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BASE Biotechnol. Agron. Soc. Environ.201418(3),383-396 Focus on:

Non-isothermalmoisturebalanceequationinporousmedia:areviewofmathematicalformulationsinBuildingPhysicsSamuelDubois(1),ArnaudEvrard(2),FrédéricLebeau(1)(1)UniversitédeLiège-GemblouxAgro-BioTech.DépartementdesSciencesetTechnologiesdel’Environnement.PassagedesDéportés,2.B-5030Gembloux(Belgium).E-mail:[email protected](2)UniversitéCatholiquedeLouvain.ArchitectureetClimat.PlaceduLevant,1.B-1348Louvain-La-Neuve(Belgium).

ReceivedonOctober4,2013;acceptedonMarch17,2014.

Understandingheatandmasstransfersinporousmaterialsiscrucialinmanyareasofscientificresearch.Mathematicalmodelshaveconstantlyevolved,theirdifferenceslyingmainlyinthechoiceofthedrivingpotentialsusedtodescribemoistureflows,aswellasinthecomplexityofcharacterizingthephysicalphenomenainvolved.ModelsdevelopedinthefieldofBuildingPhysics (HAMmodels) are used to describe the behavior of envelopeparts and assess their impact onuser comfort andenergyperformance.Thewaterbalanceequationcanbedescribedinmanyways;itisafunctionoftheboundaryconditionsconsideredandthefacttheyinducehighorlowwatercontentintheporousmaterialsused.ThispapergivesanoverviewofvariousformulationsforthisequationthatarefoundintheBuildingPhysicsliterature.Itfocusesfirstonthephysicallybased formulation ofmoisture balance, drawing on theRepresentativeElementaryVolume (REV) concept, coupledwiththermodynamicflowratesdescription.Thisisthenreformulatedinlinewithvariousmainmoisturestatevariablesofferingawidevarietyofexpressionsthatarecomparedwithavailablemodels.Thisapproachprovidesaccesstoallsecondarytransportcoefficientsassociatedwiththeprocessofmathematicaltransformation.Particularemphasisisplacedonthemoisturestoragefunctionchoiceanditsimpactonthefinalmathematicalformulations.Keywords.HAMmodeling,porousmedia, capillarity, sorption,diffusivity, isostericheat,mathematicalmodels,moisturecontent.

L’équation de bilan d’humidité dans les milieux poreux : synthèse des formulations mathématiques en physique du bâtiment.Lacompréhensiondestransfertsdechaleuretdemassedanslesmilieuxporeuxestcrucialedansdenombreusesdisciplines scientifiques. Les modèles mathématiques ont constamment évolué pour se différencier principalement parles potentielsmoteurs utilisés pour décrire les flux d’eau, ainsi que par la complexité de la description des phénomènesphysiquesimpliqués.EnPhysiqueduBâtiment,lesmodèleshygrothermiques(modèlesHAM)sontimportantspourdécrirelecomportementdesparoisetainsidéterminer leur impactsur leconfort intérieurautantquelaperformanceénergétique.L’équationdebilanhydriquepeutyêtredécritedemultiplesmanièresetestfonctionducasétudiéetlefaitqu’ilinduisedesbassesouhautesteneurseneaudanslesmatériauxdeconstruction.Cetarticleproposeunesynthèsedesdifférentesformesquepeut prendre cette équationdans la littérature scientifique.Nousproposons de partir de l’équationde bilanhydriquephysiquementbasée,héritéeduconceptdeVolumeÉlémentairereprésentatifetdeladescriptionthermodynamiquedesflux.Celle-ciestensuitereformuléeenutilisantdifférentesvariablesd’étatprincipales,offrantunegrandevariétéd’expressionsfinalesqui sontcomparéesauxmodèlesdisponibles.Ladémarcheoffre l’accèsà l’ensembledescoefficientsde transportsecondairesliésauprocessusdetransformationmathématique.Unaccentparticulierestégalementportésurlechoixdelafonctiondestockaged’humiditéetsursonimpactsurlesformulationsfinalesobtenues.Mots-clés.ModélisationHAM,milieuxporeux,capillarité,adsorption,diffusivité,chaleurisostérique,modèlemathématique,teneureneau.

384 Biotechnol. Agron. Soc. Environ. 201418(3),383-396 DuboisS.,EvrardA.&LebeauF.

1. INTRODUCTION

Heat, air and moisture (HAM) transfers acrossenvelope elements play a crucial role in buildingperformance and durability as well as in usercomfort (Fang et al., 1998; Padfield, 1998;Toftumet al., 1999; Adan et al., 2004; Salonvaara et al.,2004;Osanyintolaetal.,2006;Mudarrietal.,2007).Theabilityofporousmaterials to storeand releasemoisture through capillarity and hygroscopicityeffectshasan impactonbuildingbehaviorandcancause damage when not sufficiently controlled.Modeling this complex behavior in multilayeredporouswallshasbecomeanimportantresearchtopicinBuildingPhysicsandperformanceanalysis.

Moistureflowsinporousmediahavebeenstudiedfora longtimeinvariousresearchfields, includinggroundwater hydrology, agricultural, chemical andpetroleumengineering,anddryingtechnologies.Thefirst work on drying modeling described moisturetransfersviaadiffusion-likeequationlinkingtheflowratetotheproductofamoisturecontentgradientandaconstanttransportcoefficient(Sherwood,1929).Atthe same time, soil scientists were discovering theimportanceofcapillaryactioninmoisturemovement.Contemporary mathematical models have theirrootsintheworkofPhilipandDeVries(1957)andLuikov (1966), who were the first to consider thethermal effect onmoisture transfer and to separateliquid and vapor flows.At that point, however, thegoverning equations remained phenomenologicalbecause theywere inferred from continuousmediadescriptions in a rather “intuitive” way. Whitaker(1977) rationalized the theory by detailing the gapbetween microscopic and macroscopic descriptionthrough the Representative Elementary Volume(REV) concept. Moisture transfer description wasthereafterwidelydiscussedandfurther investigated(Millyetal.,1982;Nielsenetal.,1986;Milly,1988).Themaindifficultyremainedinaccuratelydescribingnon-isothermalmoisturetransferanddeterminingthetransfercoefficientsexperimentallybecauseoftheirhighdependenceonthetransferpotentials.

Initially,all theseadvances in theunderstandingofporousmediahada limited impactonmodels intheBuildingPhysicsfield.Othermethodshadbeendeveloped,mainly tocharacterize theaccumulationofwater inwallsbyvapordiffusion insteady-stateconditions (Glaser, 1959). In this context, someauthors showed, prematurely, the potential ofincorporatingexternalconceptsfromsoilscienceintoBuildingPhysics(VanDerKooi,1971;Hall,1977)andthefirstbuilding-orientedmodels incorporatingtransientheatandmoisturetransferswithcapillarityeffectsappearedinthe1980s.Theywereallorientedtowardsenvelopepartdescriptionin1D.Inthe1990s,

the first computer models were commercialized(including Rode, 1990; Künzel, 1995). As thefundamental understanding of the phenomena andcomputerpowerhaveimproved,manyothermodelshavebeendevelopedoverthepasttwodecades,withincreasing accuracy and capabilities. There is anexhaustive list of available software on theEnergyToolswebsiteof theUSDepartmentofEnergyandinDelgadoetal.(2013).

Inreality,thereisnouniquemeansofdescribingmoisture transfer in porous buildingmaterials.Themathematical expression of the moisture balanceequation and its complexity depends on the initialdefinition of driving potentials for moisture flows,the simplificationhypothesis, thechosendependentvariables and the consequent secondary transportcoefficients. This diversity is related to the widevariety of building materials, the hygrothermalconditions they face and typical cases studied inthe laboratory or in thefield. It is obvious that themoisture buffering of a homogeneous sample atambient temperature is not dominated by the sametransport mode and would not require the sameassumptions for its description as rain absorptionin a composite wall. Traditionally, HAM modelswere first designed to treat unsaturated problems,whereas soil-dedicated tools were first optimizedfor near-saturation processes. This explains whytheir sets of preferred variables often differ. Bothmodel types are now trying to broaden moisturecontent range description, while also addressingnon-isothermal case studies. There is an urgentneed for standardization of the mathematicalformulation.

Inthispaper,wefocusonclassicnon-isothermal,non-hystereticmoisturetransferinaporousmedium.Initially, we look at the moisture balance equationderivedfromtheaveragingprocessoveraREV,withallmoisturefluxesdescribedaccordingtotheactualthermodynamic driving potential.No other balanceequationisconsideredherebecausetheemphasisison moisture transfer characterization. Linking thisbalanceequationwithaphysicallybaseddescriptionof moisture fluxes, we show how different HAMformulationsareobtainedbyreducingthenumberofvariables and further simplifying the mathematicalexpressions.Severalwaysofcharacterizingmoisturestorage functions are also presented, includingtemperature effects, which are seldom discussedin Building Physics. The various mathematicalexpressionspresentedinthispapercouldbeusedtodevelopmodels for general numeric computationaltools, such as Comsol Multiphysics. Given theirinter-operability, these tools have great potentialfor solvingmanyHAMbuilding problems (Duboisetal.,2013).

Moisturebalanceequationinporousmedia 385

2. PHYSICAL BACKGROUND AND MATHEMATICAL TRANSLATION

2.1. Macroscopic modeling

A porous medium is a heterogeneous systemcharacterizedbyacomplexmattercombination:asolidstructurewhoseporesarefilledwithliquidandgaseousphases.InordertousethetraditionalcontinuousPartialDifferentialEquations(PDEs)formulationtodescribethe structure’shygrothermalbehavior, it is necessaryto make some simplifications by averaging thephenomenaandvariablesoveraREV(Whitaker,1977;Bear,2013).Thiscreatesequivalencebetweentherealdispersedenvironmentandafictitiouscontinuum.Atanypointofthisnewmathematicalspaceareassignedvaluesofthevariablesandparametersthatareactuallyaveragedvaluesoveravolumearoundthislocalpoint(Bear,2013).

When the porousmedium containsmultiple fluidphases, as in most cases in Building Physics, it isreplacedby theoverlapofseveralfictitiousaveragedcontinuums.Eachofthesecontinuumsisassignedtoaphaseandfillstheentiredomainoftheporousmedium.It can then be described with typical conservationequations for extensive quantities: inner energy andmass.Inordertoreducethenumberofunknowns,someadditional information isneededand this isprovidedbytheconstitutiveequations,whicharerelationshipsbetweenthefluxdensitiesoftheconservedquantitiesandsomedrivingforcesexpressedas thegradientofa state variable. These equations are also sometimereferred as phenomenological equations (Bear,2013) because of their dependence on experimentalobservations.Finally,byassigningareferencevelocityfrom the selectedEulerianpointofview,all thefluxdensities of the conserved quantities can be dividedintoconvectiveanddiffusiveflows.

2.2. REV conservation equation

General assumptions.ItisassumedinthefollowingsectionsthattheporousmediumΩ⊂R³isamultiphasesystem consisting of the solidmatrix, a liquidwaterphase and a gaseous phase, comprising dry air andwatervapor.Thetotalgaspressureintheporousmediaisexpressed:

pg=pa+pv . Eq.1

The following additional assumptions are madefor themathematical developments described in thispaper:– thematerial isnon-deformableandisotropic;fora non-isotropicmaterial,standardtransfercoefficients havetobereplacedbytensors;

– the fluid phases do not react chemically with the solidmatrix;– themoisturecontentofthematerialstoredasvapor rvθv is considered negligible compared with the correspondenttermintheliquidphase;– thedryairpressure isconstant in thematerial (no airadvection)andthetotalgaspressuregradientis considerednegligible;– nosolid-liquidphasechangeisconsidered;– thereisalocalthermodynamicequilibriumbetween the different phases, which means that water has the same thermodynamic potential in the gaseous, adsorbedandcapillaryphases;– there are no thermal effects caused by friction or compression;– theSoreteffectisneglected(Janssen,2011);– nohysteresisphenomenaaretakenintoaccount;– nogravityeffectisconsidered.

ThePDEdescriptionrespectsthefollowinggeneralscalarform:

da∂u∂t= –∇⋅Γ+F Eq.2

whereuisthedependentvariable,Fanddaarescalarcoefficients andΓ is the fluxes vector.The differentcoefficientscanbefunctionsofthespatialindependentvariables, the dependent variable and the spacederivativesofthedependentvariable.

Moisture balance. Equations3 and 4 show theequationsofmassconservationaveragedovertheREVforthetwowaterphasespresentintheporousmaterial.Notethatinthispaperweomittheaveragednotation.

(Liquid) Eq.3d ρlθl( )

dt= –∇⋅

j Ml − m

d ρvθv( )dt

= –∇⋅j Mv + m

(Vapor).Eq.4

Thesymbol jMl

(kg.m-2.s-1) is the totalmassfluxdensity of liquid water, j

Mv (kg.m-2.s-1) is the totalvapor mass flux density and m (kg.m-3.s-1) is thephasechangerate.Sincethevapormoisturecontentisoftenconsiderednegligibleincomparisonwithliquidmoisturecontent,wecanwrite:

m =∇⋅j Mv .

Eq.5


The two balance equations are added to give thetotalmoisturebalance:

ρl∂θl∂t

= –∇⋅j Mv +

j Ml( ) . Eq.6

When assuming negligible total gas pressuregradientsinthematerial,thereisnoconvectivevaporflux.Equation7showstheresultingbalanceequationwithdetailedtransportmechanisms:

ρl∂θl∂t

= –∇⋅jdMv +jcMl +jsurfMl( ) Eq.7

withthedifferentmassfluxdensitiesbeing:

jdMv

thevapordiffusion,

jcMl theliquidwatertransportthroughcapillarity,

jsurfMl

theliquidwatertransportinthesorbatefilm(orsurfacediffusion).

In a porous system containing a binary mixtureof dry air and water vapor under constant total gaspressure, the observed diffusive flow of vapor is acombination of Fick’s diffusion, Knudsen diffusionand a transition between both modes, depending onthepore size distributionof thematerial (Descamps,1997).Asaresult,thetotalvapordiffusionmassfluxdensitycanbeexpressedas:

jdMv = –δv ⋅∇pv Eq.8

where pv (Pa) is the partial vapor pressure and δv(kg.m-1.s-1.Pa-1)isthevapordiffusioncoefficientofthematerial,alsocalledvaporpermeability.Thelatterisacomplexfunctionofrelativehumidityandtemperature.

In an unsaturated porous building material, theliquidwaterissubjectedtothesuctionofthemediumthrough capillary forces. In a pore, the capillarypressurepc(Pa)representsthedifferencebetweengasandliquidpressureoverthemeniscus:

pc=pg-pl . Eq.9

This variable is always positive for hydrophilicmaterialsandtheliquidpressureisnecessarilylessthantheatmosphericpressure.InSoilPhysics,itiscommontofixtheatmosphericpressuretozero,whichresultsin

anegativeliquidpressureassuringthecontinuitywiththe positive water pressure met in saturated regionsunderawatercolumn.Inunsaturatedmaterials,wateristhussubjecttoapressuredeficitandinordertodealwithapositivequantityitiscommonforsoilscientiststo define the suctionψ as the negative of the liquidpressure:

ψ = – pl − pg( ) .Eq.10

The suction and the capillary pressure are thusrigorously identical and the use of one or the othertermisonlyamatterofchoiceorscientifichabits.Thecapillaryliquidflowrateisthendrivenbythecapillarypressure or suction gradient following Darcy’s lawand again assuming a negligible total gas pressuregradient:jcMl = −Kl∇pl = Kl∇pc Eq.11

withKl (s) the liquid transfer coefficient (called the“unsaturated hydraulic permeability”; also referredto as “liquid conductivity”). It is important to notethatmoisturetransfersduetocapillaritygofromlowsuction(highmoisturecontent,highrelativehumidity)to high suction (low moisture content, low relativehumidity),which is shown through the sign of righthandsideofEquation11.

Thelastmoisturetransportphenomenontoconsideris the flow occurring in the liquid film adsorbed onthe surface of pores. For building materials understandardoperating conditions,multilayerdiffusion isexpected to prevail because strictmonolayer surfacediffusion is active only in a very dry state. Thistransportmodeisrathercomplexanditisparticularlydifficult to isolate from capillary transport becauseboth phenomena can occur simultaneously in poresofdifferentsizes(Uhlhorn,1990).Thetransferoccursfrom high to low concentration of adsorbed water,whichdependsontemperatureandrelativehumidityinahomogeneousmaterial.InHAMmodels,multilayerdiffusionisneverfeaturedindetail.Inthispaper,wetakethisphenomenonintoaccountthroughtheliquidconductivity function definition, which implies thatthetemperatureeffectonmultilayeradsorptionisnotconsidered.TherearemoredetailedreviewsofsurfacediffusionmodelinginUhlhorn(1990)andChoietal.(2001).

In consequence, Equation12 shows the generalmoistureconservationequation,withphysicallybaseddrivingpotentials:

liquidpressureinSoilSciencereferrential


ρl∂θl∂t

= –∇⋅ Kl∇pc −δv∇pv( ) .Eq.12

2.3. Closing relationships

In order to solve the macroscopic balance equation(Equation12), relationships have to be formulatedbetween thedifferentmoisture-relatedvariables (i.e.,themoisturecontentθl,thevaporpressurepvandthecapillary pressure pc) in order to limit the numberof unknowns. First, as the local thermodynamicequilibrium hypothesis is assumed to be valid aboutthe meniscus in a pore, Kelvin’s equation (Defayetal.,1966)relatesthecapillarypressuretothevaporpressurethroughrelativehumidity:

pc = f ϕ,T( ) = −ρlRvT ⋅ ln ϕ( )

Eq.13

withtheinverserelationship:

ϕ = f −1 pc,T( ) = exp −pcρlRvT⎛

⎝⎜

⎞

⎠⎟ . Eq.14

TheMoistureStorageCurve(MSC)ofthematerialprovides the second necessary relationship. It is amaterial-dependentcontinuousfunctionthatexpressesthe equilibriummoisture content present in theREVfor a given thermodynamic humidity condition. Thestorageprocesscanbeexpressed inrelation toeitherrelative humidity or capillary pressure due to localequilibrium hypothesis. The function is complex tocharacterize on the whole moisture content rangebecauseitisdeterminedmainlybyphysicaladsorptionat low moisture content and capillary condensationat high moisture content, with a transition pointdependingontheporesizedistributionofthematerial,aswellasontemperatureorhysteresiseffects,whichalterequilibriummoisturecontentforagivenrelativehumidity/capillarypressure.As the studyofbuildingmaterialsbehaviortakesrootinseveralscientificfields,themoisture storage functions found in the literatureon HAM models take many forms, depending onthe moisture content range of interest. They alwaysincorporate adjustable parameters that are eitherphysicallybasedorpurelyempirical.

Scientistsstudyinglowmoisturecontentprocessesor non-capillary materials often use an isothermalfunction in the formθl = gϕ

(ϕ). This formulation iscommon in chemical or food process engineeringstudies,wheretheemphasisisonmono-andmultilayeradsorption,whicharesurfacephenomena.Inthispaperwe refer to these storage models as “type1 MSC”.

Someofthemaresemi-empiricalgϕformulations,such

as the BET model (Brunauer et al., 1938), whereasothers offer a truly empiricalg

ϕ adsorption function,

suchastheOswinformulation(Oswin,1946).Incontrast,otherscientistsemphasizethecapillary

condensation phenomenon linked to high moisturecontent. Soil scientists, for example, will often usea storage function that is dependent on suction. It isthencalledthe“MoistureRetentionCurve”andcanbeeasilyexpressedasθl =gp(pc).Inthispaper,thistypeof formulation is referred toas“type2MSC”.Manyof these models evolved from the VanGenuchtenexpression(VanGenuchten,1980)andofferempiricalgp description.Somephysicallyaccurategp functionsarebasedonporesizedistributionmodels.Thesemaythemselvesbedividedinto“BundleofTubesModels”(Carmeliet et al., 2002;Grunewald et al., 2003) and“Network Models” (Dullien, 1979; Carmeliet et al.,1999).Whenusingapore-spacemodel,theisothermalmoistureretentioncurvemathematicalfunctionisoftenachievable based on pore-filling theory. There is anexampleofthisapproachinHäupletal.(2003).

For materials showing mono- and multilayeradsorption as well as capillary condensation, type1and type2MSCs,with a physically based approach,could be combined by choosing relative humidity orcapillary pressure as the thermodynamic potentialandperformingvariabletransformationsofoneoftheMSCsusingEquations13or14.

Thetemperatureeffectonmoisturestorageisstillnot perfectly understood, nor has it been studied indetailinBuildingPhysicsbecauseitiswidelyexpectedto have a minor impact. For some products such aswood-basedmaterials, however, itmight play anon-negligible role (Rode et al., 2004).There are severalapproaches for incorporating thermal effects intomoisturestoragefunctiondescription.Forhygroscopic-orientedmodels, the temperature effect is sometimesconsideredbyintroducingatemperaturedependenceofthefunctionparametersorbyinterpolationonthebasisofseveralindividualisothermalcurves.Thistypicallygives a direct function in the formθl = gϕ,T (ϕ,T) or“type3 MSC”. According to Poyet et al. (2009),temperature affects equilibrium relative humiditythrough the Clausius-Clapeyron relationship, whichdescribestheheatinvolvedintheadsorptionprocess.The equilibrium relative humidity dependence ontemperatureandmoisturecontentisthenwrittenthus:

ϕ θl,T( ) = ϕ[ ]Tref ⋅expqst θl( )Rv

T −TrefT ⋅Tref

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Eq.15

whereqst (J.kg-1)isthenetisostericheatandtherelative

humidity at reference temperature is obtained by theinverseoftheisothermalsorptionfunction:


ϕ[ ]Tref = gϕ−1 θl( )⎡⎣ ⎤⎦Tref

. Eq.16

Such a ϕ(θl,T) function can be used to obtaindifferent isotherms in order to construct the type3MSCinterpolation.

In over-hygroscopic and capillary condensationorientedmodels, it isoftenassumedthat, foragivenmoisture content, temperature affects equilibriumcapillarypressuremainlybymodifyingwatersurfacetension. Equation17 provides an example of thisapproach,asproposedbyMillyetal.(1980):

Eq.17pc pc,T( ) = pc ⋅exp a T −Tref( )( )

whereisassumedtobeconstant.

When the temperature-dependent expression ofcapillarypressurePc(pc,T)isincorporatedintoatype2MSC with parameters corresponding to referencetemperature,itgivesanewfunction:

θl=gp(Pc)=gp,T(pc,T).

In this paper, this is referred to as “type4 MSC”.Again, it isworthnotinghere that the representationofphysicalphenomenavariesconsiderably,dependingon the original scientific field. In fact, the effect oftemperature on moisture storage is a combinationof sorption and capillary condensation equilibriummodification.Currently,thereappearstobenomodelthataccountsforbotheffects,anotherindicationoftheneedforstandardization.

2.4. Material moisture transport coefficients

Theliquidconductivityandvapordiffusioncoefficientare,inreality,functionsofthetwochosendependentvariables giving typically non-linear PDEs. In a fewparticularcases theycouldbeconsideredasconstantparameters,butwhenlargerangesofmoisturecontentor temperatureareexpected tooccur in thematerial,these simplifications can easily lead to incorrectpredictions. As a result, these functions have to becharacterized properly when going for so-called“full-range modeling”. There is an example of suchcharacterization in Grunewald et al. (2003), whichillustratesthecomplexityofdefiningphysicallybasedmaterialwatertransportfunctions.Wedonotgointomoredetailon this subjectherebecause it isbeyondthescopeofthispaper.

3. FORMULATIONS ACCORDING TO A MAIN MOISTURE DEPENDENT VARIABLE

3.1. Method

We showed how the theoretical moisture balanceequationisobtainedfromamacroscopicdescriptionofaporoushomogeneousmaterialusingtheREVconcept,withaphase-dividedapproach.Thecapillarypressureandvaporpressureemergedasthetwophysicallybaseddrivingpotentials for thedescriptionofwaterfluxes.The model is based on pressure-driven flows (Funket al., 2008).As shown by Funk andWakili (2008),the total moisture flux can be described using thecombinationofthetemperaturegradientandoneofthemainmoisturevariablegradient.Given thatmoisturevariablesarelinkedtogetherduringthetransferprocessthrough closing relationships, and assuming localequilibrium,wecanperformvariable-transformationson each of the constitutive relationships that defineliquidorvaporflux(Equations8and11).Thisisdonethroughpartialderivativeexpressions:

∇pv =∂pv∂Xi

T∇Xi +∂pv∂T Xi

∇T

∇pc =∂pc∂Xi

T∇Xi +∂pc∂T Xi

∇T

⎧

⎨

⎪⎪

⎩

⎪⎪

Eq.18

whereX =(ϕ,pv,pc,θl)isthemainmoisturevariables

vector. We can see immediately that if the mainmoisture variable is vapor pressure or capillarypressure,thereisnoneedtoreformulatethevapororliquidflux, respectively. In thefinal stage, the vaporandliquidfluxescanbeexpressedintheform:

.Eq.19jcMl +jdiffMv = DXiXi

∇Xi +DXiT∇T

Bydevelopingthebalanceequationintermsofone

or other mainmoisture state variable, new transportcoefficientswillarise,DXiXi andDXiT

,called“secondarytransportcoefficients”associatedwiththegradientofthemainmoisturevariableXiandtemperatureTintheXi-Tsystem,respectively.Theyarephenomenologicalfunctionsdependentonthetwodependentvariables,aswellasonprimarytransportfunctions.Inaddition,bychoosingaphase-dividedapproachasastartingpoint,thevaporandliquidtransportsub-functionsareeasilyidentifiedthroughEquation18.

Inthispaper,thepressure-drivenmoisturebalanceequationisre-formulatedusingtwomodelingstrategies.Thefirststrategyistochooseoneofthethermodynamicvariablesasthemainmoisturestatevariable,Xi≠θl.Inthiscase,thelefthandsideofEquation12alsohastobereformulatedthroughmoisturestoragefunctions:


∂θl∂t

=∂θl∂Xi

T

ξXiXi

∂Xi

∂t+∂θl∂T Xi

ξXiT

∂T∂t

Eq.20

where ξXiXi is the isothermalmoisture capacity of the

material(alsoreferredtoasspecificmoisturecontentinanalogytospecificheat)andξXiT

is itsnon-isothermalmoisture capacity. The second strategy is to use themoisturecontent,whichisanempiricalvariablethatcanbedefinedonlyatREVlevel.Itimpliesusagelimitationsofthebalanceequationobtained.WheneverpossibleinthispaperwehighlightthelinksbetweentheobtainedformulationsandsimilaroridenticalapproachesfoundintheliteratureonBuildingPhysicsmodels.Foreachvariablechoice,mathematicaldevelopmentsareprovidedintheformofasynthesistable.Thistablecontainsthe

partialderivativesneededforthereformulation,aswellas the full mathematical developments of secondarytransportfunctionsthatmightbeusefulwhencomparingmodels.Foreachstrategyandmainvariablechoice,themostcommonlyusedMSCsarementionedandusedtodeveloptheconnectionsbetweenmoisturecontentandthermodynamicvariables.

3.2. Strategy 1: thermodynamic variables formulations

Relative humidity. Table 1 shows the mathematicaldevelopmentsforrelativehumidityasthemainmoisturevariable,with themost frequent expressions ofMSCrelatedtothisformulation.Here,onlytype3MSCtakesaccount of the influence of temperature on moisturestorage.

Table 1. Moisture balance equation formulated with relative humidity and temperature as the main dependentvariables—Équation de bilan hydrique formulée en prenant l’humidité relative et la température comme variables dépendantes.


The expression of moisture balance with respect torelativehumidityisverycommoninBuildingPhysicsresearch.Thismoisturefieldformulationisparticularlyrelevantwhenmodeledphenomenaarepredominantlyin the hygroscopic range or have a simplified liquidtransfercharacterization.Inaddition,relativehumidityis easily measured with widely used and affordablesensors. The sorption isotherm dependence ontemperature is rarely taken into account in softwarebased on ϕ-formulation, which almost always usesa type1 MSC. With this common assumption, thebalanceequationbecomes:

ρlξϕϕ∂ϕ∂t

=∇⋅ DϕTl +DφT

v( )∇T + Dϕϕl +Dϕϕ

v( )∇ϕ⎡⎣

⎤⎦

Eq.21

whichisfoundinTarikuetal.(2010).Künzel(1995)andKünzeletal.(1996)chosetoneglecttheeffectsoftemperatureonliquidtransport,i.e.

DϕTl = 0

and

jcMl = Dϕϕ

l ∇ϕ .Eq.22

Accordingtotheseauthors,thetransportcoefficientDϕϕ

l canbederivedfromwet-cupvaporpermeabilitytests or, more commonly, calculated from theisothermalliquiddiffusivityDθθ

l (m2.s-1),whichisthetransportcoefficientderivingfromthemoisturecontentgradientdescriptionforliquidtransfer(seeStrategy2,Table 4). With this second method, the isothermalliquidtransportcoefficientisexpressedas:

Dϕϕl = Dθθ

l ⋅ξϕϕ . Eq.23

Without liquid transport, the equation is furthersimplified(Roelsetal.,1999)andfinally,inisothermalconditions,itbecomes:

ρlξϕϕ∂ϕ∂t

=∇⋅ Dϕϕv ∇ϕ⎡⎣ ⎤⎦ .Eq.24

Thissimplifiedbalanceequationisfrequentlyusedfor studying isothermal behavior in the hygroscopicregion, such as during anMBV experiment (Duboiset al., 2013). It should be noted that the notion ofisothermalbehavior ispurelytheoreticalbecauseanyrelative humidity change in the material will resultin heat transfer through the latent heat involved inthe adsorption process (Hens, 2012). In some cases,however,thesimplificationcanbejudgedsatisfactory.

Vapor pressure.Table 2 shows the development ofthe moisture balance equation with vapor pressureas themainmoisturevariable.As thevaporpressure

is the thermodynamic variable for vapor diffusion,only the liquid transport and storage terms have tobe reformulated. The pv-based formulation, like theϕ-formulation, is more appropriate for describingenvelope materials subjected mainly to hygroscopicbehavior. The chosen illustration MSC types arethereforethesameasfortable 1.

Afterthemathematicaltransformation,thethermalmoisturecapacity

ξ pvT=∂θ∂T pv

consistsoftwodistinctparts.Oneaccountsformoisturestoragedependenceon temperature, theother for thevariation of moisture content with temperature atconstantvaporpressurebecauseofthemoisturestorageequilibrium condition with relative humidity. Whenthe temperature increases, as the saturation pressureincreases collaterally, the material releases watermolecules in thepore space in order tomaintain theequilibriumrelativehumidityvalue.Asaresult,withanMSCconsideredtobeindependentoftemperature,thethermalmoisturecapacitydoesnotdisappearandshouldbeexpressedas:

ξ pvT

= −ξϕϕϕpsat

dpsatdT

. Eq.25

A similar balance equation was proposed byDosSantosetal.(2009),withthedifferencethattheseauthors did not provide the complete mathematicaldevelopmentsofthedifferentpartialderivatives.SuchtransformationshavebeendescribedbyGalbraithetal.(2001), providing adetailed expressionof secondarytransport functions.Only a type1MSC, however, isusedthere.InQinetal.(2009),theeffectoftemperatureonvaporpressureequilibriumseemedtobeconsiderederroneously in that the authors did not mention anythermalmoisturecapacity.Nevertheless,theseauthorspresentedanoriginalapproachfordefiningthethermaldiffusioncoefficientDpvT

l asapropermaterialfunctionand provided an experimental method in order tomeasureit.

Inisothermalconditionsthepv-balance-equationisreducedto:

ρlξ pvpv

∂pv∂t

=∇⋅ Dpvpvl +δv( )∇pv⎡

⎣⎤⎦.Eq.26

Galbraith (1992) introduced the concept ofdifferential permeability δ* in order to identify theexperimentallyliquidandvaportransportfunction.Hewrote:


δ * ϕ( ) = Dpvpvl +δv( ) = A+BϕC

Eq.27

where A, B and C are constants determined froma set of experiments based on the gravimetric cuptest. A review of other mathematical expressionsfor describing differential permeability is found inGalbraithetal.(1998).

Capillary pressure. The capillary pressure variableis particularly useful when working with capillarymaterialsincasestudieswhereliquidtransportisthemaintransportmechanism.Suchapc-formulationfindsitsrootsinSoilPhysicsstudies,withMillyetal.(1980)

being the first to reformulate the well-known PhilipanddeVries equation correctly in termsof capillarypressure head. In Building Physics, capillary-basedformulation is fairly unusual and is used by authorswho have a particular interest in the near-saturationbehavior of materials (e.g., through rain absorption/redistribution or direct soil contact). In this type ofresearch, the hydraulic conductivity and moistureretentionfunctionsneedtobeproperlydefinedintheover-hygroscopicrange,whichcanoftenbeachievedwithaporesizedistributionmodel.

Mathematical developments and thefinal balanceequationintermsoftemperatureandcapillarypressureareshownintable 3.Asimilarbalanceequationwasformulatedby Janssenet al. (2002),who studied the

Table 2.Moisture balance equation formulatedwith vapor pressure and temperature as themain dependent variables—Équation de bilan hydrique formulée en prenant la pression de vapeur et la température comme variables dépendantes.


influenceofsoilmoistureonheatlossesviatheground.When considering a negligible variation ofmoisturestorage with temperature, the following balanceequationisobtained:

ρlξ pcpc

∂pc∂t

=∇⋅ DpcTv( )∇T + −Kl +Dpcpc

v( )∇pc⎡⎣

⎤⎦ .Eq.28

ThisexpressionhasbeenreportedbyHäupletal.(2003),Janssenetal.(2007)andLietal.(2009).Thesemodels usually incorporate type2 MSC, which isillustratedinthesynthesistable.

3.3. Strategy 2: moisture content formulation

Philip et al. (1957) and Luikov (1966) originallychose to develop moisture fluxes according to

moisture content and temperature gradients.Moisturecontent isonlyanempiricalquantityas itcannotbedefinedatthemicroscopicscale.Itsvalueis discontinuous acrossmaterial interfaces,makingit an “improper” driving force (Hens, 2012).Whenstudyingassembliesofmaterialsatthewallscale,asitisoftenthecaseinBuildingPhysics,itisnecessaryto reformulate the equation using thermodynamicpotentials.

Table 4providesallthemathematicaldevelopmentsrelatedtomoisturebalanceexpression,withmoisturecontent as the main dependent variable. We choseto illustrate the case where the moisture storagefunction is expressed in relation to relative humidity(i.e., types1and3MSCs).Themaindifferencewiththe thermodynamic variable strategy can be seen inthe presence of moisture capacity terms inside themoisturediffusivity.Becausetherelationshipbetween

Table 3. Moisturebalanceequationformulatedwithcapillarypressureandtemperatureasthemaindependentvariable—Équation de bilan hydrique formulée en prenant la pression capillaire et la température comme variables dépendantes.


the thermodynamic variables and moisture contentdepends on the porousmaterial and wetting history,thisvariablechoicecouldleadtocomplexexpressionsof moisture diffusivities. If the hysteresis is takeninto account, the transport functionswould typicallydependonthepreviousstate.

A moisture balance equation expressed withtemperature and moisture content as dependentvariablesisrareinBuildingPhysics.Forconstitutiverelationshipsdefinition,however,someauthorschoose

a hybrid method, where liquid flux is originallyexpressed as dependent on the moisture contentgradientandvaporfluxonvaporpressure(Simonsonet al., 2004).The liquid transport coefficient is thencalled “liquid diffusivity”, for which many authorshave proposed empirical expressions (Häupl, 1987;Pel, 1995; Krus et al., 1999). For more informationondiffusivityapproaches,Scheffler(2008)providesapertinentreview.Fromthemathematicaldevelopmentsshownintable 4,itappearsthatliquidtransfershould

Table 4.Moisturebalanceequationformulatedwithmoisturecontentandtemperatureasthemaindependentvariables,wherethemoisturestoragecurveislinkedtorelativehumidity—Équation de bilan hydrique formulée en prenant la teneur en eau volumique et la température comme variables dépendantes.


beassumedtoalsobelinkedtotemperaturegradientthrough non-isothermal liquid diffusivity, which isequal to zeroonlywhen theMSC is independent oftemperature.

In isothermal conditions, the equation furthersimplifiesinto:

∂θ∂t

=∇⋅ Dθθl +Dθθ

v( ) ⋅∇θl⎡⎣

⎤⎦

Eq.29

whichshouldberelatedtothefollowing,stillwidelyusedinresearchpapers:

∂θ∂t

=∇⋅ D ⋅∇θl[ ] Eq.30

The coefficient D (m2.s-1) is then referred to as“moisturediffusivity”. In thisform, theequationhasthegreatadvantageofofferingananalyticalsolutionthrough the Boltzmann transformation. It is worthmentioningheretheworkofPeletal.(1996),whousedthis equation and computed the moisture diffusivityfunctionfromisothermalwaterabsorptionanddryingexperiments performed at thematerial sample scale.Such an expression is not valid when a significanttemperature gradient is involved because a complexcombinationofphenomenacanhardlybereducedtooneuniquetransportmechanism.

4. CONCLUSION

This paper has presented a synthesis of possiblemoisture balance equation formulations found incoupledheatandmoisturetransfermodels.Thesetoolsareintendedforthestudyofporousbuildingmaterialsandcancoveravarietyofphenomena,sometimeswiththe focuson thehygroscopicdomainandsometimeson the capillary behavior.As a result, they differ interms of the choice of the main dependent variableandinthelevelsofsimplificationthatunderliethem.Allthesemodelsare,infact,rootedinolderscientificdisciplines that have influenced them in variousways,(e.g.,inthechoiceofmoisturestoragefunctionexpression).

The work presented began with the physicallybasedbalanceequation,drawingontheREVconcept.Assumptionsusedcoveredthefrequentnon-hystereticandnon-isothermalcasewithout convective transfer.It is clear that themoisture balance equation can bereformulatedinmanyways.Bydevelopingthebasicequation in terms of each moisture-related variable,including thermodynamicvariables and themoisturecontent pseudo-variable, we provided an exhaustivelist of possible final mathematical formulations.We

discussed the moisture storage function in detailand proposed a general classification ofmodels.Weshowed how temperature can affect thermodynamicequilibrium and how thermal moisture capacitiescan be expressed depending on the main variablechoice. All the secondary transport functions, andtheirrelationshiptothetwoprimaryfunctions,vapordiffusion coefficient and liquid conductivity, areavailable.

Nomenclature

DXT: secondary transport coefficient linked to temperaturegradient in theequationdevelopedwithXas themainmoisturevariable

DXX: secondary transport coefficient linked to variable X gradient in theequationdevelopedwithXas themainmoisturevariable

j Mα :totalmassfluxdensityoftheα-phase(kg.m-2.s-1)jcMl : mass flux density of liquidwater through capillarytransport(kg.m-2.s-1)

jdMv: mass flux density of vapor through diffusion(kg.m-2.s-1)

Kl:liquidwaterconductivityoftheporousmaterial(s)m :phasechangerate(kg.m-3.s-1)

pc:capillarypressure(Pa)psat:saturationvaporpressure(Pa)pα:pressureoftheα-phase(Pa)qst:netisostericheat(J.kg

-1)Rv:specificgasconstantforvapor(J.kg

-1.K-1)T:temperature(K)δv: vapor diffusion coefficient of the porous material

(kg.Pa-1.m-1.s-1)δ*:differentialpermeability(Galbraith,1992)(kg.Pa-1.m-1.s-1)ε:totalopenporosityoftheporousmaterial(-)θα:volumetricfractionoftheα-phase(m³.m

-3)ξXT: non-isothermal moisture capacity in the equation

developed with X as the main moisture variable(m³.m-3.K-1)

ξXX:isothermalmoisturecapacityintheequationdevelopedwithXasthemainmoisturevariable(m³.m-3)

rα:massdensityoftheα-phase(kg.m-3)

ϕ:relativehumidity(-)ψ:suction(Pa)

Subscripts

l:liquidv:vaporg:gaseousa:dryair


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