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FOR MULTICOMPONENT ISOTHERMAL DIPHASIC EQUILIBRIA

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Niveau: Supérieur, Doctorat, Bac+8
STATISTICAL THERMODYNAMICS MODELS FOR MULTICOMPONENT ISOTHERMAL DIPHASIC EQUILIBRIA FRANC¸OIS JAMES ? Mathematiques Appliquees et Physique Mathematique d'Orleans URA CNRS 1803, Universite d'Orleans, 45067 Orleans Cedex 2, France MAURICIO SEPULVEDA † IWR der Universitat Heidelberg Im Neuenheimer Feld 368 D-69120 Heidelberg, Germany PATRICK VALENTIN Centre de Recherches Elf-Solaise B.P. 22, 69360 Saint-Symphorien d'Ozon, France We propose in this paper a whole family of models for isothermal diphasic equilibrium, which generalize the classical Langmuir isotherm. The main tool to obtain these models is a fine modelling of each phase, which states various constraints on the equilibrium. By writing down the Gibbs conditions of thermodynamical equilibrium for both phases, we are lead to a constrained minimization problem, which is solved through the Lagrange multipliers. If one of the phases is an ideal solution, we can solve explicitely the equations, and obtain an analytic model. In the most general case, we have implicit formulæ, and the models are computed numerically. The models of multicomponent isotherm we obtain are in this paper designed for chromatography, but can be adaptated mutatis mutandis to other cases. 1. Introduction In this paper, we give a systematic description of a family of multicomponent diphasic equilibria at constant temperature, which contains as its simplest case the classical Langmuir11 isotherm. Such models, which we shall call isotherms for short, encounter very often in Chemical Engineering; one can also mention polynomial ?E-mail : james@cmapx.

  • physique mathematique d'orleans ura

  • mobile phase

  • ideal lattice-gas

  • gas-liquid equilibria

  • cellular species

  • gas model

  • stable isothermal equilibrium


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STATISTICALTHERMODYNAMICSMODELSFORMULTICOMPONENTISOTHERMALDIPHASICEQUILIBRIAFRAN¸COISJAMESMathe´matiquesApplique´esetPhysiqueMathe´matiqued’Orle´ansURACNRS1803,Universite´d’Orle´ans,45067Orle´ansCedex2,FranceMAURICIOSEPU´LVEDAIWRderUniversita¨tHeidelbergImNeuenheimerFeld368D-69120Heidelberg,GermanyPATRICKVALENTINCentredeRecherchesElf-SolaiseB.P.22,69360Saint-Symphoriend’Ozon,FranceWeproposeinthispaperawholefamilyofmodelsforisothermaldiphasicequilibrium,whichgeneralizetheclassicalLangmuirisotherm.Themaintooltoobtainthesemodelsisafinemodellingofeachphase,whichstatesvariousconstraintsontheequilibrium.BywritingdowntheGibbsconditionsofthermodynamicalequilibriumforbothphases,weareleadtoaconstrainedminimizationproblem,whichissolvedthroughtheLagrangemultipliers.Ifoneofthephasesisanidealsolution,wecansolveexplicitelytheequations,andobtainananalyticmodel.Inthemostgeneralcase,wehaveimplicitformulæ,andthemodelsarecomputednumerically.Themodelsofmulticomponentisothermweobtainareinthispaperdesignedforchromatography,butcanbeadaptatedmutatismutandistoothercases.1.IntroductionInthispaper,wegiveasystematicdescriptionofafamilyofmulticomponentdiphasicequilibriaatconstanttemperature,whichcontainsasitssimplestcasetheclassicalLangmuir11isotherm.Suchmodels,whichweshallcallisothermsforshort,encounterveryofteninChemicalEngineering;onecanalsomentionpolynomialE-mail:james@cmapx.polytechnique.frE-mail:sepulved@cumin.iwr.uni-heidelberg.de1
2F.JAMES,M.SEPU´LVEDA&P.VALENTINmodels,and,forinstance,Sips21andFreundlich3isotherms,whicharedesignedforadsorptionequilibria.WementionhereasageneralreferenceforisothermsusedinchromatographyabookbyGuiochon&al.4,whichcontainsalsoalargebibliographyonthesubject.Thecommonfeaturetothemodelsweareabouttodescribeisthattheyarebuiltupfromstatisticalthermodynamicsmodelling.Eventhoughtheymaylookquitedifferentonefromanother,theysharesomestructuralproperties,inparticularthethermodynamicalconsistencywhichisdefinedbelow.Theideaofthemodeloriginatedfromgas-solidadsorptioninchromatography,andwasgivenbyJ.M.MoreauandP.Valentinin1983,butunfortunatelywasnotpublishedatthistime(seeRef.8).Theformalderivationoftheisotherm,andageneralizationtogas-liquidabsorptionweregiveninRef.7,aswellasseveralmodelsofchromatographicpropagation.Bothmodelswerelimitedbythefactthatoneofthephaseswasassumedtobeanidealsolution.Thiswasjustifiedforthemobilephaseinthecontextofgaseouschromatography,butwasofcourseforbiddingthemodellingofliquidchromatography.InRef.9,wegaveageneralizationforthisisothermusingalattice-gasmodelforbothphases,whichallowstheapplicationtononidealsolutions.Theaimofthispaperistosummarizetheseresultsandgiveafewexamplesofone-componentisotherms.Theansatzofthemethodisthefollowing:takeforbothphasesanymodelfromstatisticalthermodynamics:idealsolution,ornon-ideallattice-gasmodel,inviewoftakingintoaccountpossibleinteractionsbetweencomponents.Wecanobtainexplicitelythecorrespondingthermodynamicalpotentials(energy,enthalpy,...).Thenassumethatthesystemconstitutedbythetwophasesisatastateofstableisothermalequilibrium.Thisleadstominimizethetotalenergyofthesystem,underseveralconstraintswhicharisenaturallyinthemodellingofthephases.TheseconstraintsaredealtwithusingLagrangemultipliers,andweobtainasystemofequationsfortheequilibriumstate,whichplaysthesamepartastheclassicalGibbsequalitiesforunconstraineddiphasicequilibrium.Theseequationscanbeformallysolved,andgivetherequestedisotherm.Ifbothphasesaremodelledbyalattice-gas,theequilibriumequationscannotbesolvedexplicitely,butthenumericalresolutionisquitestraightforward.Ifweassumeoneofthephasestobeanidealsolution,theseequationsareexplicitelysolvedwithrespecttoquantitiesintheidealsolution,sowegetananalyticexpressionfortheisotherm.Animportantfeatureofthatfamilyofisothermsisthefollowingintrinsicprop-ertyofthermodynamicalconsistency.Supposetheisothermbetweenphase1andphase2isgivenbyN2=h(N1),whereNjisthevectorofquantitiesinphasej.Fromtheequationsofequilibrium(Gibbsequalities),onecanshowthatnecessarilythejacobianmatrixJ=h(N1)isdiagonalisable,itseigenvaluesbeingpositive(seeRef.10,7).Thispropertyisnotobviouswhengivenaphysicalmodelofdiphasicequi-librium,andthelackofitmayleadtophysicalincoherences.ItappearsalsointhesimulationofchromatographiccolumnsbysystemsofPartialDifferentialEquations(seeRef.18,7).Indeedthediagonalizationensuresapropertycalledhyperbolicity