modelling of coupled ion and moisture transport in porous building materials

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Construction and Building Materials 22 (2008) 2185–2195

and Building

MATERIALS

Modelling of coupled ion and moisture transport in porousbuilding materials

T.Q. Nguyen a,b,*, J. Petkovic b, P. Dangla b, V. Baroghel-Bouny b

a Laboratoire Central des Ponts et Chaussees, Division for Concrete and Cement Composites, 58 Boulevard Lefebvre, F-75732 Paris, Franceb Universite Paris-Est, Institut Navier, LMSGC, 2 allee Kepler 77420 Champs sur Marne, France

Received 15 November 2006; received in revised form 19 August 2007; accepted 21 August 2007Available online 15 October 2007

Abstract

In this paper, a physically based model describing the coupled ion and moisture transport is developed by combining existing theoriesof liquid water and water vapour transport with aqueous electrolyte theory. We derive the set of governing differential equations describ-ing simultaneous movement of water in the vapour and liquid phases and consequent transport of ions in unsaturated porous media. Theequations are developed in one-dimension, assuming isothermal conditions. It is also assumed that the movement of water in both thevapour and the liquid phases are first-order phenomena described by Fickian diffusion and Darcy’s law, respectively. The influence ofions on liquid–vapour equilibrium is modelled here by considering the water activity term. The effect of salt crystallization on the trans-port properties is considered. The diffusion of the ions present in the system is modelled by solving the Nernst–Planck/local electroneu-trality set of equations. A computer program has been developed to solve this highly non-linear problem. The validation of the model hasbeen performed on the basis of a comparison between predicted kinetics of moisture or ion concentration profiles and measurementobtained by NMR on plaster/Bentheimer sandstone system during drying.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Ionic transport; Convection; Durability; Modelling; Building material

1. Introduction

Moisture and in water soluble salts are notorious origi-nators of structural deterioration, defacing by efflorescenceand growth of all kinds of micro-organic matter when por-ous building materials are exposed to wetting–drying envi-ronment. Detailed information on the deterioration ofconstruction and building materials can be derived fromthe fast amount of the literature (see, e.g. the prominentwork and numerous publications of Binda et al. [1,2]).

The deterioration process determines maintenance needsand, finally, the service lifetime as elaborated by Bekker[3,4], see also the references therein. He initiated and

0950-0618/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.conbuildmat.2007.08.013

* Corresponding author. Address: Laboratoire Central des Ponts etChaussees, Division for Concrete and Cement Composites, 58 BoulevardLefebvre, F-75732 Paris, France. Tel.: +33 140435301.

E-mail address: [email protected] (T.Q. Nguyen).

guided the word of Pel [5]. Bekker dealt with a life-cycleapproach including economic consequences as well as theimplications on scarce (natural and financial) resources.Furthermore, he formulated a logical relationship betweendeterioration and durability resulting in a clear and concisedefinition: durability is the resistance that follows from theinherent property or capacity to prevent decay. Dynamicmoisture and ion transport is highly important topic in thatrespect.

In the building structures, plasters are often present asa finishing layer. The salt resistant plaster works eitherby accumulating the salts in the inner layer, whereenough space should be provided for their crystallization,either by transporting of the salts to the surface of theplaster, where efflorescence occurs. But the performanceof specially developed salt resistance plasters is notalways as good as expected. In some cases salt crystal-lizes at the interface between two materials (masonry

mailto:[email protected]

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Gas in Bentheimer sandstoneRel

ativ

e pe

rmea

bilit

y [-]

Degree of saturation Sl [-]

Water in plaster

Water in Bentheimer sandstone

Gas in plaster

Fig. 1. Relative permeabilities to liquid water and to gas versus degree ofsaturation.

2186 T.Q. Nguyen et al. / Construction and Building Materials 22 (2008) 2185–2195

and plaster), causing the plaster layer to detach. A betterknowledge of the properties of the plaster (composition,physical properties and moisture transport behaviour)together with a sound awareness of the field situation(plaster/substrate combination, moisture and salt load,environmental conditions) may help in avoiding unsuc-cessful applications in practice. To understand these pro-cesses in more detail, we have investigated how transportand accumulation of salt in a plaster depends on theunderlying masonry material. The object of this paperis to model the moisture and ion transport in the plas-ter/substrate system.

In fact, salt transport can be subdivided at least intotwo processes based totally on different mechanisms. Ionsof dissolved salts can be transported with the migratingwater by advection. The second transport mechanism isthe ion diffusion driven by a concentration gradient. Inaddition, the previous experimental researches [6–9] showthat the presence of salt strongly influences the kinetics ofdrying of water-containing porous media and lead to areduction in the drying rate. Furthermore, the crystalliza-tion near the surface reduces the effective surface areaavailable for the evaporation, leading to a decrease ofthe drying rate. Then, the coupled transport of moistureand ions involves complex chemical and physical model.Here, in the first part of this paper, the theoretical modelis presented. It deals with the analysis and simulation ofcoupled ion and moisture transport in layered porousbuilding materials taking into account insoluble salt pre-cipitation. A physically based model describing these phe-nomena is developed by combining existing theories ofliquid water and water vapour transport with aqueouselectrolyte theory. We derive the set of governing differen-tial equations describing simultaneous movement of waterin the vapour and liquid phases and consequent transportof ions in unsaturated porous media. The equations aredeveloped in one-dimension, assuming isothermal condi-tions. It is also assumed that the movement of water inboth the vapour and the liquid phases are first-order phe-nomena described by Fickian diffusion and Darcy’s law,respectively. The diffusion of the ions present in the sys-tem is modelled by solving the Nernst–Planck/localelectroneutrality set of equations. The mass balance prin-ciples obtained for the different elements are supple-mented with constitutive assumptions making thesystem of equations complete. The finite volume methodhas been used to discretize the coupled ion–moisturetransport equations. A computer program has been devel-oped to solve this highly non-linear problem. Then,results of an experimental test (moisture and sodium pro-files), which have been measured non-destructively with anuclear magnetic resonance (NMR) technique during dry-ing of the plaster/Bentheimer sandstone system, wereselected as a benchmark to test the computer programdeveloped using the methodology described above, show-ing an excellent agreement between the numerical and theexperimental results.

2. Governing equations of transport and mass balance

equations

In this section a model is proposed that accounts for theisothermal drying process. The basic laws that describe theporous medium with simultaneous movement of fluid andions through it will be presented. It is assumed that themean velocity of the solid phase is zero; that is, no defor-mation of the solid phase is taken into account.

The transport of the liquid–ion mixture and that of thegas (mixture of water vapour and dry air) is assumed to begoverned by the extended Darcy’s law, which reads (a = lor g, for liquid and gas, respectively) for an isotropic por-ous medium:

va ¼ �Kga

kra Slð Þgradpa ð1Þ

In Eq. (1), K is the permeability of the porous material (inm2), which is an intrinsic property of the solid matrix andindependent of the specific fluid characteristics. Sl is the li-quid saturation rate. va, pa, ga and kra(Sl) are, respectively,the Darcy filtration velocity (in m s�1), the pressure (in Pa),the dynamic viscosity (in Pa s) and the relative permeabilityassociated with fluid a.

The relative permeabilities to liquid and to gas, whichaccount for the variation of saturation degree due to mois-ture transfer, are usually approached using Mualem’smodel and Van Genutchen relationship [10]

krl Slð Þ ¼ Snl 1� 1� S1=m

l

� �mh i2

and

krg Slð Þ ¼ 1� Slð Þp 1� S1=ml

h i2mð2Þ

where n and p are fitting parameters. The best value for thesoils corresponds to n = p = 0.5 [5]. Using these expressionsand the value of parameter m, which is derived from non-linear curve fitting of the capillary pressure curve (see Eq.(17)), the relative permeabilities to water and gas for theanalysed material are calculated and presented in Fig. 1.

T.Q. Nguyen et al. / Construction and Building Materials 22 (2008) 2185–2195 2187

The relative diffusion process of water vapour and dryair phases relative to the gaseous mixture is assumed tobe governed by Fick’s law, which reads (a = a or v, forair and water vapour, respectively)

Ja ¼ �qg

Maf /; Slð ÞDg grad

qa

qg

!with

Dg ¼ 2:17� 10�3 p0

pg

TT 0

� �1:88

ð3Þ

where Ja, Ma and qa are the molar flux (in mol m�2 s�1),the molar mass (in kg mol�1) and the density (in kg m�3)of constituent a, respectively.

Fick’s law as formulated by Eq. (3) agrees with themicroscopic expression, which can be derived from thekinetic theory of gases [11]. Furthermore, Eq. (3) accountsexplicitly for the inversely proportional dependence of therelative flow of constituent a to the total pressure pg. Thisdependence comes from the expression adopted for the dif-fusion coefficient Dg (in m2 s�1) of the water vapour in theair given by De Vries and Kruger [12] (with T0 = 273 Kand p0 = patm = 101,325 Pa).

In addition, the factor f(/,Sl) accounts simultaneouslyfor the tortuosity effects and the reduction of space offeredto gas diffusion in porous medium. Numerous expressionshave been proposed by researchers for the resistance factor,which is involved in Fick’s law (Eq. (3)). The form Milling-ton’s works is used here [13]:

f /; Slð Þ ¼ /2x 1� Slð Þ2xþ2 ð4Þwhere x is a fitting parameter and / the porosity.

Thanks to measurements of diffusion coefficient of gasesO2 and CO2 in mortar [14], the value x = 1.4 can beobtained for parameter x. Functions fitted using Eq. (4)are reported in Fig. 2 and can be compared with the exper-imental data [14,15].

0.4 0.6 0.8 1.00.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

Data exp.

φ = 0.17φ = 0.15φ = 0.09

Res

ista

nce

fact

or [-

]

Liquid water saturation Sl [-]

Analytical formula

0.0 0.2

Fig. 2. Resistance factor: experimental measurements [14,15] are pre-sented with open symbols for different porosity values. Correspondingfunctions fitted using Eq. (4) are presented with lines.

The moisture flow is divided into vapour and liquidflows. Ions will be transported only with the liquid flow.Here, the ionic transport is described by separate diffusiveand convective terms. Moreover, the ions diffuse not onlydue to concentration gradient. When two ionic speciesare present in solution, such as sodium and chloride ions,chlorides tend to diffuse faster than sodium ions, as a resultof the higher chloride diffusion coefficient. Nevertheless,chlorides are charged negatively, whereas sodium ions arecharged positively. An electrical field therefore takes place,which slows down the faster ions and accelerates the slowerones. All these physical phenomena, which involve ionicdiffusion in the pore solution, can be taken into accountthrough the Nernst–Planck equation, as in [16,17]

J i ¼ �Di Slð Þ gradci þziFRT

ci gradw

� �ð5Þ

where Ji, Di, ci, and zi are the flux (in mol m�2 s�1), theeffective diffusion coefficient (in m2 s�1), the free concentra-tion in the pore solution (in mol m�3), and the valencenumber of each ionic species i, respectively. F is the Fara-day constant (9.64846 · 104 C mol�1), R the ideal gas con-stant (8.3143 J mol�1 K�1), and T the absolute temperature(in K). The electrical interactions between ions are de-scribed by the local electrical potential w (in V), which isdetermined here through local electroneutrality condition[17,18]:X

cizi ¼ 0 ð6Þ

The two terms on the right-hand side of Eq. (5) representthe two mechanisms of the ionic transport. They are thediffusion of ions due to ionic concentration gradient andthe migration of ions due to electrostatic potential gradi-ent, respectively.

The effective diffusion coefficient is a function of thetotal open porosity and of the degree of water saturation.The importance of concrete moisture state on the rate ofionic transport has been recognized [19], but it is difficultto determine experimentally the chloride diffusion coeffi-cient in non-saturated conditions. Saetta et al. [20] haveproposed an equation for the effect of relative humidityon the rate of ionic diffusion in cementitious materials.As the ions have to diffuse in the liquid phase, the degreeof saturation can therefore strongly affect the transportof ions. Buchwald [21] measured the diffusion coefficientfor different materials as a function of the degree of satura-tion using the impedance spectroscopy technique. Heshowed that the diffusion coefficient decrease non-linearlywith the degree of saturation

Di Slð Þ ¼ D0i Sk

l ð7Þ

where D0i is the diffusion coefficient of the saturated mate-

rial (in m2 s�1), and k is the saturation exponent.Thanks to measurements of chloride diffusion coefficient

of masonry materials with different degrees of saturation[21]; results can be summarized on a relative diffusivity


curve, which describes the diffusivity of ionic species, nor-malized to its value at full saturation (see Fig. 3). We fitparameter k for different materials.

It has to be underlined that Eq. (7) is very important interms of characterizing the coupling between the moisturetransfer and the chloride penetration.

Finally, once the diffusive flux Ji of all ions is known, itis possible to calculate the diffusive flux of water Jw (inmol m�2 s�1) using the relation

MwJ w þXi 6¼w

M iJ i ¼ 0 ð8Þ

where Mw and Mi are the molar mass of water and of theion i (in kg mol�1), respectively.

The mathematical description of transport processes forthe ions, water vapour, dry air and water can be obtainedby adding up the components of advection and diffusion:

wa ¼ vlca þ Ja with a ¼ w or i;

for water and ions; respectively

or

wa ¼ vgca þ Ja with a ¼ a or v;

for air and water vapour; respectively

8>>>>>><>>>>>>:

ð9Þ

where wa is the total molar flux (in mol m�2 s�1), va (a = lor g) is the Darcy filtration velocity that can be calculatedas Eq. (1), and Ja is the flux of each component in the mix-ture that can be calculated from Eqs. (3), (5) and (8) for thewater vapour, dry air, ions and liquid water, respectively.ca is the molar concentration with the condition thatX

taca ¼ 1 with

a ¼ w or i; for water and ions; respectively ð10Þ

where ta is molar volume of component a in solution (i.e.tw = 18.0 cm3 mol�1, tNaþ ¼ �1:5 cm3 mol�1).

0.0 0.2 0.4 1.00.0

0.2

0.4

0.6

0.8

1.0

Curve fitting

Plaster Sandstone

D/D

s l=10

0%

Liquid water saturation Sl [-]

Data exp.

0.6 0.8

Fig. 3. Relative values of diffusion coefficients versus degree of watersaturation. Open symbols represent experimental data [21]. Lines repre-sent functions fitted using Eq. (7).

Now, the mass balance equations that govern the statevariables describing the phenomena under considerationwill be described. A mass balance equation is consideredfor each ionic species. This mass balance equation includesthe terms to account for ionic exchange between the solu-tion and the solid (crystallization of the salt). The mass bal-ance equations for the water (as the component of theliquid (w) and the gas (v) phases), the dry air (a) and thesalt ions i (Na and Cl), respectively,

oot /Slcw þ / 1� Slð Þ qv

Mv

� �¼ �div ww þ wvð Þ

oot / 1� Slð Þ qa

Ma

� �¼ �div wað Þ

oot /SlcNa þ sNaClð Þ ¼ �div wNað Þoot /SlcCl þ sNaClð Þ ¼ �div wClð Þ

8>>>>>><>>>>>>:

ð11Þ

In Eq. (11) sNaCl is the amount of crystallized salt (inmol m�3 of material).

On the other hand, a salt crystallization model is heredeveloped and coupled with the ion transport equationdescribing the ion movements and reactions in porousmedia. It has to be underlined that most of the modelsfound in the literature have disregarded the crystallizationof salts. This is easily understandable since the crystalliza-tion phenomenon is complicated to model, because of thediscontinuous aspect of the chemical equilibrium equation[22]. In this model, the chemical equilibrium of the solidphase present in the material is satisfied unless the solidphase has totally disappeared. For example, the equilib-rium constant of NaCl (Keq) is given by

Keq ¼ ½Naþ�½Cl�� ð12Þwhere [Na+] and [Cl�] are the sodium and chloride concen-trations in the pore solution, respectively.

If one notes the amount of NaCl in the solid phase assNaCl, the instantaneous dissolution of NaCl can thus bedescribed by the following conditions [23]:

sNaCl P 0

Keq P ½Naþ�½Cl��sNaCl Keq � ½Naþ�½Cl��

� ¼ 0

8><>: ð13Þ

It should be noted that this approach assumes the existenceof a local chemical equilibrium throughout the system. Itmeans that the dissolution/precipitation rate of the variousspecies in solution is assumed as much faster than thetransport rate. More information on this approach canbe found in [22].

Moreover, during the ionic transport process, chemicalreactions can modify the transport properties of the mate-rial by affecting its pore structure. In fact, the crystalliza-tion of sodium chloride may contribute to local decreaseof the porosity, thus decreasing the section across whichions are able to diffuse. The effects of the salt crystallizationon the moisture and ionic transport are taken into accountby correcting the material transport properties as a func-tion of the porosity in each time step.

1E-3 0.01 0.1 1 100.0

0.2

0.4

0.6

0.8

1.0

Curve fitting

liqui

d w

ater

sat

urat

ion

[-]

capillairy pressure [MPa]

Bentheimer sandstone plaster

Exp. Data

Fig. 5. Liquid water saturation versus capillary pressure.


Indeed, the influence of chemical reactions on the poros-ity of the material can be calculated as follows:

/ ¼ /0 � mNaCl sNaCl ð14Þwhere mNaCl (in m3 mol�1) is the molar volume of the salt insolid phase and /0 the initial capillary porosity.

This change of porosity due to chemical reaction mayaffect the transport properties of the material. Here, weassume that the intrinsic permeability coefficient and thediffusion coefficient depend on the total porosity by the fol-lowing expression, drawn from Refs. [24,25]:

Di /ð Þ ¼ D0i

//0

/� 0:18

/0 � 0:18

� �2

and

K /ð Þ ¼ K0//0

� �31� /0

1� /

� �2

ð15Þ

where K0 and D0i denote the intrinsic permeability (in m2)

and diffusion coefficient (in m2 s�1), respectively, corre-sponding to the initial porosity /0.

Finally, assuming that the gas is an ideal mixture, itspressure pg is the sum of the pressures of its constituentspg = pv + pa. The water vapour and the dry air are assumedto be ideal gases, while the liquid pressure is governed bythe capillary curve already introduced in [26]. Therefore,the state equations read

paMa ¼ RT qa and pg � pl ¼ pcðSlÞ ð16Þ

where pc is the capillary pressure (in Pa) and a = a or v, forair and water vapour, respectively. The state equation (Eq.(16)), also called capillary pressure curve, can be derivedfrom mercury intrusion porosimetry results (see Figs. 4and 5) by applying a set of increasing pressures to a dryspecimen and measuring the corresponding mercuryinstruction volume. According to the Laplace equationand assuming cylindrical pores, a contact angle mercury-plaster of 140�, a surface tension mercury vacuum of0.485 N m�1 and the mercury density qm = 13,549 kg m�3,the obtained ‘mercury retention curve’ is recalculated to the

0.01 0.1 1 10 1000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00

0.02

0.04

0.06

0.08

0.10

0.12

incr

emen

tal p

ore

volu

me

[ml/m

l]

cum

ulat

ive

pore

vol

ume

[ml/m

l]

diameter [μm]

Bentheimer sandstone plaster

Fig. 4. Pore size distribution of plaster and Bentheimer sandstonemeasured by mercury instruction porosimetry [33].

moisture retention curve. The calculated capillary pressurecurves for two materials are presented in Fig. 5 with theopen symbols.

A good fit of the capillary pressure curve is obtainedthrough the analytical formula proposed by Van Genuch-enten [25]

pc ¼ a S�1=ml � 1

h i1=nð17Þ

where a, m and n are the fitting parameters dependent onthe microstructure of the material. The results of fittingfor two materials are presented in Fig. 5.

To complete the model, an equation that describes theliquid–vapour equilibrium should be considered, takinginto account the influence of the ionic concentration. Thistopic is addressed in the following section.

3. Liquid–vapour equilibrium

To model the liquid–vapour equilibrium, it would seemstraightforward to use the well-known isothermal Clapey-ron’s law as in [26]. However, the development of this rela-tionship is made for the case of pure water in a porousmaterial. When ions are present, the vapour pressure abovethe solution is lower than the above pure water [27]. Rep-resentation of liquid–vapour equilibrium for electrolyticsolutions is a complex operation, because it involves phys-ico-chemical phenomena such as electrostatic interactionbetween ions and ion solvation. The calculation is morecomplex if the liquid phase is a solvent mixture, with differ-ent dielectric characteristics. In order to explain the influ-ence of the salt concentration on the liquid–vapourequilibrium, a detailed study of the thermodynamic equa-tions is carried out.

We begin by obtaining the chemical potential of a solu-tion. The general expression for the chemical potential ofliquid water is [28]

lwðp; T ; awÞ ¼ l0wðp; T Þ þ RT ln aw ð18Þ


where aw is the activity of water, l0wðp; T Þ is the chemical

potential of pure water (in J mol�1) in the same aggregatestate as the solution (i.e. the chemical potential of pure li-quid water at temperature T and pressure p). l0

wðp; T Þ is re-ferred to as the standard state chemical potential which isindependent of aw.

The chemical potential of pure water can be obtainedfrom the Gibbs–Duhem equation:

dlw ¼ �sw dT þ tw dp ð19ÞIn unsaturated porous building materials, the pressures areusually not high enough to affect the partial molar volume,therefore, the compressibility of solution can be neglected(i.e. tw = const). Eq. (19) can be integrated as follows:

l0wðp; T Þ ¼ l�wðp0; T 0Þ �

Z T

T 0

swðT ÞdT þZ p

p0

tw dp

¼ l�wðT Þ þ twp ð20Þ

Thus the chemical potential of non-ideal solution is writtenas

lwðp; T ; awÞ ¼ l�wðT Þ þ twpl þ RT ln aw ð21ÞFor the water in vapour phase (assuming that it behaves asan ideal gas), the chemical potential is given by

lv ¼ l0v þ RT ln

pv

p0v

ð22Þ

By combining Eqs. (21) and (22) and using the fact that inequilibrium the chemical potentials of water in the vapourphase and in the liquid phase are equal, the following rela-tion is derived:

l�w þ RT ln aw þ twpl ¼ l0v þ RT ln

pv

p0v

ð23Þ

but for pure water in atmospheric condition, we must have

l�w þ twpatm ¼ l0v þ RT ln

pvs Tð Þp0

v

� �ð24Þ

where pvs(T) is the vapour pressure at saturation in atmo-spheric condition pl = patm and when there is no salt insolution.

Subtracting Eq. (24) from (23), we find that

RT lnpv

pvsðT Þ

� �¼ tw pl � patmð Þ þ RT ln aw

¼ Mw

qw

pl � patmð Þ þ RT ln aw ð25Þ

The first term on the right-hand side of Eq. (25) is a correc-tion to the chemical potential of water to take into accountthe presence of a concave meniscus inside a porous mate-rial. In fact, when the solution–vapour interface is locatedinside a pore, it becomes curved due to the fact that waterpreferentially wets the surface. This curvature modifies theexpression for mechanical equilibrium. The saturated pres-sure in this water meniscus depends on the surface. For asalt solution with an infinite curvature radius, the first term

is neglected, and the chemical potential of water is definedby ionic interaction. For a diluted solution and pure water,the second term is neglected and, therefore, Clapeyron’srelation can be obtained [26].

On the other hand, numerous expressions have beenproposed by researchers for the water activity term, whichis involved in Eq. (25). For a dilute ideal solution, the activ-ity would be just the mole fraction, which can be calculatedfrom the following equation:

aw ¼cw

cw þPi 6¼w

ci

ð26Þ

Aqueous solutions with solute concentration exceeding afew millimoles depart from the ideality, which is assumedin the above relation. To represent accurately the molefraction of the water in solutions, which approach unityas the dilution is increased, it is common to tabulate datain terms of the osmotic coefficient P. The osmotic coeffi-cient for any aqueous solution can be obtained from thefollowing equation [29]:

P ¼ � ln aw

Mw

Pimi

� ð27Þ

where mi is the molal concentration of ion i (in mol kg�1).Alternatively, one can calculate the osmotic coefficient

for many solutes over a wide range of concentrations usinga semi-empirical formula. One of the most widely used for-mulas for the estimation of the osmotic coefficient is Pit-zer’s method presented in a series of papers beginning in1973 [30,31]. For NaCl electrolytes Pitzer’s equation is

P ¼ 1� A/I0:5

1þ 1:2I0:5þ I b0

NaCl þ b1NaCl exp �2I0:5

� �þ I2C/

NaCl

ð28Þ

where b0, b1 and C/ are parameters specific to NaCl elec-trolyte (b0 = 0.0765 kg mol�1, b1 = 0.2664 kg mol�1 andC/ = 0.00127 kg2 mol�2), I the ionic strength (inmol kg�1). The osmotic Debye–Huckel coefficient A/ (inkg0.5 mol�0.5) is computed as follows:

A/ ¼ 1

2:3032pN 0qwð Þ0:5 e2

4pere0kT

� �1:5

ð29Þ

where N0 is Avagadro’s number (in mol�1), qw is the den-sity of pure water (in kg m�3), e0 is the permittivity of vac-uum (in C2 N m2), er is the relative permittivity water, k isBoltzmann’s constant (in J K�1), and e is the elementarycharge (in C).

From Eq. (25) it can readily be shown that the vapourpressure above a solution can be decreased by increasingsolute concentrations (i.e. salts promote vapour condensa-tion at lower relative humidity values). Assuming that theliquid water and its vapour remain permanently in localthermodynamic equilibrium and that salt solution remainsa long time within porous media, the rate of evaporation isgoverned by Eq. (25) at each time. Similar to the Kelvin

1.0

1.5

2.0

V [c

m3 ]

Bentheimer sandstone - model Bentheimer sandstone - epx. Plaster - model Plaster - exp.


model, the proposed model describes that, from the ener-getic point of view, for materials with small pores, evapo-ration is unfavourable. This is especially true forconcentrated solutions due to the fact that water activitydecreases because of ionic interactions. Therefore, thismodel can be used to explain the evaporation processobserved in experiments, taking into account both saltpresence and pore structure.

4. Numerical resolution

To solve such a complex system of non-linear equations(Eqs. (11), (16) and (25)), a numerical algorithm has to beused. The spatial discretization of this coupled system isperformed here through the finite volume method, sincethe discrete form of the mass balance laws (Eq. (11)) isstrictly satisfied. At each step, the mass balance equations,the state equations, the local thermodynamic liquid–vapour equilibrium law and electroneutrality of pore solu-tion constitute the whole system of equation. The unknownvariables in the adopted discretization scheme are the con-centration for each ionic species taken into account, theelectrical potential, the water vapour pressure and the dryair pressure. An implicit approximation of the normalderivative, ensuring the best stability of the scheme, is used.The non-linear set of equations is solved by means of astandard Newton algorithm [32]. The output includes ionicconcentration profiles, electrical potential profile, the mois-ture content distribution and also drying kinetics.

5. Comparison between model and experiment

Some drying experiments on plaster/substrate systemreported in [33] have been simulated with the model.Experiments were done on a plaster/Bentheimer sandstonesystem, which is schematically presented in Fig. 6. Theplaster had a composition of lime:cement:sand = 4:1:10.The sample was cylindrical with a diameter of 19 mmand a total length of 50 mm. Initially, the sample was uni-formly capillary saturated by immersing it in water or aNaCl solution, c = 4 mol L�1, for about 20 h. To be ableto model the observed drying and salt transport as 1D pro-cesses, the samples were sealed with teflon tape at all sidesexcept at the top (the plaster/air interface), over which thedry air is blown (see Fig. 6). The relative humidity of the airwas less than 1%. An NMR (Nuclear Magnetic Resonance)

drying

Sandstone Plaster

isolation

Fig. 6. Plaster/Bentheimer sandstone system isolated at all sides exceptthe air/plaster interface, where the water vapour can escape.

method is used to measure the moisture and dissolvedsodium profiles during drying process.

First, the drying behaviour of plaster/Bentheimer sand-stone system saturated with pure water has been studied.Experimentally and numerically obtained volumes of waterin the plaster and Bentheimer sandstone versus drying timeare shown in Fig. 7. By comparing numerical and experi-mental curves we deduce the intrinsic permeabilities ofboth materials. This comparison consists in determiningthe value of K, which leads to the best account of theobserved data. It is known that the effective diffusion coef-ficient of ions can be derived from the value of the intrinsicpermeability by using the Katz–Thompson relation [34].Note that the relevance of the proposed method of determi-nation of K can be appreciated a posteriori through itsaccuracy to represent a lot of experimental plots.

Fig. 8 shows the comparison of the calculated and mea-sured water content distribution in the sample. The waterprofiles in the plaster/Bentheimer sandstone system areplotted for several times during the drying process. Wenote a good agreement. Our numerical model is capableof predicting the existence of two drying stages, which wereobserved experimentally. During stage 1, the plasterremains saturated and the Bentheimer sandstone dries.The plaster starts to dry during the second drying stage(t > 1 h, see Fig. 7). This drying behaviour was explainedby the dependence of the capillary pressure on the poresizes in two materials. During drying water remains inthe smallest pores, in which the capillary pressure is high-est. Therefore the Bentheimer sandstone dries first, sinceits pores are an order of magnitude larger than the poresin the plaster (see Fig. 4). In addition, our numerical resultsconfirm experimental observation that after about 6 h areceding drying front enters the plaster layer.

The deviations of numerical results from the experimen-tal are explained by the variations of the pore sizes with the

0 10 20 300.0

0.5

t [h]

Fig. 7. Total volumes of water present in the plaster and Bentheimersandstone as a function of the drying time. The sample was initiallysaturated with water. Experimental data [28] are presented with opensymbols, while the numerical results are presented with lines.

0.00 0.01 0.02 0.03 0.04 0.050

100

200

Qua

ntity

of c

ryst

alliz

ed N

aCl [

mol

/m3 ]

x [m]

Plaster Bentheimer sandstone

270 h

200 h 270 h

185 h

Fig. 10. Quantity of crystallized NaCl, predicted numerically (in mol ofNaCl per volume of material) in the plaster/Bentheimer sandstone systemduring drying.

0.00 0.01 0.02 0.03 0.04 0.050.0

0.1

0.2

Wat

er c

onte

nt [m

3 /m3 ]

x [m]

Bentheimer sandstonePlaster

0 h

2h20 h

10 h

2 h

0 h

4 h6 h

Fig. 8. Water content (in volume of water per volume of material) versusposition in plaster/Bentheimer sandstone system. The sample was initiallysaturated with water. Experimental data are presented with open symbols,for several times during the drying process. Results obtained numericallyare presented with lines.

0.00 0.01 0.02 0.03 0.04 0.050.00

0.05

0.10

0.15

0.20

0.25

Wat

er c

onte

nt [m

3 /m3 ]

x [m]


0 h

6 h

10 h

25 h

0-25 h

50 h

75 h

150 h200 h

270 h

Fig. 9. Water content (in volume of water per volume of material) versusposition in plaster/Bentheimer sandstone system. The sample was initiallysaturated with NaCl solution, c = 4 mol L�1. Experimental data [33] arepresented with open symbols, for several times during the drying process.Numerical results are presented with lines.


height (x axis in Fig. 8) of the plaster and Bentheimer sand-stone. In our model, we have assumed that the pore sizesare isotropic and that only porosities change with theheight (distribution of the porosity is presented as a firstprofile in Fig. 8, for t = 0 h, when the sample is completelysaturated with water). This assumption is not true, and inthe case of the plaster, pore sizes near to the substratemight be much smaller than at the drying surface. Namely,it is known that the water is extracted from the mortarlayer during brick laying [35], which results in the decreas-ing of the pore sizes of the mortar near the interface withthe brick. In fact, the measure of water content by a scan-ning technique presumes that the moving liquid frontshould remain horizontal, perpendicular on the adsorp-tion–desorption direction. This requires an almost flatfront in the space within two scanning steps. But, if thesolid porous material is neither homogeneous nor isotro-pic, this assumption is not satisfying.

To study the salt transport in the plaster/Bentheimersandstone system, the sample was initially saturated withNaCl solution, c = 4 mol L�1. In Fig. 9 we have plottednumerically obtained water profiles for several times duringthe drying process and we have compared it with the exper-imentally obtained results. A satisfactory agreementbetween the calculated and experimental data can beobserved. As in the case of pure water saturation, the exis-tence of two drying stages is observed. In addition, ourmodel confirms experimentally observed slower drying ofthe salt saturated sample in comparison to the water-satu-rated sample. This is partly explained by the decrease of therelative humidity near the liquid–air interface in the pres-ence of salt, which leads to the decrease of the drying rate(see Eq. (29)). Our numerical model also takes into accountthe decreasing of the drying rate due to blocking of thepore by salt crystals, which reduces the effective surfacearea that is available for evaporation. The quantities of

crystallized NaCl versus position in the sample are pre-sented in Fig. 10 for several times during the drying pro-cess. Crystallization of sodium chloride inside thematerial contributes to local decrease of the porosity, thusdecreasing the section across which the transport propertiesof the material are modified.

In Fig. 11 we present numerical concentration profiles ofdissolved Na in the Plaster/Bentheimer sandstone system atdifferent times of the drying process. During the initial dry-ing, Na ions are transported by advection to the surface,where they accumulate. There, NaCl concentration slowlyincreases to 6 M, which is the saturation value for a NaClsolution. At this point additional advection will result incrystallization at the top of the sample, which was experi-mentally observed as a white efflorescence. After 75 h the

0.00 0.01 0.02 0.03 0.04 0.05

4000

4500

5000

5500

6000D

isso

lved

Na

ions

con

cent

ratio

n [m

ol/m

3 ]

x [m]


0 h0 h

75 h50 h

75 h 150 h

270 h200 h

Fig. 11. Numerical concentration profiles of dissolved Na in the plaster/Bentheimer sandstone system (in mol of Na per volume of solution).


drying rate decreases and so will the advection. At thispoint, the advection competes with the diffusion of the ionsand diffusion starts to be a dominant transport process.The NaCl concentration profile in the sample starts to leveloff until the total sample is at 6 M. The simulations havedemonstrated that the slow moisture and salt solutiontransport in the plaster layer lead to salt accumulationinside the plaster layer. In this case the plaster behavesaccording to the ‘salt accumulating’ working principle. Ifthe evaporation is slow or/and the plaster layer thin, thesalt accumulates and crystallizes near the surface. There-fore, a sufficient thickness of the plaster layer is necessaryto avoid surface damage and to achieve the salt accumula-tion inside the plaster layer (under conditions of slow mois-ture and salt transport).

In Fig. 12, the Na profiles are presented for severaltimes during the drying process. Numerical results are

0.00 0.01 0.02 0.03 0.04 0.050

200

400

600

800

1000

1200

1400

Na

cont

ent [

mol

/m3 ]

x [m]


0 h

6 h

10 h25 h

0 h

25 h

50 h

75 h

150 h

Fig. 12. Measured and calculated content profiles of dissolved Na ions inthe plaster/Bentheimer sandstone system (in mol of Na per volume ofmaterial).

given with lines, and experimental with open symbols.The difference between measured and calculated valuescan be attributed to the loss of measured Na signal inmaterials with small pores. In fact, in the Bentheimersandstone there appeared to be no significant Na signalloss, while in the plaster not all Na could be measured,due to the fast transverse relaxation of Na in the smallpores [36]. The signal loss in the plaster was roughly cor-rected for by multiplying the measured signal with a cor-rection factor, which is the ratio between the initial Naconcentration presented and measured by NMR [33]. Itis assumed that the transverse relation rate of Na doesnot change with moisture content.

The total amount of dissolved Na in the plaster and theBentheimer sandstone as a function of the drying time isshown in Fig. 13. A rather good agreement in the orderof magnitude can be observed between the numericalresults and the experimental values, which are correctedaccording to the procedure described above.

The numerically determined total Na content is com-pared to the experimental values, which have beenobtained by ion chromatography method. Fig. 14 showsthe measured and predicted amount of Na in 8 slices,which were cut after the drying of the sample, perpendic-ularly to the drying axis (x axis). These Na contents ineach slice have been divided by the total content of Naions in the sample. As can be seen in Fig. 14, the numer-ical results suggest that salt mainly accumulates and crys-tallizes in the plaster layer. A good agreement between thecalculated and the experimental data can be observed. Inaddition, the numerical model confirms the conclusionsderived from measured results, in the sense that after dry-ing almost all salt is present in the plaster layer. Thisconsistency is a good argument to validate the proposedmodel.

0 20 40 60 80 100 120 1400

1

2

3

4

5

6

7 Bentheimer sandstone - exp. Plaster - exp. Plaster - exp. corrected (f = 1,9) Bentheimer sandstone - model Plaster - model

Na

cont

ent [

m m

ol]

Time [h]

Plaster

Bentheimersandstone

Fig. 13. Total amounts of dissolved Na ions in the plaster and Bentheimersandstone as a function of the drying time. Open symbols are originalexperimental data and lines represent our numerical data. The correctedexperimental values of dissolved Na in the plaster are represented bysymbol (·).

0.0

0.1

0.2

0.3

0.4

experiment model

Na

cont

ent/t

otal

Na

cont

ent

x [m]


0-5 5-10 10-15.5 15.5-21 21-27.3 28.1-33.6 34.4-40.3 41.1-47.7

Fig. 14. Amount of Na in 8 slices of a plaster/Bentheimer sandstonesample calculated by the model and measured by ion chromatography.The Na content in each slice is divided by the total content of Na ions inthe sample.


6. Conclusions

Theoretical and experimental studies of coupled waterand salt transport in porous building materials are newtopics and they are not yet very frequent in the literature.In this paper, an attempt towards better understanding ofmechanisms driving the coupled water and ion transportin porous medium was done. Comparison with the experi-ment also confirmed that an application of the aqueouselectrolyte theory in mathematical models of water and salttransport is unavoidable for a realistic description of pro-cesses taking place in the porous medium. Neglecting theosmotic effect of ionic concentration on the transport ofwater can lead to a departure from reality. Although over-simplified models neglecting the above phenomenon havebeen used, their application should always be done for rel-ative purposes only and bearing in mind that importantfactors are not taken into account.

A new calculation model for the coupled moisture andion transport in a porous building material is proposedhere. This model can be used to predict the transport ofall ions present in the system. The water transport (in thevapour phase and the liquid phase) is described by employ-ing Fickian diffusion and Darcy’s law. The influence of ionson liquid–vapour equilibrium is modelled by consideringthe water activity term. The ions move in the material underthe combined effect of diffusion and advection. The electri-cal coupling between the various ionic species in solution istaken into account. Here, the electrical potential, arisingfrom the electrical coupling between the ions to maintaina neutral solution, is calculated with electroneutrality con-dition. In addition, the salt crystallization phenomenon istaken into account. The influence of the precipitation ofthe salt on the transport properties of the materials is con-sidered. The simulations reported in this paper point outthat the presence of ions affects significantly the moisturetransport in concrete by changing the local thermodynamicequilibrium between liquid water and vapour. Compared tothe traditional simplified models, the present model is more

detailed and takes into account the individual effects of allthe mechanisms involved (diffusion, convection, ionic cou-pling, salt crystallization and the effect of salt presence onequilibrium liquid–vapour). It can be applied successfullyto various different cases with unified ionic diffusivities.

The concordance of the model with the experimentalresults confirms the reliability and the effectiveness of theproposed numerical model. We hope that this work will leadto formulation of a simulation tool of greater general appli-cability and attainment of a more detailed understanding ofthe mechanism of ion transport in partially saturated mate-rials. To achieve this aim, our work gives some preliminaryresults. In addition, some questions are raised concerningthe interpretation of field data by empirical models. Thesequestions demonstrate the interest of using physical modelsas a complement for understanding and quantifying thetransport mechanism in porous building materials.

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modelling of coupled ion and moisture transport in porous building materials

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