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Hyperbolic models in gas solid chromatography

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Niveau: Supérieur, Doctorat, Bac+8
Hyperbolic models in gas-solid chromatography C. Bourdarias, M. Gisclon Laboratoire de Mathematiques, UMR CNRS 5127 Universite de Savoie 73376 Le Bourget-du-Lac Cedex, France S. Junca Laboratoire JAD, UMR CNRS 6621 IUFM et Universite de Nice Parc Valrose, 06108 Nice Abstract We present different models arising in chemical engineering and essentially related to isothermal gas chromatography. These models describe a fixed bed adsorption process of separation of a gaseous mixture: each compound can exist either in a mobile phase or a solid and static one, with a finite or infinite mass-exchange kinetics. Many authors, in the fields of chemical engineering and mathematics, have investigated these models under various assumptions, from a theoretical or numerical point of view. We explain first the relations between some of these approaches. Next, we present some results related to these models, some of them being new, particularly in the case of a monovariant system with one or two active compounds for the Cauchy problem. Lastly, we mention some open problems. Keywords: gas chromatography, nonlinear chromatography, adsorption, mass-transfer kinetics, systems of conservation laws 1 Introduction Chromatography is the collective term for a family of laboratory techniques for the se- paration of mixtures. It involves passing a mixture dissolved in a ”mobile phase” (liquid or gaseous) through a stationary phase, which separates the analyte to be measured from other molecules in the mixture and allows it to be isolated (source: Wikipedia).

  • can also

  • transfer kinetics

  • mobile phase

  • no additional mobile

  • chromatography

  • ?1 ?2n

  • carrier gas

  • bet isotherm


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Hyperbolicmodelsingas-solidchromatographyC.Bourdarias,M.GisclonLaboratoiredeMathe´matiques,UMRCNRS5127Universite´deSavoie73376LeBourget-du-LacCedex,FranceS.JuncaLaboratoireJAD,UMRCNRS6621IUFMetUniversite´deNiceParcValrose,06108NiceAbstractWepresentdifferentmodelsarisinginchemicalengineeringandessentiallyrelatedtoisothermalgaschromatography.Thesemodelsdescribeafixedbedadsorptionprocessofseparationofagaseousmixture:eachcompoundcanexisteitherinamobilephaseorasolidandstaticone,withafiniteorinfinitemass-exchangekinetics.Manyauthors,inthefieldsofchemicalengineeringandmathematics,haveinvestigatedthesemodelsundervariousassumptions,fromatheoreticalornumericalpointofview.Weexplainfirsttherelationsbetweensomeoftheseapproaches.Next,wepresentsomeresultsrelatedtothesemodels,someofthembeingnew,particularlyinthecaseofamonovariantsystemwithoneortwoactivecompoundsfortheCauchyproblem.Lastly,wementionsomeopenproblems.Keywords:gaschromatography,nonlinearchromatography,adsorption,mass-transferkinetics,systemsofconservationlaws1IntroductionChromatographyisthecollectivetermforafamilyoflaboratorytechniquesforthese-parationofmixtures.Itinvolvespassingamixturedissolvedina”mobilephase”(liquidorgaseous)throughastationaryphase,whichseparatestheanalytetobemeasuredfromothermoleculesinthemixtureandallowsittobeisolated(source:Wikipedia).TheprincipalmethodsareFrontalChromatography:aprocedureinwhichthesample(liquidorgas)isfedcon-tinuouslyintothechromatographicbed.Infrontalchromatographynoadditionalmobilephaseisused.DisplacementChromatography:aprocedureinwhichthemobilephasecontainsacom-pound(theDisplacer)morestronglyretainedthanthecompoundsofthesampleunderexamination.Thesampleisfedintothesystemasafiniteslug.ElutionChromatography:aprocedureinwhichthemobilephaseiscontinuouslypassedthroughoralongthechromatographicbedandthesampleisfedintothesystemasafiniteslug.1
Chromatographymaybepreparativeoranalytical.Preparativechromatographyseekstoseparatethecompoundsofamixtureforfurtheruse(andisthusaformofpurification).Itisaprocessthathaslatelybecomeofconsiderableinterestinthepharmaceuticalin-dustry:onlychromatographyissufficientlyflexibleandpowerfulltosatisfythepracticalrequirementsencounteredinmostdifficultseparationsofpharmaceuticalintermediates.Analyticalchromatographynormallyoperateswithsmalleramountsofmaterialandseekstomeasuretherelativeproportionsofanalytesinamixture.Thetwoarenotmutuallyexclusive.Anothermethodofseparatingchemicalsubstancesisdistillation,basedondifferencesintheirvolatilitiesinaboilingliquidmixture.Itisaprocessusedforinstanceinpetroleumindustry.Acommonfeaturebothtochromatographyanddistillationisthattheseparationfollowsfromtheinteractionbetweentwophasesinmotiononewithrespecttotheother.Adistillationcolumnisheatedatthebottom,thusseparatingthemixtureinagaseousphasemovingupwards,andaliquidonemovingdownwardsbygravity.Instandardchromatographythemixture,ingaseousorliquidform,isinjectedinacolumnfilledwithsomeporousmedium.Thechemicalcompoundsarepartiallyretainedbythepores,thusgeneratingstationaryphaseinthecolumn(fixedbedadsorption).Inthemodellingofsuchprocess,twotypesofphenomenaaretobeconsidered.Ontheonehand,thepropagationofthemobilephaseisruledbythelawsoffluiddynamics,gasdynamics,porousmedia,...Ontheotherhand,therepartitionofmatterbetweenthetwophasesreliesonthermodynamicsandthenotionofdiphasicequilibriumisinvolved.TherearemanyreferenceworksinthefieldofChromatography.G.GuiochonandB.Lin[17],forinstance,describethedifferentmathematicalmodelsofchromatography,examinetheassumptionsonwhichtheyarebased,considertheirpropertiesanddiscusstheirsolutions.In[18],onecanfindthefundamentalsofthermodynamics,mass-transferkineticsandflowthroughporousmediathatarerelevanttochromatography.Theauthorspresentthemodelsusedinchromatography,theapplications,describethedifferentprocessesusedandthemethodsofoptimizationoftheexperimentalconditions.Inthisarticlewemainlyconsiderthecaseofisothermalgas-solidchromatography,aprocedureinwhichthetemperatureofthecolumniskeptconstantduringtheprocess.Themixtureanalyzedisvaporizedattheentranceofacolumnthatcontainsasolidsubstance(theadsorber)calledthestationaryphaseandthenistransportedacrossitbyacarriergas.Thecarriergas,orvectorgas(usuallynitrogen,sometimeshydrogenorhelium),isthemobilephase(seeFig.1).Inmostcasesithastobeinertedvisavisthesolutesandthestationaryphase.Thepaperisorganizedasfollows.First,weexamineanddiscusssomemodelsarisingingas-solidchromatography(orcloselyrelatedprocesses).Nextwegivesomemathematicalresultswhichseemtoussignificantinthatfield.Weemphasizeinparticularthecaseoftheso-calledPressureSwingAdsorpionprocess(PSA)inthecaseoftwocompounds,forwhichwegivesomenewresultsbytheauthors.Lastlywementionsomeopenproblems.Wespecifythatwedonotseektobeexhaustive,neitherfromthepointofviewofmo-delling,norofthatofthemathematicalanalysis.Ourgoalistoconcentrateonsomeproblemswhichappearmoreparticularlyinterestingtousintermsofpotentialdevelop-ments,pointingoutvariousopenproblems.2