF 0 l 0 quotgt0 e 1 1 0 0 1 3 November 3 2003 start Presentation A I MIME 449549 Continuous 35M latitudinal Volume and Entropy Storage 0 Let us consider once more the situation discussed in the previous lecture S 0 It was no accident that I drew the two I capacitors so close to each other In Cth reality the two capacitors together C form a twoport capacitive eld After t all heat and volume are only two 1 different properties of one and the 0 A 1 H 0 same material I November 3 2003 start Presentation AL I 449549 Gluntinnuus System whaling The Internal Energy of Matter I 0 As we have already seen there are three different though inseparable storages of matter I Mass l Volume I Heat 0 These three storage elements represent different storage properties of one and the same material 0 Consequently we are dealing With a storage eld 0 This storage field is of a capacitive nature 0 The capacitive field stores the internal energy of matter November 3 2003 start Presentation I MIME 449549 Continuous 35M autumnal A The Internal Energy of Matter II 0 Change of the internal energy in a system ie the total power flow into or out of the capacitive field can be described as follows Igginiculljoten al Flow of U T 3 39 P V 1V Jlolur internal energy T quotuns ow Heat aw Mass nw Volume ow 0 This is the Gibbs equation November 339 2003 start Presentation I 449549 Gluntinnuus System whaling A The Internal Energy of Matter III The internal energy is proportional to the the total mass n By normalizing with n all extensive variables can be made intensive U quot 7 S V N s 7 v 7 Iii 71 Therefore t 1n s p nev tt n ni K November 3 2 J03 Start Presentation ltnrgt A I MIME 449549 Cuntinnuns 35M whaling The Internal Energy of Matter IV This equation must be Valid independently of the amount n therefore Finally here is an M explanation why it internal energy was okay to compute with funny derivatives Internal energlz Navember 3 2003 start Presentation Asia I 449549 Gluntinnuus System whaling The Internal Energy of Matter IV 2 UT Szpl i hT SIi V i Q T spV i 7i 39 This is the GibbsDuhem equation November 3 2003 start Presentation A I MIME 449549 Cuntinnunx 35M mmm The Capacitive Field of Matter T r C g39cv39NF c 5 November 3 2003 Start Presentation AL I 449549 Gluntinnuus System whaling Simpli cations In the case that no chemical reactions take place it is possible to replace the molar mass ows by conventional mass ows In this case the chemical potential is replaced by the Gibbs potential November 3 2003 Start Presentation A l we 449549 continuous system mnaml Bus Bond and Bus OJunction The three outer legs of the CFelement can be grouped together November 3 2003 start Presentation l was 449549 toutian System motelmg Once Again Heat Conduction November 3 2003 start Presentation A low 449549 ouunnous spam mowingl Volume Pressure Exchange Pressure is being equilibrated just like temperature It is assumed that the inertia of the mass may be neglected relatively small masses andor velocities and that the equilibration occurs Without friction The model makes sense if the exchange occurs locally and if not too large masses get moved in the process November 3 2003 start Presentation l we 449549 Ginn nunnx System unwind General Exchange Element I The three ows are coupled through RS elements This is a switching element in bondgraph notation This element has not yet been introduced November 3 2003 start Presentation A I ma 449549 continuous system loheliugl General Exchange Element II 0 In the general exchange element the temperatures the pressures and the Gibbs potentials of neighboring media are being equilibrated This proess can be interpreted as a resistive eld November 3 2003 start Presentation l we 449549 Glontinunnx System mnamg Multi phase Systems 0 We may also Wish to study phenomena such as evaporation and condensation November 3 2003 start Presentation A I 449549 Continuous pstem mnamgl Evaporation Boiling Mass and energy exchange between capacitive storages of matter CF elements representing different phases is accomplished by means of special resistive elds RF elements 0 The mass ows are calculated as functions of the pressure and the corresponding saturation pressure 0 The volume flows are computed as the product of the mass ows With the saturation volume at the given temperature 0 The entropy ows are superposed With the enthalpy of evaporation in the process of evaporation the thermal domain loses heat latent heat November 3 2003 start Presentation I we 449549 Continuum System ooding l Condensation On Cold Surfaces Here a boundary layer must be introduced November 3 2003 start Presentation 10 A I 1396 449549 Cuntinnnux pKt2m mmml Thermodynamics of Mixtures When uids gases or liquids are being mixed additional entropy is generated 0 This mixing entropy must be distributed among the participating component uids 0 The distribution is a function of the partial masses 0 Usually neighboring CFelements are not supposed to know anything about each other In the process of mixing this rule cannot be maintained The necessary information is being exchanged November 3 2003 start Presentation l we 449549 Glontinunnx System mtteltngl Entropy of Mixing November 3 2003 start Presentation ll A 449l549t ou nnnux system mowingl November 3 2003 start Presentation ltIgt 2A l we 449549 mummy System mmw Convection in Multielement Systems November 3 2003 start Presentation ltnrgt 12 A Imammammal system mowingl Twoelement Twophase Two compartment Convective System November 3 2003 start Presentation l we 449549 Ginntinunnx System moh ingl Concentration Exchange It may happen that neighboring compartments are not completely homogeneous In that case also the concentrations must be exchanged November 3 2003 start Presentation l3 A MIME 449549 Continuous 35M whalingl References I Cellier F E 1991 Continuous Svstem Aodeling SpringerVerlag New York Chapter 9 Greifeneder J and FE Cellier 2001 Modeling convective ows using bond graphs Proc ICBGM OI Intl Conference on Bond Graph Modeling and Simulation Phoenix Arizona pp 276 284 November 3 2003 start Presentation Asa 449549 Gluntinnuus System mending l References II Greifeneder I and FE Cellier 2001 Modeling multi phase systems using bond graphs Proc ICBGM OI Intl Conference on Bond Graph Modeling and Simulation Phoenix Arizona pp 285 7 291 Greifeneder I and FE Cellier 2001 Modeling multi element SVsterns using bond graphs Proc ESS 01 European Simulation Symposium Marseille France pp 758 7 766 November 3 2003 start Presentation 14
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