saetta

CEMENTandCONCRETERESEARCH.Vol.23,pp.761-772,1993.PrintedintheUSA.0008-8846D3.$6.00+00.Copyright©1993PergamonPressLtd.THECARBONATIONOFCONCRETEANDTHEMECHANISMOFMOISTURE,HEATANDCARBONDIOXIDEFLOWTHROUGHPOROUSMATERIALSAnnaV.Saetta,BernhardA.SchreflerandRenatoV.VitalianiIstitutodiScienzaeTecnicadelleCostruzioniFacolt~diIngegneria-Universit~diPadova-ViaMarzolo9-35131-Padova-Italy(Refereed)(ReceivedFeb.18,1992;infinalformMarch29,1993)ABSTRACTThegoverningequationsofmoisture,heatandcarbondioxideflowsthroughconcretewithintheframeworkofadistributedparametermodelaredescribed.Thecouplingtermsandthenon-linearityoftheproblemaretakenintoaccountandanumericalprocedurebasedonthefiniteelementmethodisdevelopedtosolvethesetofequations.Theinfluenceofrelativehumidityandtemperatureisinvestigatedandtypicalresultsarepresented.Comparisonswithexperimentaltestsarealsocarriedoutandoneexampleispresentedindetailinordertoshowthereliabilityandtheeffectivenessoftheproposednumericalmodel.Ialr.aaalgliaaThedurabilityofplainandreinforcedconcretestructureshasrecentlyreceivedmuchattention.Severalexperimentalstudies[e.g.1to5]aswellastheoreticalmodels[e.g.6to9]toanalyzethedifferentmechanismofmaterialdeteriorationhavebeendeveloped.Thestudyofthedurabilityofconcretestructuresneedsamultidisciplinaryapproachbasedonthephysicsandchemistryofmaterials.Inparticularthedurabilitydependsbothontheabilityofconcretetoresistthepenetrationofaggressivesubstancesfromtheenvironment,andonitsabilitytoprotectembeddedsteelreinforcement.Thetransportofaggressivespeciesmayfollowdifferentmechanisms,dependingontheporestructureoftheconcrete,theexposureconditionsandthecharacteristicsofthediffusingsubstances.Thepenetrationofcarbondioxideoroxygen,aswellasofmanyothergaseoussubstances,mayoccuronlyiftheporesofconcretearealmostdry,whilethediffusionofchlorideorsulphateionstakesplaceonlyinwater.Thereforeallthesemechanismsshouldbestudiedtogetherwiththeevolutionofmoisturecontentinsidetheconcrete.Oneoftheprocesseswhichtakesplaceintheporesofconcreteandwhichmaylimittheservicelifeofreinforcedconcretestructuresisthecarbonationofmaterial,i.e.thechemicalprocessdenotingthereactionofcalciumhydroxidewithcarbondioxideresultingintheformationofcalciumcarbonateandwater[1,11,12].ThemainconsequenceofcarbonationisthedropofthepHoftheporesolutionofconcretefromthestandardvaluesbetween12.5and13.5,toavalueofabout8.3inthefullycarbonatedzones,sothatthepassivelayerthatusuallycoversandprotectsthereinforcingsteelagainstcorrosionbecomesnolongerstable[8,10].761762A.V.Saettaetal.Vol.23,No.4Actuallytherealmechanismofcarbonationiscomplexandinvolvesdifferentstepsthataremutuallyinterdependent,hencemakingnecessaryacarefulreviewoftheexperimentalresultslistedinliterature[see13,14].Thenumericalmodelspresentedinliterature[e.g.6]haveprovedtobepowerfultools,allowingtotreatthesecomplexinteractiveprocessesinaquantitativeway.Obviously,theparametersusedinthesimulationmustbelinkedwithconcretecharacteristicsandwithexposureconditionsbymeansofsuitable,andalwaysnecessary,experimentalstudies.Thecarbonationrateisessentiallycontrolledphysicallybythecarbondioxidediffusionprocessandchemicallybythereserveofcalciumhydroxideoftheconcrete.Thereforeaneffectivenumericalmodeltosimulatethewholephenomenonshouldconsidertheinteractionbetweenmanyprocesses,namelytheCO2diffusion,themoistureandheattransfer,themechanismofCaCO3formationandtheavailabilityofCa(OH)2inporesolutionduetoitstransportbywatermovement.Sincethecarbonationofconcreteisonly"onepieceinthejigsawpuzzle"[17],itshouldbestudiedincontextwithotherprocesses,suchaschloridepenetration,sulphateattack,crackformation,freezingofporesolution,oxygendiffusionandwatersorptivity,thatundernormaland/orextremeclimaticconditionsmayaffectthedurabilityofthestructure.Inparticularcarbonationisusuallyassociatedwithvolumechanges,whichcancausemicrocrackingandcracking,whichinturncanchangetheeffectivepermeability.Thereforecarbonationprobablycauseschangesinpermeabilityanddiffusivitydirectly.Allthesephenomenawouldaffectthediffusionequationsandshouldbetakenintoaccountinthedevelopmentofthisresearch.Onepossiblewaytoensuredurabilityofthestructurecouldbetospecifyandassessdirectlytherequiredqualityofconcrete,namelyitsresistancetochemicalandphysicalattackswhichareattheoriginofdamage.Inthisstudythefundamentalrelationshipsdescribingthecarbonationofconcreteatvariabletemperaturearepresentedwithreferencetointernalandexternalinfluences.ThegoverningpartialdifferentialequationsaredevelopedonthebasisoftheworksofWittmannetal.[6,18to20],andaresolvednumericallytocalculatethedepthofcarbonationasfunctionoftime.AssuggestedbyWittmann,materialparametersareincludedintheanalyticaldescriptionsoallowingtorepresenttheinfluenceofconcretecharacteristicsonthechemical-physicalprocesses.Inparticular,sincethediffusionofcarbondioxideintoconcretedependsonthemoisturecontentintheporesandonthetemperature,aswellasoncarbondioxideatmosphericconcentration,theequationofCO:diffusioniscoupledwiththedifferentialequationsgoverningtheheatandmassflowinthedryingprocessofthematerial[21to27].Thismassflowisexpressedintermsofporehumidity,takingalsointoaccountthenon-linearitytypicalofthesephenomena.Itisworthnotingthatthemathematicalanalysisofdiffusionphenomenaandchemicalreac-tionsissimilartothecorrosionproblemofsteelinconcreteandthefreeze-thawproblem[28to30]BasicHvuothesesandModelDescriptionThecarbondioxidediffusesfromtheatmosphereintothecapillaryporesofconcreteandcombineswithwaterformingcarbonicacidwhichthenreactswiththealkalihydroxide,sodium,potassiumandcalcium,formingcarbonates.Sincetheconcentrationofcalciumhydroxideinnormalconcreteisusuallyhigherthantheconcentrationoftheotherhydrationproducts,thereactionofCO2withCa(OH)2predominates,andthecarbonationprocesscanbesimplydescribedbythefollowingchemicalreaction:CO2+Ca(OH)2H~o)CaCO3+H20(1)Actuallythisequationrepresentsonlythefinalresultofseveralstepsthroughwhichthetruereactionoccurs.Furthersimplificationistoconsiderthereaction(1)irreversibleandinstantaneous,sothatthecalciumcarbonateimmediatelyprecipitatesoncethetworeactantscoexistinsolution.Iftheporesarefilledwithwater,thepenetrationofcarbondioxideishindered'becauseofthelowrateofdiffusionofCO2inwater,whileiftheporesarecompletelydrythereactionofcarbondioxidewithwatermoleculesisabsent(inpracticethisisthecaseofoven-driedconcrete).Vol.23,No.4CONCRETE,CARBONATION,MOISTURE.HEATFLOW,MECHANISMS763Inbothcasescarbonationwouldbeverylargelyruledoutbecauseoftheabsenceofnecessaryconditions.Actuallythecaseofporesonlypartlyfilledwithwaterisnormallytheconditionoftheconcretesurface,andthecarbonationcanproceedonlytothedepthatwhichtheconcreteporeshavepartiallydriedout.AccordingtoBa~ant[21to25],theformulationofdryingcoupledwithheattransferisdevelopedintermsofrelativehumidityratherthanwatercontentandthediffusivitiesofwaterandcarbondioxideareassumedtobestronglydependentonporehumidity,temperatureandalsoonthedegreeofhydrationofconcrete.Notethattheuseofevaporablewatercontent,wo,asbasicvariableactuallyinvolvescertainerrors[22],sinceFick'slawintermsofw,cannotbeappliedtoobtainthedifferentialequationgoverningthedryingphenomenonwhenhydrationproceeds.Thisisbecausetheporositybecomesnon-uniformintimeandequalvaluesofevaporablewatercontentdonotcorrespondtoequalvalueofporehumidityinthesamepoints.RateofcarbonationTheamountofcalciumcarbonate[CaCO3]thatformsinaunitoftimedependsonthedegreeofcarbonation(i.e.theavailabilityofcalciumhydroxide[Ca(OH)2]),thetemperatureT,thecarbondioxideconcentration[COd,andtherelativehumidityhintotheporestructureofconcrete.Assumingthatreaction(1)isoftheFirstorderwithrespecttoCO2andCa(OH)2concentrations,itsratevcanbewrittenasb[CaCO3]=f(h,T)x[Ca(OHh]x[CO2]v=~tandthefunctionf(h,T),canbeexpressedasfollows:f(h,T)=fl(h)xOqxkinwhichk=A.e-E°/RTrepresentstheArrheniusequationforthermicallyactivatedprocesses,¢z~isamaterialparameter,andfl(h)takesintoaccountthemoistureinfluenceoncarbonation.Thereforethecarbonationratecanbewritteninthefollowingway,verysimilartothatproposedbyWittmann[6,7]:Oco._t_=v={x~×fl(h)×fz(g)×f3(c)xf4(T)(2)Inthisequationf3(c)describestheinfluenceofthedegreeofcarbonationc,andvariesbetween0and1accordingtothefollowingrelationship:fi(c)=1-(c--~.~)~(3)inwhichCm~xisthemaximumamountofcalciumcarbonateandmisusuallytakenequalto1.Alsothefunctionf2(g)rangesbetween0,inthezoneswherecarbondioxidehasnotyetpenetrated,and1,wherecarbondioxideconcentrationisequaltoitsmaximumg....Thefollowinglinearrelationshipcanbeassumed:f2(g)=ggm*~(4)Thefunctionfx(h)isintroducedtotakeintoaccountthatreaction(1)mayoccuronlyinpresenceofwater.Foralmostdryconcrete(h<0.5)thisfunctionisequaltozero,hencestoppingtheprogressofcarbonation:764A.V.Saettaetal,Vol.23,No.4f,(h)=(h-0.5)0<h<0.50.5<h___0.90.9<h<l(5)Thefourthfunctionf4(T)=A.e-E*/RTistheabovementionedArrheniusequation,inwhichE.istheactivationenergyofthecarbonationprocess,RistheconstantofgasandTistheabsolutetemperature.Finallythematerialparameter0q,dependingontheconcretecharacteristics,takesintoaccountthatthecarbonationreactionoccursinanopensystem,i.e.theconcrete,andthisfactmaymodifytherateofreactiondeduciblefromsimplechemicalconsiderations.Waterdiffusionandheattransferinnon-saturatedconcreteSincethediffusionofwaterthroughconcreteisaveryslowphenomenon,thedifferentphasesofwaterinthesingleporecanbeconsideredinthermodynamicequilibriumatanytime.Thereforethedesorption/sorptionisothermsgoverntherelationshipsbetweentherelativehumidityhandthewatercontentw(dh=kdw).If,asusual,theslopekofthedesorptionisothermatroomtemperatureisapproximatelytakenasconstantinawiderangeofrelativehumidity(i.e.0.15<h_<0.95),itmaybecombinedwiththepermeabilityofconcretec,yieldingthediffusivityC=kc.Inthishypothesis,theequationgoverningthedryingofconcretecanbewritten[21,22]:~h3___t_=div(Cgradh)+~t.~t,+K~T(6)wherehistherelativehumidity,Tisthevariable,butalmostuniform,temperature,Kisthehygrothermiccoefficientgivenby:)(l-h)~h=0.0135xhx(1.25-h)K(h)=w.u(7)andh,(i.e.theself-dessication)isanempiricalfunctionthatrepresentsthegradualdecreaseofporehumidityfromtheinitialvalueof1.00(initiallywetspecimen)toabout0.96-0.98afterlongconservationandinsealedconditions(i.e.withoutexternaldrying).Thediffusivity,ordiffusioncoefficient,C=kcofconcretedependsonhandT,accordingtothefollowingrelationship[22,24]:C(h,T)--Cl(to,T)[a0+1--(X0(1-hV(8)whereCto,nandhoareconstantparametersrepresentingrespectivelytheratiominC/maxC,rangingbetween0.025to0.10;thespreadofthedropinthediffusivityC,variablebetweenn=6ton=16and,finally,thehumidityatwhichthediffusivityCdropshalf-waybetweenitsmaximumandminimumvalues(ho=0.75).AsuitableexpressionforC~,thatstandsforthediffusivityathumidityh=1,isthesemi-empiricalformula,proposedbyBa~ant[23,24]:TexP(RQ~0QT)Cl(T'tc)=C0[0"3+~t@]~0(9)InthisequationQistheactivationenergyofthediffusionprocess,RisthegasconstantVol.23,No.4CONCRETE,CARBONATION,MOISTURE,HEATFLOW,MECHANISMS765(usuallyQ/R=4700K),Toisthechosenreferencetemperature(inKelvin,normally296K),C0isthediffusivityforT=To,after28days,andt~istheequivalenthydrationperiod,definedby:witht.=~/~r~dt(10)[-Uh(--I_=exPLT°-XT)](11)fl,=[I"I-(£~--~h)4]-IinwhichUhistheactivationenergyofhydrationandthcratioU~RcanbeexpressedbytheempiricalrelationUh/R=4600[30/(T-263)]°.39.Thevalueofparameteraisobtainedbytestdataanditisroughlyabout5.Inordertotakeintoaccountthewaterliberatcdduringthereaction(i),thcdifferentialequation(6)mustbereplacedbythefollowingone:Oh0TOhodiv(Cgradh)+~+K+~--~0t(12)where:Oh0c--~=ct2-~-isthechangeoftherelativehumidityduetocarbonationinaunitoftime;-tx2isannondimensionalparameterthatvariesbetween0and1,accordingtotheconcretecharacteristics,liketheparameterintroducedbyWittmannintheequationgoverningthemoisturediffusionintermsofmoisturecontentw[6].Thiscoefficientaccountsforthepossibleinteractionbetweenthewaterliberatedduringcarbonationandtheconcretecomponents;-0c/0tistherateofcarbonation,givenbyequation(2).Theuseofporehumidity,h,asbasicvariableallowsustoneglecttheself-dessicationterm0hd0tintheequation(6),sinceforusualwater/cementratiosthedropinhduetothisprocessisrathersmall(h,>0.95).Onthecontrary,intheequationgoverningdryingphenomenawrittenintermsofevaporablewatercontent[22],therateoflossinfreewaterduetohydrationreaction(0wdOt)mustbeincluded.Thedifferentialequationthatdescribestheheatdiffusioncoupledwiththedryingprocessinconcretecanbewrittenintheusualwayasfollows[26]:pCq-~=div(bgradT)+OQhot(13)wherepandCqrepresentrespectivelythemassdensityandtheisobaricheatcapacityoftheconcrete,Q~istheoutflowofheatperunitvolumeofsolid,bistheheatconductivity.Similarlytothecaseofmoisturediffusion,equation(13)mustbemodifiedtotakeintoaccounttheheatdevelopedduringcarbonation,anditcanberewrittenas:0T(bgradT)+~+0TopCq~=divOt(14)where:-=oq-~-representsthechangeofthetemperatureduetocarbonationinaunitoftime;-ct3isannondimensionalparameter,likeix2inequations(12),thatvariesbetween0and1,accordingtotheconcretecharacteristics.766A.V.Saettaetal.Vol.23,No.4CarbondioxidediffusionprocessTheprocessofCO2diffusionintoporousmaterials,suchasconcrete,canbedescribedbymeansoftheusualdiffusionequation,resultingfromlcksfirstandsecondlaws:F.t•Og=div[Dggrad(g)]Ot(15)whereDgstandsforthediffusivityofcarbondioxide,thatdependsontemperatureandrelativehumiditycontentofconcrete.Inordertotakeintoaccountthelossofcarbondioxidetakingplaceduringreaction(1),thedifferentialequation(15)mustbereplacedbythefollowingone:Og=div[Dggrad(g)]og~D--i-~t(16)where:__~t~=iX0c4~representsthechangeofCO2concentrationduetocarbonationinaunitoftime;-0~isanondimensionalparameter,like¢z2usedinequation(12),thatvariesbetween0and1,accordingtotheconcretecharacteristics.ThisparameteristhesameasthatintroducedbyWittmann,[6].NumericalExample,sDuetothecomplexityresultingfromthenon-linearitiesandthecouplingofthedifferentialequations(2),(12),(14)and(16),analyticalsolutionsareverydifficulttoobtainandanumericalapproachshouldbepreferred[e.g.31].Forthemathematicalexpressionofthediffusionequationsandchemicalreactionswecanobservethatitissimilartothemathematicalformulationofthecorrosionproblemofsteelinconcreteandofthefreeze-thawproblem[28to30]Inthisstudy,thestandardfiniteelementmethodisappliedbothintimeandinspace.AfterspacediscretizationusingaGalerkinprocedure,weobtainthefollowingsystemofcoupledordinarydifferentialequationintime:Qh+Idh_THdT_cHdtdtLg+Idg+cGdC=0dtdtIdC+Sc-CJ=0dtdcOHGdt0t--=0(17)TRT+TsdT_cTde~TG=0dtdt~twhereh,e,gandTarerespectivelythediscreterelativehumidity,thediscretecarbonateconcentration,thediscreterelativecarbondioxideconcentrationandthediscretetemperaturevectors;Q,I,TH,ltG,TR,TS,TGaretheusualmatricesofthediscretizedmassandheattransferequations[27];CH,CGandCTarethematricesrepresentingrespectivelythecouplingbetweenthecarbonationprocessandthehumiditydiffusionphenomenon,thecarbondioxidediffusionphenomenonandtheheattransfer;Listhecarbondioxidediffusionmatrix;SandCJarethematricesthatrepresenttheinfluenceofrelativehumidityandtemperatureattheprevioustime-steponthecarbonationprocess.Forthedetailedexpressionofthematrices,seeReferences27and33.Thesystem(17)canbewritteninthefollowingmoreconciseform:Vol.23,No.4CONCRETE,CARBONATION,MOISTURE,HEATFLOW,MECHANISMS767Bx+Cx=F(18)wherethemeaningofthetermsofequation(18)followsimmediatelyfromequation(17).Thissystem(18)aftertimediscretization(i.e.one-stepalgorithm),becomes:[B+CotAt]xn+]=[B-C(1-(x)At]xn+Fn+c~At(19)withtheusualmeaningofthesymbols[32].Thealgebraicnon-linearequation(19)issolvedbymeansofadirectapproach.Thenecessaryconvergenceanalysishasbeenmadebyimposingthelimitationofthetotalerror.Inparticularthestabilityconditions,theconvergenceconditionsandthecompatibifityconditionshavebeencarriedout[33,34].Thevalidationofthenumericalmodelpresentedinthispaperhasbeenmadebyusingseveralexperimentaldata,bothtakenfromliteratureandfromnewexperimentaltests.Forthesakeofbrevity,inthefollowingonlyonecomparisontestispresentedtoshowgoodagreementbetweennumericalandexperimentaldata.Somenumericalexamplesarepresentedtoshowtheinfluenceofvariabletemperatureandrelativeexternalhumidityonthecarbonationphenomena.Thematerialparameters~,ct2,ix3and~introducedinthedifferentialequations(2),(12),(14)and(16)andalsothevaluesofthediffusivitiesofcarbondioxideandwater,mustbeevaluatedbyfittingexperimentaldata.Inparticulartheinfluenceofwater/cementratio,concretecomposition,initialdegreeofhydrationandcuringtimeconditionshavetobetakenintoaccount.Moreoverthechangesinpermeabilityanddiffusivityduetothemicrocrackingandcrackingphenomenadirectlyduetothecarbonationprocess,havetobebealsoevaluatedandincludedinthenumericalmodel.Atthisstageoftheresearch,somevaluesoftheseparameterscq,tz2,ct3andtx4anddiffusivitiesofcarbondioxideandwateraretakenfromliteratureandothersarecalibratedbyavailabletestdata.ComparisontestFigure1showsthecomparisonbetweenexperimentaldataobtainedbyHoandLewis[15],andthenumericalresultsobtainedfromthesolutionofthesetofequations(2),(12),(14)and(16).Thedifferenttypesofconcrete,characterizedbydifferent28-daysstrengthlevel(i.e.25MPa,30MPaand38MPa),aretakenintoaccountinthenumericalmodelbymeansofdifferentvaluesforthediffusivitiesofcarbondioxideandwater.Alsotheparameteralis35-30-E25-o~20-~15-~105°I...........................................4""~i........l.........:.-'7"i.....It""i..............:.7i..........[=3a[.........................~"'"t,*1,52.533.5l/f-(tinweeks)FIG.ICarbonationdepth-exposuretimerelationshipsasinfluencedbythe28-daysstrengthlevels.Comparisonbetweennumericalandexperimentalresults(dots)[15]768A.V.Saettaetal.Vol.23,No.4supposedtovary(thelowervalueisusedfortheconcretewith28-daysstrengthlevelequalto38MPa),whileor4,ct3and~2arekeptconstantforalltheconcretes.Itisworthemphasizingthatthevaluesoftheparameterstxt,otz,tx3andor4havebeenestimatedinthepreliminaryphaseoftestingofthenumericalmodel,byusingbothexperimentaldataandthevaluesgiveninliterature.Inthisexample,thedifferentvaluesoft~lassumedfordifferentstrengthlevel,dependonthedifferentrateofreactionobservedintheexperimentaltests.Onthecontrarytheparameters~,t~3andtx2canbeassumedconstantwiththestrengthlevel,asshownbothbytheexperimentaltestsandbythedataavailableinliterature.Notethattheagreementisgood,andthattheshorttermtestsofreference[15]followtheclassicalsquarerootrelationshipbetweendepthofcarbonationandtime.VariabletemperatureandrelativeexternalhumidityinfluencesFigure2depictstheinfluenceofvariableexternaltemperature(i.e.1°C,20°Cand90°C)onthedepthofcarbonation.Figure3dealswiththecaseofvariablehumidity(i.e.RelativeHumidityRH=40%,60%and80%)andshowsthetrendofcarbonate(Figure3a),carbondioxide(Figure3b)andporehumiditycontent(Figure3c)after250,500,750and1000days,foraconcretemadewithOrdinaryPortlandCement,withawater/cementratioequalto0.50andinitialrelativehumidityequalto0.98.Alltherelationshipsshowninfigures2and3havebeenobtainedbyusingthenumericalmodelpresentedinthispaper.ConclusionsandDiscussionTheneedtodesigndurableconcretestructuresinchemicallyaggressiveenvironmentsleadstoevermoresophisticatedmodellingofdeteriorationphenomena.Inthesemodelsitisessentialtoconsidercouplingandnon-linearitiesofalltheprocessesinvolvedinthedamagemechanism.Sinceconcreteisaporousmaterial,thepenetrationofaggressivespecies,suchascarbondioxideorchlorideions,isbasicallydeterminedbytheporestructureofthematerialandbythelevelofporewaterfilling.Thereforethecouplingbetweendryingphenomenaofconcreteandcarbondioxidediffusionprocessmustbeconsidered,togetherwiththeinfluenceoftemperature,tomakethesimulationclosetotheexperimentalreality.,0j......L9......IiJ'........."]IOrdinaryPortlandCementIi/..""/i.:'.":s:84......IRatiow/c0.50l.-:........,,(....-'Lt..........~.........'l•s;I|IRelativeHumidity60%I/i:'-/'::7-]......ICO2concentration0.035%1"""'j/'-::i'S'""i.........i.........o=II/..:,.'ii63.........;.........~.........,.........;..........;.:..........~.~,.~.........:.........~.........:/...,,::::::."s:::.g5...........:.........,.......~":'}/';:."/......~.........:.........'....