Modelling hydraulic conductivity for porous building materials based on a prediction of capillary conductivity at capillary saturation

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International Journal of Heat and Mass Transfer

journalhomepage:www.elsevier.com/locate/hmt

Modelling hydraulic conductivity for porous building materials based on a prediction of capillary conductivity at capillary saturation

Jon Ivar Knarud

^a,∗

, Tore Kvande

^a

, Stig Geving

^b

aDepartment of Civil and Environmental Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

bDepartment of Architecture, Materials and Structures, SINTEF Community, NO-7465 Trondheim, Norway

a rt i c l e i nf o

Article history:

Received 23 October 2021 Revised 10 December 2021 Accepted 19 December 2021

Keywords:

Liquid conductivity Moisture diffusion Thin film surface diffusion Capillary absorption coefficient

a b s t r a c t

Liquidmoisturetransportplaysakeyroleinperformanceofmanybuildingassemblies.Forhygrothermal simulationmodels,used toassess suchassemblies, itisimportant toincluderealisticliquidtransport propertiesforthespecificporousbuildingmaterialsinvolved.Unfortunately,comprehensiveexperimen- talandmodelingmethodsassociatedwithdeterminingthehydraulicconductivitylimitwidespreadap- plicationofmaterial-specificdetermination.Toeaseapplicability,thispaperinvestigateshowtosimplify conductivitypredictionandmodelingbybuilding onabundleoftubesapproach. Incorporatinganew expressionvariantforthecapillaryabsorptioncoefficient(Aw),anovelpredictionexpressionforthecon- ductivityatcapillarysaturation(Kc,cap)isderived.modelingofunsaturatedcapillaryconductivity(Kc)can thusbescaledtoKc,capinsteadofthetraditionalapproachofscalingtoconductivityatover-capillarysat- uration(Ksat),avoidingsomecomplexityandconcernsonetraditionallyhasfaced.Hence,incontrastto mostmodelsforKc,whichapply Ksat,thispaperappliesKc,capas referencetoscaletheconductivityat unsaturatedconditions.Tomodelthehydraulicconductivity(K)forthefullmoisturerange,Kciscoupled withathinfilmmodel(Kfilm)andahygroscopiccorrectionmodel(Khyg).Thepredictionmodelisevalu- atedagainstawiderangeofporousbuildingmaterialdatasetsfoundinliteratureaswellascompared toacommonalternativeapproach,withreasonableresults.Thefindingsofthisstudycanhelpforbet- terunderstandingofchallengesinanalyticalcalculationofAwand ofwhybundle oftubemodelshave accuracyissuesinpredictingKc,withthestudysuggestingremediesforsomeoftheseissues.

© 2021TheAuthor(s).PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Hygrothermalsimulationhasbecomean importanttoolforas- sessingthehygrothermalperformance ofbuildingdetailsorparts, either it concerns new designs or retrofits, renovations or im- provements toexisting buildings. When involvingcapillary mois- ture transport, it isimportant that capillaryproperties ofporous materials arerealisticallycaptured.Ofkeyinterestisthemoisture retentioncurveandthehydraulicconductivitycurve.Ofthesethe latter is the most challenging, as it is difficult to experimentally determineintheunsaturatedregion[1],andrelativelyresourcein- tensive todetermine (accurately)inthe saturatedregion.Usually, one oftwoapproachesareusedtoidentifythehydraulic conduc- tivity forthe full rangeof moisturecontents: 1) modelingwhich usuallyincludescalingtothesaturatedconductivity,or2)calcula- tionfromthemoisturediffusivity.Althoughthemoisturediffusiv-

∗Corresponding author.

E-mail address: [email protected] (J.I. Knarud).

ityisrelativelyeasiertodetermineoverunsaturatedcapillarycon- ditionsit isstill resource intensive,traditionallyinvolvingexperi- mentallydetermining moisture profiles andforinstance applying the Boltzmanntransform methodto determine the moisture dif- fusivityfunction, e.g. [2].Thus, forpractical applicationsit is not obviousthat thediffusivityapproachisrealistic toutilize[1].Re- gardingmodeling,hydraulic conductivityhasoftenbeenmodeled bybundleoftubesmodels,withthemostwell-knownmodelcon- tributions,originallydevelopedforpetroleumandsoilscience,be- ingBurdine[3],Mualem[4]andVanGenuchten[5].Analternative tobundleoftubesmodelshavebeenthemoreadvancednetwork models,whichincorporatepercolationtheory[6,7].

Althoughbundle oftubes models are not without flaws, with their oversimplification of the pore system and flow paths (e.g.

[6]), their relative simplicityprovides an approach less laborious andeasieraccessible toutilizethantheir networkmodelalterna- tives[7].Bundleoftubesmodelsarecommonlyscaledfrommea- suredcapillaryconductivityat saturationKsat oratzerocapillary pressure K₀; however, thesehave shown to be difficult to deter- mine accurately [8]. Furthermore, it has been reported difficulty

https://doi.org/10.1016/j.ijheatmasstransfer.2021.122457

0017-9310/© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Nomenclature(excludingTable1)

A area(m²)

A_int,v crosssectionalareaofinternalvoidsperunitarea (m²/m²)

Aw capillaryabsorptioncoefficient(kg/(m²s^1/2)) B_A,Bc,B_f area, curvature and flow rate correction factors

duetoporeshapeirregularity(-)

C_int,v circumference of internal voids per unit area (m/m²)

D_w diffusioncoefficient(m²/s) fcurvature filmcurvaturecorrectionfactor(-) f_d factorofdeviation(-)

f_l mechanisticscalingfunction(-) j moistureflux(kg/(m²s))

K hydraulicconductivity(kg/(msPa))

L length(m)

lw,cw,nw coefficients for retention curve expression, (-), (Pa⁻¹),(-)respectively

m mass(kg)

m˝ massuptakeperunitarea(kg/m²) m˙˝ massrateperunitarea(kg/(m²s)) n cumulativeporenumber(-) p pressure(Pa)

r capillaryporeradius(m)

r₀,r_eff,rae average, effective and average/effective pore ra- dius(m)

Rw gasconstantforwater(J/(kgK))

S_wi,S_wf wettingphasesaturation,initialandbehindimbi- bitionfrontrespectively(-)

t time(s)

T temperature(K) V volume(m³)

V˙ volumetricflowrate(m³/s)

V˙ˊ volumetricflowrateperunitlength(m³/(ms)) V˝ absorbedvolumeperunitarea(m³/m²) w moisturecontent(kg/m³)

x spatialcoordinate(m) Greeksymbols

α

^{correctionfactor(-)}

α

^p dimensionlesspressure(-)

δ

^{filmthickness(m)}

δ

^{v vapordiffusioncoefficient(kg/(msPa))}

ε

^porosity(-)

η

Aw,

η

^{cap variousscalingfactors(-)}

η

^sp,

η

φ variousexponents(-)

θ

^{moisturecontent(m3/m3)}

μ

^{vapordiffusionresistance(-)}

μ

^{w dynamicviscositywater(kg/(ms))}

^,e,m, disjoining pressure, with electrostatic and molecularcomponents(Pa)

ρ

^{w densitywater(kg/m3)}

σ

w surfacetensionwater(N/m)

τ

^{tortuosity(-)}

ϕ

^{contactangle(°)}

φ

^{relativehumidity(%)}

Subscripts

a air

abs absorption

ad adsorbed

c capillary

cap capillarysaturation

dry drycupmeasurement Dw diffusioncoefficientbased eff effective

film adsorbedmoisturefilm

g gas

hyg hygroscopic

l liquid

lim limiting

mod modified

nom nominal

p pore

red redistribution ref reference rel relative sat saturation tot total

v vapor

w water

wet wetcupmeasurement

withscaling tothe saturatedconductivity,when saturationis set equaltototalporosity,becausethemoistureretentioncurveisill- definedintheovercapillaryregionclosetosaturation[9].

Nevertheless,toaccommodatean engineeringneedforlessre- source intensive predictions of hydraulic conductivity, bundle of tubesmodelsarestillofinterest.Withabundleoftubesmodelas thefoundation,SchefflerandPlagge[7]proposedawholemoisture rangehydraulicconductivitymodel.Althoughthismodelisintrigu- ing,itreliesonacoupleofmaterialdependent parameterswhich require iterative post-processing through simulation to be deter- mined properly. Furthermore, the model still relies on scaling to an effectiveconductivityatover-capillarysaturation,whichneeds to be determined experimentally. Equipment for, and experience with,suchexperimentaldeterminationisnotparticularlyavailable forwidepracticalapplication.

Inthepresentpapertheaimhasbeentodevelop amodel,in- spiredbytheSchefflerandPlaggemodel,butwhichiseasiertoap- ply,byremovingrelianceoniterativepost-processingandreducing relianceon material property data which is particularly resource intensivetoacquire.

Specifically, theobjective ofthis studyisto derive andinves- tigatean alternativeapproachtopredicthydraulic conductivityas functionofcapillarypressure,K(p_c),notrelyingoncomprehensive testingofK(orKsat)incontrasttoexistingapproaches.

Frominitial,inspirationalideasresearchquestionswereformu- latedtosubstantiatetheobjective.Thefollowingquestionsareex- ploredinour study:1) Isit feasibleto predict thecapillarycon- ductivityatcapillarysaturation?2)CantheScheffler-Plaggemodel forK(pc)be simplified by scalingto conductivityatcapillarysat- urationinstead ofatsaturation?3) Cantheoverall procedurefor determining K(pc)be simplified andmademore practically feasi- ble,forwhenonlyanecessaryminimumofmaterialpropertytest dataisavailable. 4) Forsuch amodel,how isthepredictionper- formanceforK(pc)whenassessingawiderangeofporousbuilding materialsdescribedinpreviousstudies?

The focus of this paper is categorically limited to bundle of tubesmodels,indescriptionofthehydraulic conductivity,incon- trasttonetworkmodels.Hysteresis effectsarenotaddressed. Fur- thermore,neededinformationontheporesizedistributionwillbe estimated fromthe retention curve, andit has been outside the scopeoftheworktoassesswhetherdirectuseofameasuredpore sizedistributionwouldimprovepredictions.Thestudydoesnotin- cludeacomparativeevaluationofhowrealisticphysicsare repre- sentedincomparable, alternativepredictionapproaches;however,

aquantitativecomparativeevaluationofpredictionperformanceis included.

Thepaperisoutlinedasfollows:Firstthemodelisderivedand presented.Thenprocedureforitsapplicationandevaluationisin- troduced.Afterfollowsresultsandresultassessments,followedby furtherdiscussionandfinallyasummaryandconclusion.

2. Hydraulicconductivitymodel

Several hydraulic conductivity models for the whole moisture rangehavepreviouslybeenproposed, e.g.,withinfieldofbuilding physics [7,10] andsoilscience [11,12].Incontrastto theformer thelatterincludemodelsforthinfilmflowtotheoverallhydraulic conductivity.Withthinfilmflowmodelsseeminglyhavingbenefit- tedconductivitymodelingatlowmoisturecontentsinsoilscience, it is possiblesimilar benefits can be introduced toapplication in building physics.Thin filmflow will thereforebe incorporated in theoverallhydraulicmodelpresentedinthefollowingsections.In this studywe will limit the hydraulic conductivity modelto liq- uidconductivity,withthepresumptionthatvaportransportisad- dressedseparatelyinhygrothermalsimulationsoftware.Hence,va- portransport(vaporconductivity)isnotincluded.

Inthefollowingsectionswegothroughthesequentialstepsof deriving the hydraulic conductivitymodel.First we introducethe bundle of tube modelbased on Grunewaldet al.[10].Then the- oryonpredictingthecapillaryabsorptioncoefficientisintroduced followed bya proposed newpredictionexpression.Next,thisen- ables forminga predictionexpressionforthe capillaryconductiv- ity atcapillary saturation.Further, the Sceffler andPlaggemodel [7]isrearrangedforscalingtoconductivityatcapillarysaturation.

ThereafterfollowsathinfilmmodelbasedonLebeauandKonrad [11] and a correction modelfor thehygroscopic region basedon the Scefflerand Plaggemodel[7].The overall hydraulic model is then established. Finally,aprocedure forincorporatingthe reten- tioncurveintothemodelisgiven.

2.1. Capillaryconductivity

TheHagen-Poiseuilleequationdescribesthevolumetriclaminar flowinacylindricalpore(tube)ofradiusralongthetubepathof length L.However,pores inporousmediausually nevermeetthe idealofcylindricalgeometry[13].Therefore,aflowratecorrection factor B_f is included to account forimpact of irregular geometry (non-cylindrical), on the volumetric flow rate. In contrast to Cai et al.[13], which relatesa correction factor

α

^{directly to r,B}f is hererelatedtothevolumetricflowrate;hence,B_fequatesto

α

⁴ⁱⁿ

[13].TheHagen-Poiseuilleequationthustaketheform:

V˙

(

^r

)

^=−Bf

π

^r4

8

μ

^w

dp_l

dL (1)

where

μ

^{w isthedynamicviscosityofwaterandp}l theliquidpres- sure. Withcapillarypressurepc =pg -p_l andpresumedconstant gas (air) pressure p_g, dp_l is simply substituted with -dp_c. Here, forconvenience, positivevaluesforpc are appliedthroughout, al- though pc alternativelycan be written asnegativepressure (suc- tion).rrepresentsanequivalentcylindricradius,inpracticalterms, halfofahydraulicdiameter,ortheradiusofaninscribedcirclefor regular polygons. Flow in capillaries may be perceived to follow tortuous streamlines [13, 14]. Consequently, theflow path length dLisgreaterthanamorerelatabledimensiondx,ofacontrolvol- ume.Thiscanbeaddressedbyintroducingthetortuosity

τ

^,which

from a macroscopic perspective represents the ratio of effective capillarypathlengthtolengthdx(thicknessofacontrolvolume).

Thatis;dL=

τ

•dx,seee.g.[15].Withthesechanges,Eq.(1)trans-

formsto:

V˙

(

^r

)

^=Bf

π

^r4

8

μ

w

1

τ

^{ddxpc (2)}

Thevolumetricflowrateinabundleofcapillariescanbefound by integrating Eq. (2) over the pore size distribution density [7, 10],i.e.integratingwithrespecttoradiustheproductofvolumet- ric flow rateand corresponding incremental number of pores at respectiveradius.Adaptedfrom[7,10]thecapillarymoisture flux thenbecomes:

jw,x=

ρ

^w

R

V˙

(

^r

)

^dn

(

^r

)

dr·dAdr (3)

wherejwisthemoistureflux,anddAindicatesunitareadA=dydz ofacontrolvolumedV=dxdydz=1m³.

Theexpressionfortheincrementalnumberofporescanbees- tablishedasfollows:

dn

(

^r

)

⁼

^V

(

^r

)

Ap

(

^r

) τ

^{dx =}

θ

^c·dV

B_A

π

^r2

τ

^{dx =}

dA B_A

π

^r2

τ

d

θ

^c

(

^r

)

dr dr (4)

where dn(r) is the increment number of pores at a radius, ^{V(r) change in moisture filled capillary volume as function of} r,Ap(r) porecross section,

θ

^{c volumetriccapillary retainedmois-}

turecontent. B_A area correction factorfornon-circular cross sec- tionB_A = A_p/Ap,cylindrical (e.g.B_A = 1.27square, B_A = 1.65equilat- eraltriangle). Including thefactor B_A is important toaccount for the “extra” water in each pore which is not included in the in- scribed circlewhich r represents.For non-circularcross sections, notincludingB_A willoverestimatethenumberofpores.Rearrang- ingEq.(4)wehave:

dn

(

^r

)

dr·dA= 1

BA

π

^r2

τ

d

θ

^c

(

^r

)

dr (5)

Notethat by includingB_A and

τ

^{,Eqs.(4)and(5) differsfrom}

theapproachin[7,10].InsertingEqs.(2)and(5)intoEq.(3)and integratingover all radii involvedat a capillarymoisture content

θ

^c: jw,x=B_f

BA

ρ

^w

8

μ

^w

τ

²

_θc

0

r

( θ

^c

)

^2d

θ

^cd_dx^{pc (6)}

Eq.(6)issimilartowhatisreportedin[7],butwiththeaddi- tionalinclusionoftortuosityandthefactorsB_f andB_A.Theradius canberelatedtothecapillarypressurethroughtheYoung-Laplace equation,whichwhengivenbyEq.(7)includesacorrectionfactor Bc forporeshapeirregularity[13,16]. Thisirregularityaffectsthe meniscuscurvature,see[17].

pc=2

σ

wcos

( ϕ )

r/Bc (7)

where

σ

^{wisthesurfacetensionofwaterand}

ϕ

^{thecontactangle.}

FollowingWongetal.[17]thegeneralYoung-Laplaceequationcan bearranged:

pc=

σ

r

∇

^·nˆ→r

σ

^{pc ≡}

α

^p=

∇

^{·nˆ (8)}

where

α

pisadimensionlesspressureand

∇

•nˆisthedimension- lessmeancurvature.

Bccanthenbedefinedas:

Bc=

α

p,actual

α

p,cylindrical

(9)

ConsideringEqs.(8)to(7),

α

^p =2cos(

ϕ

^)foracylindricalpore.

Assuming0°contactangle;foracylindricalpore

α

p =2(Eq.(9): Bc =1.0),foranequilateraltriangleshapedpore

α

^p= 1.7776[17, 18](Bc =0.8888),andforasquare shaped pore

α

^p = 1.8862[17, 18](Bc =0.9431). Notethat thecorrectionfactorassignedby Cai et al. [13] as

α

^{, would hereequate to}

α

= B_f^1/4 = 1/Bc. For an equilateraltriangle

α

=1.186andasquare

α

=1.094[13](

α

^can

be found fromassessingcalculated analytical solutionsofHagen- Poiseuille flow for respective pore shapes). These

α

^{-values coin-}

cidewith

α

=B_f^1/4=1/Bc.Thus,in[13]thetwocorrectionfactors, respectively fortheHagen-PoiseuilleandYoung-Laplace equation, areincorrectlyconflatedintooneandthesame.

InsertingEq.(7)solvedforrintoEq.(6)gives:

j_w,x= B_fB²_c BA

ρ

^w

σ

w²cos²

( ϕ )

2

μ

^w

τ

²

_θc 0

1 pc

( θ

^c

)

^2d

θ

c

dpc

dx (10)

withthecapillaryconductivityinEq.(10)being:

Kc=B_fB²_c BA

ρ

^w

σ

w²cos²

( ϕ )

2

μ

^w

τ

²

_θc

0

1

pc

( θ

^c

)

^2d

θ

^{c (11)}

K_cbecomesK_c,cap,i.e.capillaryconductivityatcapillarysatura- tion,when integratedup to

θ

^c =

θ

^{c,cap, Byincluding poreshape}

correction factors and tortuosity Eq. (11) distinctly differs tradi- tional approaches.Although, it isnot particularly usefulsince B_f, Bc andB_A arestillunknownfactors.However,Kc,cap haspreviously been suggestedto bepredicted fromthecapillaryabsorption co- efficient A_w [19],asK_c,cap = 10⁻⁸

η

AwA_w²,where

η

Aw beingama- terial dependent coefficient reported to be inthe interval 0.95– 16.0.Thisexpressionisneither specificallysophisticatedinits in- tuitiveness (non-correctorhiddenunits) norconvincingly related to physical characteristics ofthe material andfluid. Furthermore, withacoefficientspanningoveroneorderofmagnitudeprediction accuracysufferswithoutexperienceinchoosingthecoefficient.

Nevertheless, ifassuming K_c,cap could be predictedfrom A_w,a dimensional analysis of K through the Rayleigh method [20] re- veals that an expressionof K could be a functionof Aw2 divided byadensitycharacteristic,units[kg/m³],andapressurecharacter- istic,unit[Pa].(Thisdoesnotnecessarilyexcludeotherpossibilities ofphysicalparameters.)Guessingthecorrectappearancehowever wouldnotnecessarilybestraightforward,riskingbecomingheav- ilyreliant ona nonsensicalcoefficient.A plausibleapproachisto presume moreinformationisneededregardingAw tounderstand therelationtoK.

2.2. Capillaryabsorptioncoefficient

AnexpressionforAw wasderivedbyBeltranetal.[21]tobe:

Aw=

ρ

^w

σ

^w

μ

^w

1/2

ε

^cap

τ

^r01/2

_cos

ϕ

2

1/2

(12)

where

ε

^{capis thecapillaryporosity andr}0 isan averagepore ra- dius. Eq.(12) can alsobe directly derived from the early Handy imbibition model [13, 22], with the liquid permeability k_w =

ε

^capr02/(8

τ

^{)[21,23].Asseen,Eq.(12)doesnotincludecorrection}

for pore shape irregularity; however a similar expression by Be- naventeetal.[24],hasonesuchcorrectionincluded.

Aw=B¹c^/2

ρ

^w

σ

w

μ

^w

1/2

ε

τ

^{1/2r1e f f/2}

_cos

ϕ

2

1/2

(13) where

ε

^{is the porosity and r}eff is an effective radius which re- quirestobecalculatedbyNewton’siterationmethod,see[24].Hy- pothetically,withmeasurementofAwonecantherebyestimateBc. Unfortunately,Eq.(13)suffersfromsomeshortcomings,including;

incorrecthandlingofthetortuosity,notincludingacorrectionfac- tor inthe Hagen-Poiseuilleequation andnotaddressing the wet- tingliquidsaturation [13].Afurtherdevelopedexpression canbe foundfromanimbibitionmodelderivedbyCaietal.[13]: Aw=

α

^3/2

ρ

^w

σ

^w

μ

^w

1/2

ε

Sw f−Swi

τ

^r1ae/2

_cos

ϕ

2

1/2

(14)

where S_wf is thewetting phase saturation behind the imbibition front, S_wi the initial wetting phase saturation, and rae an aver- age/effective pore radius. Ifassuming S_wf equals capillarysatura- tion andS_wi isnegligible,i.e.foraninitially drymaterial orfora

relativelynon-hygroscopicmaterial,then

ε

^(Swf-S_wi)≈

ε

^cap.Even

though Eq. (14) is a considerable improvement from Eq. (12) it also has its issues. As previously mentioned, the correction fac- torsfortheHagen-PoiseuilleandYoung-Laplaceequationshavein- correctlybeenconflated in

α

^.Furthermore,formaterials havinga pore structure ofhighly varying pore size it isdifficult to assess r_ae.

Thereby,toaccommodatetheseissuesarevisedderivation ap- proachtoAwiswarranted.

2.3. ProposednewAw-expression

InthefollowinganexpressionforA_wisderivedwithderivation stepsfromCaietal.[13]coupledwithapproachesfromSection2.1. Specificallyaddressingimbibitionwhereafaceofaporousma- terial is put in contact with a free water surface, andassuming sharp-front theory of capillary absorption [25], there will be a sharpmoisture front which movesthrough thematerial. We fur- thermoreassume dealingwithmaterials andasettingwhich fol- low linear cumulative absorption with respect to square root of time, i.e., mʺ = Aw√t, where mʺ is cumulative liquid mass ab- sorptionperunitarea.Proportionalityto√

tcorrespondstoaspe- cific time-dependent imbibition regime in which neither inertia nor gravitationalforces are significant. A thorough review of the imbibitionregimesisprovidedbyDejametal.[26].

InEq.(1)dp_l isreplaced with-dp_c asbefore,butwithdLnow thedistanceL(porelength)theimbibitionmoisturefronthastrav- eled.Similarto[13],pc isfurthermorereplacedwithEq.(7).Con- sequently,thevolumetric flowrateofone pore canbe expressed as:

dVp

dt =B_fBc

π

^r3

4

μ

^w

σ

^wcos

( ϕ )

L (15)

Assuming Eq.(15) only addresses capillarypores, initially be- ingdry,whichthroughtheimbibitionprocessbecomingfullysat- urated between the free liquid surface and the moisture front, we haveVp = L•Ap.Substituting LinEq. (15)withVp/Ap enables Eq.(15)tobeintegratedwithregardtoVpandt.Integratinglimits are;fort=0,Vp=0,andfort=t,Vp=Vp.Hence:

1

2V_p²=B_fBc

π

^r3

4

μ

^w

σ

^wcos

( ϕ )

^{Apt (16)}

The absorbed volume of waterVp can be integrated over the bundle ofcapillaries involvedby repeating thesame approachas inEq.(3).SolvingEq.(16)forVpandintegratingovertheporesize distributiondensity:

V =

R

B_fBcBA

π

^2r5

2

μ

^w

σ

^wcos

( ϕ )

^t

1/2

dn

(

^r

)

dr·dAdr (17) ApplyingEq.(5),Eq.(17)becomes:

V = _θc,cap

0

B_fBc

B_A r

2

μ

^w

σ

^wcos

( ϕ )

^t

1/2

1

τ

^d

θ

^{c (18)}

MultiplyingEq.(18)withthewaterdensity,andrearranging:

m =B¹_f^/2B¹_c^/2 B¹_A^/2

ρ

w

σ

w

μ

^w

1/2

_cos

ϕ

2

1/21

τ

_θc,cap

0

r^1/2d

θ

c·t^1/2 (19) FromEq.(19)itfollowsthat:

Aw=B¹_f^/2B¹_c^/2 B¹_A^/2

ρ

^w

σ

^w

μ

^w

1/21

τ

_θc,cap

0

r^1/2d

θ

^c

_cos

ϕ

2

1/2

(20)

Comparing Eqs. (20) to (14) one can see

α

^{3/2 is recovered if}

oneallowstheincorrectconflation

α

=B_f^1/4=1/Bc previouslyad- dressed.InderivationofEq.(20)itisassumed

ε

^Swf=

ε

^cap=

θ

^c,cap,

andS_wiisthroughtheintegral steptoEq.(16)implicitlyassumed to be negligible (S_wi = 0). Both can be included by multiply- ing Eqs. (17)-(20) with

ε

^(Swf - S_wi)/

ε

^cap.Nevertheless, thebiggest difference to Eq. (14) is the treatment of the pore structure, in Eq. (20) with the integral of pore radii involved in capillaryab- sorption over the interval of saturation up to capillarysaturated moisturecontent

θ

c,cap.

Replacing r by means of the Young-Laplace equation, Eq. (7), Eq.(20)finallybecomes:

Aw=B¹_f^/2Bc

B¹_A^/2

ρ

^w

σ

^w

μ

¹w^/2

cos

( ϕ ) τ

_θc,cap

0

1

p¹_c^/2d

θ

^{c (21)}

2.4. Proposednovelpredictionofconductivityatcapillarysaturation

Taking thesquareofEq.(21),solving fortheunknowncorrec- tionfactorproductB_fBc2B_A⁻¹ andinsertingintoEq.(11),Kc,capcan finallybepredictedby:

Kc,cap= A²_w 2

ρ

^w

_θc,cap

0

1 p²_cd

θ

_θ

c,cap 0

1 p¹_c^/2d

θ

^c

−2

(22)

Eq.(22)satisfiesthedimensionalanalysispreviouslymentioned withthedensitycharacteristicrevealedtobethedensityofwater andthepressurecharacteristicexpressedasarelationoftwointe- gralsbothoffunctionsofpc.

2.5. Capillarymodel

WithK_c,capbeingthecapillaryconductivityatcapillarysatura- tionthecapillarymodelforsaturations0≤

θ

^c ≤

θ

^{c,cap canfollow}

thecapillarymodelofSchefflerandPlagge[7];Kc=f_l

η

^capKeff,satK_rel, in wheref_l(w_cap)

η

capK_eff,sat equalsK_c,cap.

η

cap is ascaling parame- ter to scaleKc to a measured effective (over-capillary)saturation K_eff,sat. Since Kc,cap in the present work is predicted directly and not reliantonscalingby

η

cap themodelofSchefflerandPlaggeis rewrittentoEq.(23).

Kc= f_l

fl

(

^wcap

)

^{Kc,capKrel (23)}

f_l,Eq.(24),beingthescalingfunctionofthemechanisticserial- parallelporemodeldescribedbySchefflerandPlagge[7],following theprinciplesofGrunewaldetal.[10],andf_l(w_cap)beingf_l evalu- atedatwcap(moisturecontentatcapillarysaturation).

fl=

_w

wsat

η^sp

_w

wsat

η^sp +

1−_w^w_sat

2

1−

_w

wsat

η^sp

⁽²⁴⁾

where

η

sp is aparameter to adjusttheserial-parallel relation,by modifying the volumetricfractionthat is parallelpore domain in themechanisticmodel[7],andwsatismoisturecontentatsatura- tion.K_rel beingtherelativecapillaryconductivity[7,10]givenas:

K_rel= θc 0 p⁻_c²d

θ

^c

θc,cap

0 p⁻_c²d

θ

^{c f or}

θ

^c≤

θ

^c,cap (25)

In contrast to [7, 10] the upperintegral limit belowthe frac- tion lineis

θ

^{c,cap,insteadof}

θ

eff,sat.The

η

^{sp parameterismaterial}

dependent [19]; however, we will argue it is also dependent on boundary conditions,i.e. dependent on whether it is absorption, redistributionordryingofmoisture,oracombinedrepresentation ofthese, whichisinfocuswhendeterminingcapillaryconductiv- ity(seeSection4.3).Forahypotheticalpureparallelflowbehavior

η

^sp=0;however,usuallyitresidesintheintervalupinlowersin- gledigits.

2.6. Thinfilmmodel

Surfacediffusionisaliquidtransport mechanismwhichisim- portant in pores not available forcapillary transport, due to too low moisture filling forcapillary meniscito form. Thin film flow isanapproachtoaccount forsurfacediffusion.We applypartsof the model approach described by Lebeau and Konrad [11]. Inte- grationofavelocitydistribution,arrivedfromNavier-Stokesequa- tions,overafilmthicknessyieldsthevolumetricflowrateperunit lengthoffilm crosssection [11].Adopted from[11],thethinfilm equation assuming no-slip at pore wall and negligible shear be- tweenliquidandairbecomes:

V˙ =

δ

³

3

μ

^w

dpc

dx (26)

where

δ

^{isthefilmthickness, whichcanbe expressedasfunction}

of capillarypressure. Positivevalue forp_c gives Eq.(26) without minussignincontrastto[11].

Multiplying Eq.(26)withthewaterdensityandthe poresys- temvoidcircumferenceoveracrosssectionofthecontrolvolume givesthemoistureflux:

jf ilm=

ρ

w

δ

³

3

μ

^{wCint,vddxpc (27)}

where C_int,v is the circumference of internal voids not capillary filledasfunctionofpc,withC_int,v=C_int,v,tot– C_int,v,c,whereC_int,v,tot isthetotalcircumferenceofinternalvoidsandC_int,v,cisthecircum- ferenceoffilled capillarypores.IdeallyC_int,v,c(pc)shouldbe found fromaporesizedistribution;however,ifrelyingontheretention curve,asisdoneinthecurrentpaper,itcanbecalculatedas:

Cint,v,c=

R

2

π

^rdL_dxd^dn_r·

⁽

_d^r

⁾

_A^dreq=.(5)_B²

A

_θc

0

1

rd

θ

^{c (28)}

Cint,v,tot= 2 BA

_θc,cap

0

1

rd

θ

c (29)

NotethatEqs.(28)and(29)providetheinscribedcirclecircum- ference of the capillaries, thereby constituting a simplification to filmflow. The radiusinEqs. (28)and(29)issolved fromEq.(7), where Bc needs to be approximated by comparing measured Aw

toEq.(21).Inlackofdetailedinformationabouttheporeshapes, B_f, B_c and B_A are unknown. If values for B_f and B_A are chosen, basedonsimpleassumptions,avalueforBccanbeidentifiedand Eqs.(28)and(29)canbecalculated.Wehereassumefilminover- capillary pores (not filled at capillary saturation) has negligible contributionto hydraulic conductivity,due toa relative low total circumferenceof such pores. These are thereforenot included in the calculation. Furthermore,for theintegration in Eqs. (28)and (29) we do not allow accumulated circumferencefor pores with radiusbelowtwicethediameterofawatermolecule(diameterof awatermolecule ≈ 3E-10m) asnoefficientfilm flowwill allow toformforsmallerporesizes.

FromEq.(27) thefilmcontributiontothehydraulicconductiv- itycanbeidentifiedas:

K_{f ilm}=

ρ

^w

δ

³

3

μ

^{wCint,v (30)}

Accordingto[11]thefilmthicknessisinvolvedintworelations ofdisjoiningpressurecomponents.Theoveralldisjoiningpressure isgivenas[11]:

( δ )

⁼

e

( δ )

⁺

m

( δ )

⁽³¹⁾

whereПe istheionic-electrostatic componentandПm themolec- ularcomponent.

e

( δ )

⁼

ε

^r

ε

⁰

2

π

^kBT

eZ

2

1

δ

^{2 (32)}

Table 1

Parameters for Eq. (32) and (33) adopted from [11] .

Parameter Description Value

A svl(J) Hamaker constant −6.0 ×10 ⁻²⁰

e (C) Electron charge 1.60218 ×10 ⁻¹²

k B(J/K) Boltzmann constant 1.38065 ×10 ⁻²³

T (K) Temperature 293.15

Z (-) Valence charge 1

ε0(C ²/(J m)) Permittivity of free space 8.85419 ×10 ⁻¹² εr(-) Relative permittivity of water 80.23

^m

( δ )

⁼⁻₆^A

πδ

^{svl3 (33)}

withparameterssummarizedinTable1,assessedat293.15K.

The disjoining pressure Eq. (31) is related to liquid pressure [11]:

( δ )

^=pg−pl (34)

RelatingEq.(34)tothecapillarypressurewhichisalsodefined pc=pg-p_l onehavethatthedisjoiningpressureisanalogtocap- illarypressure.

Since Eqs. (32)and(33) are functionsof

δ

^{−2 and}

δ

^{−3 respec-}

tively,itisinconvenienttoanalyticallysolvefor

δ

^{.Instead,forsim-}

plicitywepropose calculatingП(

δ

^{) fora rangeof}

δ

^{andthen plot}

log(

δ

^{) asfunctionoflog(}П(

δ

^{)).Fromsuchaplotasimple2ndde-}

greepolynomial functioncan befitted.Followingthisapproach

δ

canbeapproximatedwith:

δ

f ilm=10^{0.0116·(log|pc|)2−0.5535·log|pc|−5.7810} (35) Eq.(35)hasup to± 5% deviationtothe actualfilm thickness over the range 10⁰ < |pc| < 10⁹ Pa. The film model, Eq. (30), could seemingly model hydraulic conductivityin the hygroscopic region for non-filled pores. However, with its background stem- ming fromthe rather macroscopic perspective of solving Navier- Stokes, it lacks inhandling complexityassociated with very thin films at nanoscale.For verythinfilms, measuring ina low num- berofwatermolecule layers,limitingaspects,physicalconditions, material propertiesand porewall characteristics willimpact film existence and behavior. Forinstance,1) watermolecule diameter limitslowestfilmthicknessandsmallestporesthatareeffectively accessible to water, 2) material hydrophilicity or hydrophobicity, temperature,filmconfinement,andporewallroughnessaffectwa- ter molecule orientation, structuring ofthefluid, adhesionforces, no-sliptendencyatporewallandfluid propertiessuchasdensity andviscosity,e.g.[23,27].

Therefore,thereisneedforcorrectionstothefilmmodel,ora more advancedfilmmodelaltogether, toaddress nanoscaleprop- erly.However,insteadofaddingcomplexitytothefilmmodel,such as to some extent isdone in [11], we circumvent theissue with two simple/practicalcorrectionalsteps;1) thefilmthicknesscan- notbethickerthanwhattheadsorbedwatercontentinthemate- rial allowsfor.Hence,theoverallfilmmoisturecontent(adsorbed part of retention curve)divided by theproduct ofwater density andtheporesystemsurfaceareaofnon-filledcapillariesgivesan upper bound. 2) in the lower to middle hygroscopic region we keep the hygroscopic model fromSchefflerand Plagge, described innextsubchapter,withasmoothtransitionfromthehygroscopic modeltothefilmmodelasfunctionofrelativehumidity(RH).By takingthesetwosteps themodeling iskeptsimpler,butatacost ofrealismandaccuracy.

With step 1), resulting film thickness to be applied in Eq.(30)becomes:

δ

=min

δ

f ilm

(

^pc

)

^,w

(

^pc

)

^−wc

(

^pc

) ρ

wCint,v

(

^pc

)

(36)

Or

δ

^=min

δ

f ilm

(

^pc

)

^,w

(

^pc

)

⁻

ρ

^wBAAint,v,c

ρ

^wCint,v

(

^pc

)

where wc is the capillary retained moisture content occupying filled capillaries, and A_int,v,c the cumulative inscribed circle cross sectionalareaoffilledcapillariesgivenbyEq.(37),derivedinsame wayasEq.(28).

Aint,v,c= 1 BA

_θc

0

d

θ

^{c (37)}

2.7. Hygroscopiccorrectionmodel

SchefflerandPlagge[7]proposeaccountingforliquidconduc- tivityinthehygroscopic region byassessing thedifferencein va- pordiffusionbetweendrycupandwetcupmeasurements.Herein theyassume anegligibleliquidtransportcontributionincludedin thevapor conductivityKv (Eq.(38)[1])forthe drycupmeasure- ment. The difference;wet cup – dry cup, Eq.(39), approximates theliquidtransport fractionK_hyg actingduringthewetcupmea- surement. They propose three vapor diffusion measurements are neededtoenablelogarithmicinterpolationandextrapolation:one drycup and two differentwet cup measurements. Eq.(38) arise fromrelatingvapordiffusiontoadrivingpotentialontheformof capillarypressure.

K_v=

δ

v,a

μ

^·

φ

^pv,sat

ρ

wRwT (38)

where

δ

v,aisthevapordiffusioncoefficientofair,

μ

^thevapordif-

fusionresistancefactor,pv,satthesaturatedvaporpressure.

K_hyg

( θ

^wet

)

⁼

φ ( θ

^wet

)

wet

μ

^{wet −}

φ

dry

μ

dry

δ

v,a

p_v,sat

ρ

^{wRwT (39)}

where

θ

^{wetistheassociatedvolumetricmoisturecontentatwhich}

K_hyg isdetermined,μwetandμ_dry arevapordiffusionresistanceco- efficientsfromwetanddrycupmeasurements,and

φ

^{wet and}

φ

dry

effectiveRHassociatedwithrespectiveresistancecoefficient.Com- monly

φ

^wet = 0.715 and

φ

dry = 0.25since μwet andμ_dry are re- spectivelyfoundatstandardized conditions50/93%and0/50%RH [28].

Unfortunately,usually onlyasinglewetordryvapordiffusion resistance(inEuropecommonlydefinedby [28])issought.Rarely morethanoneoftheseisreportedinastudy.CarmelietandRoels [1]isone offew exceptionsexplicitlyhavingreportedthreemea- surements(onedrycupandtwo wetcup).Therefore,threemea- surementsneedtobepreplannedwithdeterminingK_hyginmind.If onlyoneoftheresistancecoefficientsisavailable,forinstancethe drycup,thentheotheroneassociatedwith

φ

^wet=0.715couldbe subjectedto a guesstimate. Formaterials having very low hygro- scopicity, i.e., relatively smalldifference insorption between25%

and71.5%RH,onecanassumeμwetandμ_dry toberathersimilar.If onlyoneortwovaluesareavailableone canadoptK-valuesfrom thethinfilmmodelatrelativelyhighRH values,aslong asthese arelarger valuesthantheone ortwovaluesoftheliquidpartof vaporconductivitywhichareavailable.Ifthethinfilmmodelgives larger

μ

-equivalent values(Eq.(38)solvedfor

μ

^fromKfilm) than

μ

drywerecommendsettingsuccessivelyslightlylowervapordiffu- sionresistancefactorsforthetwowetcupcalculationsofEq.(39), (e.g.

μ

^wet=

μ

dry-0.1).Then,logarithmicinterpolationstillcanbe achieved.

A2nddegreeLagrangelogarithmicinterpolation,incorporating anarbitrarythirdliquidconductivitypoint;K_hyg(

φ

dry)=10⁻²K_v,dry, couldbeapracticalandreasonableapproachforinterpolation.