Class Note for ECE 449 at UA
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Date Created: 02/06/15
A I we 449549 ontinnons System mmml Chemical Thermodynamics II In this lecture we shall continue to analyze our chemical thennodynamics bond graphs making use of bondgraphic knowledge that we hadn t exploited so far This shall lead us to a more general bondgraphic description of chemical reaction systems that is less dependent on the operating conditions The RFelement and the CFelement explained in their full complexity are November 12 2003 Start Presentation ltIJEgt I we 449549 Glontinnnns System mutualingl Table of Contents Structural analysis of chemical reaction bond graph The chemical resistive eld Multi port gyrators The chemical capacitive field lsochoric vs isobaric operating conditions Eguation of state Adiabatic vs isothermal operating conditions Caloric eguation of state Enthalpy of formation Tabulation of chemical data Heat capacity of air November 1 2 2003 Start Presentation ltgt A I we 449549 ontinnons System mutualitth A Structural Analysis of the Generic Chemical Reaction Bond Graphs Let us look once more at the generic chemical reaction bond graph Navember 12 2003 start Presentation EA I we 449549 Gluntinnons System mutualitth Relations Between the Base Variables Let us recall a slide from an early class on bond graphs Navember 12 2003 start Presentation A I we 449549 Giantinnons System mutualitth The RFElement I 0 Let us analyze the three equations that make up the RF element 1 qGibbse i The Gibbs equation is certainly a static equation relating only efforts and ows to each other It generalizes the S of the RSelement The equation of state is a static equation relating efforts with generalized positions Thus it clearly belongs to the CFelement 2 Eguatian at state November 12 2003 start Presentation EASE I we 449549 Glontinnnns System mutualitth The RFElement II 0 By differentiating the equation of state we were able to come up With a structurally appropriate equation Yet the approach is dubious The physics behind the equation of state points to the CF eld and this is Where it should be used November 12 2003 start Presentation A I we 449549 ontinnons System mutualitth The RFElement III 0 This also makes physical sense 0 The equation of state describes a property of a substance The CF field should contain a complete description of all chemical properties of the substance stored in it 0 The RF field on the other hand only describes the transport of substances A pipe really doesn t care what ows through it 0 The RF eld should be restricted to describing continuity equations 0 The mass continuity is described by the reaction rate equations The energy continuity is described by the Gibbs equation What is missing is the volume continuity November 12 2003 start Presentation I we 449549 Glontinnnns System mutualitth The RFElement IV 0 We know that mass always carries its volume along Thus 0 Using the volume continuity equation we obtain exactly the same results as using the differentiated equation of state since the equation of state teaches us that 2 thus which is exactly the equation that we had used before November 12 2003 start Presentation A I we 449549 ontinnons System mutualitth The RFElement V 0 What have we gained if anything 0 The differentiated equation of state had been derived under the assumption of isobaric and isothermal operating conditions 0 The volume continuity equation does not make any such assumption It is valid not only for all operating conditions but also for all substances ie it does not make the assumption of an ideal gas reaction energy continuity November 12 2003 start Presentation I we 449549 Glontinnnns System mutualitth The RFElement VI 3 Reaction rate eguations The reaction rate equations relate flows fvariables to generah39zed positions qvariables However the generalized positions are themselves statically related to efforts e variables in the CFelement Hence these equations are indeed reactive as they were expected to be Thus we now have convinced ourselves that we can write all equations of the RFelement as f gg In the case of the hydrogenbromine reaction there will be 15 equations in 15 unknowns 3 equations for the three ows of each one of five separate reactions November 12 2003 start Presentation A I we 449549 ontinnons System mutualitth The Linear Resistive Field We still need to ask ourselves whether these 15 equations are irreversible ie resistive or reversible ie gyrative We already know that the Cmatrix describing a linear capacitive field is always symmetric Since that matrix describes the network topology the same obviously holds true for the Rmatrix or G matrix describing a linear resistive eld or linear conductive field These matrices always have to be symmetric November 12 2003 start Presentation I we 449549 Gluntinnnns System mutualitth The Multiport Gyrator I Let us now look at a multi port gyrator In accordance with the regular gyrator its equations are defined as November 12 2003 start Presentation A I we 449549 ontinnons System whalingl The Multiport Gyrator II 0 In order to compare this element With the resistive field it is useful to have all bonds point at the element thus November 12 2003 start Presentation AISh I we 449549 Gluntinuonx System mutualitml The Multiport Gyrator III 0 In a matrixvector form skewsymmetric matrix l 0 Any matrix can be decomposed into a symmetric part and a skewsymmetric part November 12 2003 start Presentation A I we 449549 ontinnons System muslingl Symmetric and Skewsymmetric Matrices 0 Example I11 Z 2 IEQ is symmetric My Mx is skewsymmetric M m Mm Navember 12 2003 Start Presentation AISh I we 449549 Gluntinuons System mutualitml The RFElement VII 0 Hence given the equations of the RF element Navember 12 2003 Start Presentation A I we 449549 ontinnons System manslingl The RFElement VIII Conduction matrix I 1 Gyration matrix November 12 2003 Start Presentation AISh I we 449549 Gluntinuonx System mutualitml The CFElement I 0 We should also look at the CFelements Of course these elements are substancespecific yet they can be constructed using general principles 0 We need to come up With equations for the three potentials efforts T p and g These are functions of the states generalized positions S V and M 0 We also need to come up With initial conditions for the three state variables S0 V0 and M0 November 12 2003 Start Presentation A I we 449549 untinnons System mutualitth The C FElement II 0 The reaction mass is usually given ie we know up front how much reactants of each kind are available This determines M0 for each of the species and therefore no It also provides the total reaction mass M and therefore n 0 In a batch reaction the reaction mass remains constant whereas in a continuous reaction new reaction mass is constantly added and an equal amount of product mass is constantly removed 0 Modeling continuous reactions with bond graphs is easy since the chemical reaction bond graph can be naturally interfaced with a convective ow bond graph November 12 2003 start Presentation EASE I we 449549 Glontinnnns System mutualitth Isochoric vs Isobaric Operating Conditions 0 Chemical reactions usually take place either inside a closed container in which case the total reaction volume is constant or in an open container in which case the reaction pressure is constant namely the pressure of the environment 0 Hence either volume or pressure can be provided from the outside We call the case where the volume is kept constant the isochoric operating condition whereas the case where the pressure is kept constant is called the isobaric operating condition November 12 2003 start Presentation 10 A I we 449549 Giantinnons System mutualitth The Equation of State 0 The equation of state can be used to compute the other of the two volumerelated variables given the reaction mass and the temperature Isobaric conditions constant Isochoric conditions Vconstant November 12 2003 start Presentation EASE I we 449549 Glontinnnns System mutualitth Adiabatic vs Isothermal Operating Conditions 0 We can perform a chemical reaction under conditions of thermal insulation ie no heat is either added or subtracted This operating condition is called the adiabatic operating condition 0 Alternatively we may use a controller to add or subtract just the right amount of heat to keep the reaction temperature constant This operating condition is called the isothermal operating condition November 12 2003 start Presentation ll A I we 449549 ontinnons System mutualitth The Caloric Equation of State I 0 We need an equation that relates temperature and entropy to each other In general T fSV To this end we make use of the so called caloric equation of state November 12 2003 start Presentation AS I we 449549 Glontinnnns System mutualitth The Caloric Equation of State II 0 Under isobaric conditions dp 0 the caloric equation of state simplifies to or which corresponds exactly to the heat capacitor used in the past November 12 2003 start Presentation 12 A I we 449549 ontinnons System mutualitth The Caloric Equation of State 111 0 In the general case the caloric equation of state can also be written as Thus November 12 2003 start Presentation I we 449549 Glontinnnns System mutualitth The Caloric Equation of State IV 0 The initial temperature T is usually given The initial entropy S0 can be computed as S0 M0 sT0p0 using a table lookup function 0 In the case of adiabatic operating conditions the change in entropy ow can be used to determine the new temperature value To this end it may be convenient to modify the caloric equation of state such that the change in pressure is expressed as an equivalent change in volume 0 In the case of isothermal conditions the approach is essentially the same The resulting temperature change AT is computed from which it is then possible to obtain the external heat ow Q AT S needed to prevent a change in temperature November 12 2003 start Presentation l3 A I we 449549 ontinnons System mutualitth The Enthalpy of Formation Finally we need to compute the Gibbs potential g It represents the energy stored in the substance ie the energy needed in the process of making the substance 0 In the chemical engineering literature the enthalpy of formation h is usually tabulated in place of the Gibbs free energy g 0 Once h has been obtained g can be computed easily November 12 2003 start Presentation EA I we 449549 Gluntinnons System mutualitth Tabulation of Chemical Data I 0 We can find the chemical data of most substances on the web eg at httpwebbooknistgovchemistrVformserhtml Searching eg for the substance HBI we find at the address httpwebbook nistgovcgicbookcgi C1003 5 l 06ampUnitsSIampMaskl Quantity Value Units Methm l Reference Cnmment 41391quot 3529 i 0 16 kIlmol Remew Cox Wagnan et a1 1981 CODATA Review value AH gu 36 44 kJmol Revuew Chase 1998 Data last reviewed 1 September 1965 Quantity Value Units Methml Referenue Comment 5 2qu 198700 i 0 004 Jmol qi Revtew Cox Wagman et a1 1984 CODATA Remew value Sogulbir 198 70 meolK Review Chase 1998 Data last reviewed in September 1965 November 12 2003 start Presentation l4 A l we 449549 shamans System illnheliugl Tabulation of Chemical Data 11 Gas Phase Heat Capacity Shomate Equation CP ABtC Dt3Eft Ham s Amt aegm z 4 C813 4 Dam TameramrNK 298 1100 1100 600 Squot A1nt 13 CWIZ Dt33 52 t2 3171409 3288913 CF heat Capacity Iknol K 43169992 2322116 H standard enthalpy kellmol A HDMJS enthalpy offormauon at 29815 1 2335557 13 473035 standard entropy IlmolK 791008529 0 032464 t temperature IQ1000 0 028758 3 174958 45 57464 5246318 240 0428 230 8597 W kJm l 36 44305 36 44305 Chase 1998 Chase 1998 Cement last remewed 111 September 1965 last remewed 111 September 1965 November 12 2003 Start Presentation I 449549 toutinnuux System mowing I The Heat Capacity of Air I We are now able to u u undemtand the CFAir eeeeeee emergeeee Peeemegeeeen Parameter Real SEI6 Bl l lEA quotEnLrGPV If no anquot J eeeemeeee Ree1 vnn 13112221 meme 12 ee enquot eeeemeeee Ree1 1m1 8 ne eeeemeeee Ree1 e um see eeeeem e e e emcee eeeeee eemeeee Ree1 H we see eeeeeem eeeemeeee Reel eeeueu nuns venenee meee deeueemeeeme teem eeee eeeemeeee Eeeleee heegelee quotme e hummus vehee ere eeeemeeee Ree1 Lee 91 53 hemmee ceeeeeemee e ee eeeemeeee Ree1 eneeemn nemeue eeeeeuee 12 ee en Ree1 bee Meee es en Ree1 sent Emeey e em Ree1 v e quotHume e Ree1 e Beet eeeeem e en e2 meme meme Ree1 v eeeeme meme Real 5 SE2121211 eutrclpv Ree1 1m memel Meme e eeeeme meme e f equst lee A Equation afxtate 77 Caloric equation afstate T Gibbs energy affarmatian u Exist m nut fun then TxRM7ntV71nt e152 ipibet s Exist a nut flct then 293 leeXpUS 7 5 e137 7 mung 1 17245 m else Luce Dp 7 14 f Exist then Minn else 1m 2 Exist then vim else vn s 1 Exist then 57m else su November 12 2003 l Eqummns am algumhms 17111555 15 A I was 449549 ontinnoux System whalingl u rnmmr mm 5km 7altne39 The Heat Capa 11 The pressure is de ned negatively v nu elze vu g ILAM 5A9 nu m abs 1 Equalquot an gash wig start Presentation I was 449549 untinnonx System mutualitml The Heat Capacity of Air 111 ygmaJm the pump lam mth M SWMLE 7aluxe39 I Lntl eta xv My y nu else To magmmwsnmu m n v a xmsw for ideal gases start Presentation 16 A we 449549 ontinnons System whalingl References Cellier FE 1991 Continuous System Modeling SpringerVerlag New York Chapter 9 Greifeneder J 2001 Modellierung thermodynamischer Ph inomene mittels Bondgraphen Diplomarbeit University of Stuttgart Germany November 12 2003 Start Presentation 17
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