some considerations on modeling heat and mass transfer in porous media

Transport in Porous Media28: 233–251, 1997. 233© 1997Kluwer Academic Publishers. Printed in the Netherlands.

Some Considerations on Modeling Heat and Mass

Transfer in Porous Media

P. BAGGIO1, C. BONACINA1, and B.A. SCHREFLER21Ist. Fisica Tecnica, Universita di Padova, Via Venezia 1, 35131 Padova, Italy,e-mail:[email protected]. Scienza e Tecnica delle Costruzioni, Universita di Padova, Via Marzolo, 9,35131 Padova, Italy

(Received: 19 April 1996; in final form: 25 March 1997)

Abstract. In this paper some considerations are presented about the equations needed to set upa model of the process of heat and mass transfer in porous media. A clear classification is madeof the various types of equations used and of their physical meaning. Special attention is paid tothe thermodynamic equilibrium equations and to their derivation since they are too often taken forgranted. The importance of the various transport mechanisms (of mass and energy) is analyzed andthe consequences that can arise when some term is neglected are indicated.

Key words: capillary pressure, conservation equations, constitutive equations, liquid water, mod-eling, thermodynamic equilibrium.

Nomenclature

Cp

CpgCpsCplDeff

ggKKrgKrl

M

Ma

Mw

papvpv0

effective specific heat of the porousmaterialspecific heat of the gas mixturespecific heat of the solid matrixspecific heat of the liquid phase (water)effective diffusitivity of the gas mix-turemolar Gibbs functiongravity accelerationabsolute permeabilityrelative permeability of the gas phaserelative permeability of the liquidphasemolar mass of the gas mixture (moistair)molar mass of the dry airmolar mass of the waterdry air partial pressurewater vapor partial pressureliquid water vapor tension (at totalpressurePg over a flat surface)

PcPg

Pg0

PvPv0

r∗RR.H.sS

tTvx1hvap

capillary pressuretotal pressure of the gas phase (moistair)total pressure of the gas phase (moistair) over a flat surfacewater vapor pressurewater vapor saturation pressure over aflat surfacepore radiusuniversal gas constantrelative humidity(pv/pv0)molar entropyliquid phase volume saturation [liquidVol./pore Vol.]time variabletemperaturemolar volumemolar fractionmolar enthalpy of vaporization

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234 P. BAGGIO ET AL.

Greek

λeffµgµl

µvρρa

Symbols

effective thermal conductivitygas phase dynamic viscosityliquid chemical potential or dynamicviscosity (only Equation (26), (27) and(28))vapor chemical potentialeffective density of the porous mediummass concentration of dry air in the gasphase

ρgρlρv

σφψ

gas phase densityliquid phase densitymass concentration of water vapor inthe gas phasesurface tensionporosity [pore Vol./total Vol.]water potential

1. Introduction

Porous media are usually analyzed from the macroscopic point of view acceptingthe continuum hypothesis of matter and then ignoring the discrete nature of thematter itself. This is the common approach used in thermodynamics. The underly-ing assumption is that at larger scales the macroscopic quantities can be identifiedas statistical averages of molecular quantities. When the equations governing thesystem are written in a local form, it is then implied that the elementary volumeconsidered is large enough so that the thermodynamics properties (that are averagedmolecular properties) are still meaningful. Moreover, to model a process in a con-tinuum medium, where there are gradients of the thermodynamic coordinates (suchas pressure or temperature), a local thermodynamic equilibrium hypothesis must beassumed (where local refers to the order of magnitude of the mean free path of themolecules). A rigorous assessment of such hypothesis requires to enter the realm ofnonequilibrium thermodynamics[1–3] and to evaluate the relaxation times and themolecular mean free path. In other words it is necessary to establish that the timescaleof the variations of the macroscopic system are much longer than the time required toreach equilibrium locally and that in the length scale considered there is a number ofmolecules large enough so that averaged molecular properties, i.e., thermodynamiccoordinates, become meaningful. This is true when the time required for the redis-tribution of energy between the internal degrees of freedom of the molecules can beneglected (negligible bulk viscosity), such as in ideal gases, and when the mediumis not too rarefied (i.e. Knudsen number� 0.1) [3]. For the slow processes consid-ered here, such as drying, the local thermodynamic equilibrium hypothesis is usuallyaccepted. Otherwise the continuum hypothesis must be abandoned and the intensivethermodynamic quantities/coordinates such as pressure, density, temperature, etc.cannot be defined (and in such case the problem must be dealt with using a statisticalthermodynamics and/or quantum mechanical approach. For an introductory viewsee [4]).

The porous medium poses some additional problems because it is a heterogeneoussystem. In fact, it can be considered as a multiphase system where the pores of asolid matrix are partly filled with liquid water and partly with a gaseous mixture

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MODELING HEAT AND MASS TRANSFER IN POROUS MEDIA 235

of dry air and water vapor. To treat the porous medium as a system in which theproperties are continuous functions of the space coordinates, some kind of largerscale volume averaging must be used (in addition to the statistical average at themolecular scale inside each phase, needed to define local thermodynamic properties).As suggested by several authors (see Hassanizadeh and Gray [5–7], Bear [8], Bachmatand Bear [9], Bear and Bachmat [10, 11]), this can be done introducing the concept ofrepresentative elementary volume (REV). Obviously, this representative elementaryvolume has a length scale much greater than the molecular one previously mentionedand includes different components/phases that could have different mean values ofthe system coordinates (e.g., the mean temperature of the solid matrix could bedifferent form the mean temperature of the fluid phase inside the REV). In principle,many averaging/homogenization techniques take in account different properties indifferent phases/components. But, usually, during the homogenization process someterms are assumed to be negligible and some kind of integral is performed overthe REV introducing a weighted average of the relevant variable. This leads to aredefinition of the related transport coefficients that incorporates, in a more or lessaccurate fashion, the effects of heterogeneity.

For example, a weighted mean temperature can be introduced in the REV coupledto an effective thermal conductivityλeff as proposed by Nozadet al. [12], Ariault[13], Ariault and Royer [14], Ariault and Ene [15]. Such transport coefficients willbe dependent not only on the component materials but also, to some extent, onthe state of the system, its previous history and the ratio of the system size versusthe heterogeneity length scale (some insight about these aspects can be derived frompercolation theory; for an introduction see [16]). We agree with this approach becauseit is easier to compare theoretical simulations with available experimental results(e.g., measurements of the heat transfer in heterogeneous insulating materials, suchas fiber glass and polystyrene foam, lead to the evaluation of the so-called apparentthermal conductivity Bertasiet al. [17], Arduini and De Ponte [18]).

The general approach to the description of nonequilibrium processes, includingheat and mass transfer in a porous medium, is to start from an appropriate set ofconservation equations and to supplement them with thermodynamic equilibrium(or state) equations as well as with constitutive equations (see De Groot and Mazur[1], Eu [2]). The equilibrium equations can be considered a generalization of theequation of state and express the relationship existing between the thermodynamiccoordinates and/or their variations at (thermodynamic) equilibrium. They then limitthe number of independent coordinates needed to describe the state of the system. Thegeneral applicable (non-equilibrium) conservation equations are, as usual, themassconservation equation, themomentum conservation equation(for the fluid phasesand the solid matrix) and theenergy conservation equation(or first law of thermody-namics). Since we do not consider the effect of external electromagnetic fields, andhence there is not localized torque due to electrical body forces, theangular momen-tum conservation equationis not needed. The solution of conservation equationsrequires relationships connecting the fluxes due to molecular transport mechanisms

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(i.e., the total flux minus the convective one) of the conserved quantities (such asmass, momentum, and energy) with the gradients of state variables. These relation-ships are obtained through the so-called phenomenological or constitutive equationsdescribing the specific behavior of the considered material such as the Fick’s law ofdiffusion, the Fourier’s law for the heat flux, the Newton’s law of viscosity and otherrelations expressing the relative permeabilitiesKrg Krl , the effective diffusivityDeff ,the effective thermal conductivityλeff , the saturationS a function of the chosen statevariables.

A thorough account of a possible way to obtain the conservation equations forheat and mass transfer in a porous media with a rigid indeformable matrix has beenproposed by Whitaker [19, 20] and here only some brief considerations are presented.

2. Thermodynamic Equilibrium Relationships

While the conservation equations are well known and have been thoroughly dealtwith in many papers [5–11, 19, 20], much less attention has been paid to the derivationand meaning of the thermodynamic equilibrium relationships, so it is probably wortha somewhat deeper review of this topic.

For a mixture ofN components, the equilibrium between a liquid(l) and its vapor(v) can be expressed using the chemical potentialµ as

µli = µvi ; i = 1, 2, . . . , N (1)

or, since for apure substancethe chemical potentialµ is equal to the molar Gibbsfunctiong of each phase, the equilibrium can also be expressed as

gl = gv. (2)

The differential of the chemical potential for a single phase of a pure substance canbe expressed as (Defay and Prigogine [21])

dg = −s dT + v dP + µ dx − dLσ , (3)

wheres is the molar entropy,v the specific molar volume,x the molar fraction anddLσ the amount of work made by the surface tension to extend the phase separationsurface. The phase interface is assumed to have a constant area (i.e., dLσ = 0) andfor a pure substance the molar fractionx is always a constant, then the variation ofthe molar Gibbs function is

dg = −s dT + v dP. (4)

Since dµl = dµv (and thermodynamic equilibrium requires thermal equilibrium,i.e., equal temperature in the two phases dTl = dTv = dT ), we can write

−sl dT + νl dPl = −sv dT + vv dPv. (5)

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2.1. state equations

Since the range of pressures and temperatures of the analyzed process is close to theambient ones, the state equation for ideal gas and Dalton’s law can be used for dryair, water vapor and their mixture (moist air)

pa = ρaT R/Ma, Pg = pa + pv,

pv = ρvT R/Mw, ρg = ρa + ρv.(6)

wherePg is the total pressure of the mixture,pv is the partial pressure of the vaporandpa the partial pressure of the other components.

2.2. derivation of the clapeyron–clausius equation

As shown in standard thermodynamic and physical chemistry textbooks (see forexample Zemansky [22], Glasstone [23] or Denbigh [24]), the Clapeyron–Clausiusequation can be obtained from (1) and (5) remembering that at equilibrium on a flatinterface the pressures are equalPv = Pl = P and the system has only one degreeof freedom (Gibbs rule applied to a phase change), hence

dP

dT= sv − sl

vv − vl. (7)

Since the enthalpy of vaporization1hvap is equal toT (sv − sl), (7) can be rewrittenas

dP

dT= 1hvap

T (vv − vl). (8)

This is the well-knownClapeyron–Clausius Equationthat can be applied to a phasechange of a pure vapor at constant temperature and pressure. Some judgment shouldbe exercised when integrating the Clapeyron’s Equation (8) over extended tempera-ture intervals, because in such case the fact that the enthalpy of vaporization1hvap isa function of the temperature, i.e.,1hvap = 1hvap(T ), and that the specific volumesvv andvl also change with temperature, should be taken in account. More accurateresults can often be obtained using empirical correlations such as Hyland and Wexler[25].

When the specific volumevl of the liquid water is negligible with respect tothat of the vaporvv, i.e., the system is far away from the critical point (at ambienttemperatures, for example, they differ more than three orders of magnitude) andthe specific volume of vaporvv can be expressed through the ideal gas law (6), theClapeyron’s Equation (8) can be expressed as

dP

dT≈ 1hvap

T vv= P1hvap

RT 2 (9)

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or

d lnP

dT= 1hvap

RT 2 . (10)

2.3. derivation of the kelvin equation

The Kelvin’s formula can be obtained following Defay and Prigogine [21], andremembering that the capillary pressurePc is defined as

Pc = Pv − Pl (11)

and hence

dPc = dPv − dPl. (12)

Both the liquid and the vapor phase are governed by a Gibbs–Duhem equation

sl dT − vl dPl + dµl = 0,

sv dT − vv dPv + dµv = 0.(13)

At constant temperature(dT = 0) Equation (13) together with (1) becomes

vl dPl = vv dPv (14)

and substituting (12) in (13)

dPc = vl − vv

vldPv, (15)

assuming that the vapor behaves as an ideal gas, i.e.,Pvvv = RT Equation (15)becomes

dPc = dPv − RT

vl

dPvPv

. (16)

Integrating between the initial limits:Pc = 0 andPv = Pv0 (i.e., flat surface) andthe final valuesPc, Pv and assuming constant and molar volume for the liquidvl(i.e., incompressible liquid), Equation (16) becomes

Pc = (Pv − Pv0)− RT

vlln(Pv

Pv0

). (17)

When the molar volume of the liquidvl is negligible with respect to that of the vaporvv (i.e., the vapor pressurePv is not too high) the first term on the right-hand side of(17) can be omitted obtaining

Pc = −RTvl

ln(Pv

Pv0

), (18)

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that is usually known as theKelvin’s Equation.From Kelvin Equation it follows thatthe equilibrium vapor pressurePv over a concave meniscus must be less than thesaturation pressurePv0 over a flat surface(Pc = 0) and then capillary condensationof a vapor to a liquid can happen. Similar results can be found on Krischer [26].

2.4. equilibrium between water and moist air

Clapeyron’s Equation (6) and the Kelvin Equation (18) are strictly valid only fora pure vapor. For the case of a liquid in equilibrium with a mixture of ideal gaseshaving total pressurePg, to which is applicable Dalton’s LawPg = pv + pa (6),instead of Equation (17) we obtain (see Defay and Prigogine [21])

Pc = (Pg − Pg0)− RT

vlln(Pv

Pv0

). (19)

For equilibrium between water and moist air, the total gas pressure can be usuallyconsidered constant(Pg = Pg0 = atmospheric pressure) and then(Pg − Pg0) = 0.Moreover in ordinary situations (Pg ≈ atmospheric pressure, i.e., about 1 bar) sincethe molar volume of the liquid phasevl is much smaller than the specific volume ofits vaporvv, the influence of the total gas pressurePg on the vapor tension of theliquid pv0 can be neglected [23, 24].

In such a case, Equations (6) and (18) can be used, but nowPv has the meaningof the partial vaporpv pressure in the gas mixture andPv0 of the vapor tension ofthe liquidpv0 (pv = pv0 for a flat surface, whenPc = 0).

Thus Equation (19) becomes

Pc = −RTvl

ln(Pv

Pv0

). (20)

2.5. energetic meaning of capillary pressure and waterpotential

Both the Clapeyron Equation (10) and the Kelvin Equation (18) can be consideredspecial cases of a general relationship analog to the van’t Hoff Equation (see Glas-stone [23] and Denbigh [24]), of the kind

d lnP

dT= 1h

RT 2 , (21)

where1h is an energetic parameter indicating the specific enthalpy variation betweenthe two states in equilibrium (for example, vapor and liquid). Once integrated, Equa-tion (21) gives

lnP = −1hRT

+ const. (22)

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The Kelvin Equation (18) can be rewritten as

ln(Pv

Pv0

)= −Pcvl

RT, (23)

that, derived with respect to the temperatureT at constant capillary pressurePc, gives[∂

∂T

(lnPv

Pv0

)]Pc

= ∂ lnPv∂T

− ∂ lnPv0

∂T= Pcvl

RT 2 , (24)

recalling Clapeyron’s Equation (10), (24) can be written as(∂ lnPv∂T

)Pc

= 1hvap

RT 2 + Pcvl

RT 2 (25)

and it is immediately evident thatPcvl = PcMw/ρl = 1h, when the integrationis performed starting from a flat surface (i.e., the enthalpy difference1h does notinclude the latent heat of vaporization). The same reasoning can be applied to theequilibrium with a gas mixture, i.e., using Equation (20) instead of (18) and thensubstituting in Equation (25), the partial vapor pressurepv in place of the vaporpressurePv. A clear energetic meaning can then be attached to the capillary pressurePc (and to the so calledwater potential9 = −1h/Mw equal to the capillary pres-sure multiplied by a constant, i.e.,Pc = −ρl9). This redefinition allows to use thecapillary pressurePc as a state variable whatever mechanism causes the additionalenthalpy difference between the water vapor in the gas phase and the condensedand/or adsorbed water phase.

3. Conservation Equations

As mentioned before, the detailed procedure followed to obtain the conservationequations can be found, for example, in Whitaker [19, 20]. Here only the generalguidelines are presented.

3.1. the momentum conservation equation

Themomentum conservationequations for the fluids (gas and liquid) flowing in thepores are usually the well-known Navier–Stokes Equations. When the motion of thegas and liquid-phase is ‘slow’ and can be considered quasi-steady (creeping-flow), thetime dependent and convective terms can be neglected. In such case, using averagingtechniques [5–7], Darcy’s Law can be obtained as shown, for example, by Whitaker[27–28]. Such derivation, anyhow, is not straightforward and cannot be reportedhere. Since Darcy’s Law implies a linear relationship between velocity and pressuregradient, this law can be directly incorporated in the mass conservation equations thus

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eliminating explicit appearance of velocities. Thus the mass conservation equationswill also include the momentum conservation.

3.2. the mass conservation equation

Themass conservationequations can be obtained following the approach proposedby Whitaker [19–20] and assuming no chemical reactions (neglecting, for example,the hydration phenomena). As shown by Baggioet al. [29], they can be expressed,using the multiphase Darcy Equation to eliminate the velocity variable, as follows

Dry air conservation

φ∂

∂t[ρa(1 − S)] − ∇ ·

(ρaKKrg

µg∇(Pg)

)+

+∇ ·(ρgMaMw

M2 Deff∇(pv

Pg

))= 0, (26)

Water(liquid-vapor) species conservation

φ∂

∂t[ρv(1 − S)] − ∇ ·

(ρvKKrg

µg∇(Pg)

)−

−∇ ·(ρgMaMw

M2 Deff∇(pv

Pg

))

= −φρl ∂S∂t

+ ∇ ·(ρlKKrl

µl(∇(Pg)− ∇(Pc)− ρl g)

). (27)

The total mass flux is the result of the net flow of the gas phase and of the liquidphase, both governed by Darcy’s Law. Moreover there is a diffusive flux of watervapor in the gaseous phase, governed by Fick’s Law (and, obviously, a counterflowof dry air). The total moisture flux is then the sum of the liquid flux and of the vaporflux (which, in turn, is partly due to air flow and partly to diffusion).

The liquid water flow is generally not negligible. In soil science (see, for exam-ple, Bear [8], Bear and Bachmat [11]) it is usually assumed that liquid water flowcan take place (i.e., the relative Darcy’s permeability of the liquid phaseKrl > 0)when the water content is greater than a threshold value calledirreducible saturation.For such threshold a value of the saturationS (S = liquid Vol./pore Vol.) equal to0.09 has been suggested by Bear [8]. A rigorous assessment of such threshold value(as well as of another phenomena not mentioned here, i.e., the hysteresis due theirreversibilities in the imbibition/sorption – drying/desorption process affecting therelationship between the capillary pressure or the vapor pressure and the water con-tent) requires yet another different approach such as thepercolation theorytogether

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with the knowledge of the pore structure of the material [16, 30–33] and cannot befurther pursued here.

Even the classical approach (Philip and De Vries [34]), recently used by Daian[35, 36] in a study about concrete, considers the moisture flux as the sum of a liquidflux and a vapor flux but neglects the gas phase convective flow (i.e., only vapordiffusion is considered), which is probably acceptable in most cases when the totalgas pressurePg gradient is small.

The importance of liquid flux in nonsaturated concrete seems indirectly confirmedalso by Bazant and Najjar [37] who, in order to fit a purely diffusive model of watermigration to experimental data, found that the diffusion coefficient was stronglynonlinear and increased about 10 to 20 times when passing from 0.6 to 0.9 relativehumidity R.H.(R.H. = pv/pv0) inside pores. This likely corresponds to the onset ofliquid water transfer and should be noted that for concrete such relative humidityusually corresponds to a saturation much lower than 0.09 (i.e., for concrete thethreshold is even lower than for soils).

3.3. the energy conservation equation

Also the energy conservationequation may be obtained following Whitaker[19, 20]. Expressing the first law of thermodynamics as an enthalpy balance (see alsoBird et al. [38] and Baggioet al. [29]) and with viscous dissipation and reversiblework neglected the following equation results

Energy conservation(enthalpy balance)

ρCp∂T

∂t−[Cplρl

KKrl

µl(∇(Pg)− ∇(Pc)− ρlg)+,

+Cpgρg KKrgµg

∇(Pg)]

· ∇(T )− ∇ · (λeff∇(T ))

= 1htot

Mw

[φρl

∂S

∂t− ∇ ·

(ρlKKrl

µl(∇(Pg)− ∇(Pc)− ρlg)

)]. (28)

Where 1htot is the total enthalpy difference between the vapor and thecondensed/adsorbed water. This equation takes in account the heat flow due to con-duction, convection of the gas and liquid phases and latent heat exchanges due tophase change (evaporation and/or condensation of water).

The heat-transfer is in general not negligible when there are phase changes.Whenphase change takes place, thermal energy is released (condensation) or taken (evap-oration), so a purely hydrodynamic model of drying, assuming that either isothermalor isoentropic conditions are fulfilled, is probably incorrect. This has been clearlyrecognized by Philip and De Vries [34] and De Vries [39]. In the drying case, forexample, thermal energy must be supplied to the water to evaporate. Consideringa drying phenomenonisothermalmeans to assume a very high thermal capacity

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and/or a heat transfer process taking place with a very small temperature gradient.The latent heat of vaporization of water1hvap/Mw for temperatures around 300 K(27◦C) is equal to about 2500 kJ kg−1 and the specific heat in the same temperaturerange is about 4.18 kJ kg−1 K−1. This means that, if the latent heat of vaporizationis supplied by the water itself, to evaporate a mass of water (supposing the previousdata constant) its temperature should decrease of an amount of 2500/4.18 = 650 K(below absolute zero)! Even admitting a contribution from the solid matrix, supposea porous mass made 10% of water and 90% of concrete, whose specific heat in thesame temperature range is about 1.59 kJ kg−1 K−1, the temperature decrease wouldbe equal to

0.1 × 2500

0.9 × 1.59+ 0.1 × 4.18= 135 K.

It is then self-evident that the heat source, necessary for the evaporation of the water,must be at the boundary of the media (possibly the surrounding air) and that heattransfer takes place from the boundary to the evaporating water. In such conditions‘isothermal phase change’ can only mean that the temperature gradient is very smalland this, in turn, means that the heat transfer is very small and then the evaporationrate is very small. In other words,isothermal phase changein a porous media canonly be approximated by a process in which the phase change rate is strongly limitedby the mass transfer (more or less diffusive) and not controlled by the heat transfer.This is probably what happens in the final stages of concrete drying, but does notseem a good approach to model the initial period. And this approach should not beused when modeling a situation where a temperature gradient is imposed, such asthe wall of a building.

4. Constitutive Equations

The solution of the conservation equation requires constitutive equations describingthe specific behavior of the considered material. Because of the composite nature ofthe porous material the well-known Fick’s Law of diffusion, Fourier’s Law for theheat flux, and Newton’s Law of viscosity cannot be used with constant values of thetransport coefficients. Relations are then needed expressing the effective diffusivityDeff , the effective thermal conductivityλeff and the absolute and relative permeabil-itiesK Krg Krl , as a function of the materials and of chosen system coordinates(for examplePg, Pc andT ). The issues underlying the evaluation of the transportcoefficients in heterogeneous media are quite complex (Nozadet al. [12] and can-not be further pursued here. The approach often used is to resort to empirical orsemi-empirical constitutive equations [8, 11, 20, 21, 40–43] and the interested readeris referred to the cited literature.

An additional constitutive equation is required to relate the saturationS with thesystem coordinates. In the case of partially saturated porous media (assuming thatthe liquid is pure water and the gas is a mixture of standard dry air and water vapor)

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it is a well known experimental fact that the water volume saturationS (or the masswater content), for a given material, is a function of temperatureT and of the partialwater vapor pressurepv in the moist air or of the capillary pressurePc. It has beenpreviously shown that, for a given temperatureT , the vapor pressurePv (or thepartial vapor pressurepv) and the capillary pressurePc are connected through anequilibrium relationship, i.e., the Kelvin’s Equation (18) and (20). In other words,remembering Equation (22), the saturationS is a function of the enthalpy variation1h between the vapor phase and the condensed/adsorbed one. Even if this kindof relationship is sometimes calledsorption equilibrium curveit must be stressedthat the relationS = S(1h) is a constitutive/phenomenological one and not anequilibrium one as demonstrated by the fact that in reality the saturation dependsalso by the previous history of the system (i.e., hysteresis effects are present leadingto a difference between the sorption/imbibition and desorption/drying curve).

The study of building materials is somewhat halfway between the unsaturatedflow problems analyzed by soil scientists and hydrologists (e.g., Bear [8]) and theadsorption phenomena studied by chemical engineers and applied chemistry scien-tists (e.g., Gregg and Sing [44]). In soil science the relationship between saturationand capillary pressureS = S(Pc, T ) is often known aswater retention curve. Theempirical equations proposed are often inaccurate when the saturation is close to theresidual water content(or irreducible saturationorhygroscopic water content) since,in this field, not much care is paid to the behavior at low water contents (Fuenteset al. [45]). Moreover such empirical equations are unsuitable for use in numericalmodeling because they go to infinity when the residual water content is reached.Nevertheless, some authors, e.g., [19, 20], [40] and [42], have used theJ -Leverettfunction approach (Scheidegger [46] and Bear [8]) to model the capillary pressure.

In applied chemistry the main concern is about pore specific area determinationstarting from the relationship between the gas pressure and the adsorbed amount. Theequations developed in this context (e.g., BET and Halsey [35]) result in a relationbetween water content and vapor pressure (or relative humidity) and temperatureS = S(pv/pv0, T ). The application of such equations to the study of moisture inbuilding materials is not immediate even if the theoretical background is simple andwell understood [44].

The saturation of liquid waterS is then an experimentally determined function ofthe pore radiusr∗ (through porosimetry) or of the relative humidity R.H. (throughsorption isotherms). Assuming zero contact angle between liquid phase and solidmatrix, as usual for water, the capillary pressurePc can be related to pore radius withtheLaplace Equation

Pc = 2σ

r∗, (29)

whereσ is the surface tension, weakly dependent on the temperature when far awayfrom the critical point.

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Figure 1. Pore size distributions for some building materials.

Usually the pore size distribution of a porous medium is well estimated by aRayleigh or a log-normal distribution [31], and often such distribution is bi-modal oreven pluri-modal. A possible approach is to approximate the pore size distributionfor each material as a succession of one or more log-normal distributions fitting theexperimental or literature data. SomeS = S(r∗) curves obtained with such proce-dure, are shown in Figure 1, and can be used for numerical computations [29, 47–48].Through Equation (19) and (28) the relationship betweenS, T , Pc andpv can beeasily obtained. The same approach can be used when the available data are sorptionisotherms.

5. Is a Purely Diffusive Drying Model Acceptable?

As a general rule, no. It has been pointed out by Philip and De Vries [34] in 1957and by Whitaker [19] in 1977 that a diffusive theory of drying is unsatisfactory,especially when appreciable temperature gradients are present. To obtain diffusivemodels that account for the actual behavior of porous media, diffusion coefficientsthat are complicated nonlinear functions of various parameters must be used, asshown, for example, by Philip and De Vries [34], De Vries [39], Bazant and Najjar[37] and Daian [35–36]. This is particularly true for concrete, as shown by Bazantand Najjar [37].

At this point it should be clear that this happens because the moisture transportphenomena are the resulting effect of different, coupled mechanisms. From a practicalpoint of view, a diffusive approach tries to model moisture migration as if it werethe result of a single, linear transport mechanism. This is equivalent to neglect liquidtransfer and heat transfer. As noted previously, this is acceptable only when the watercontent is below the irreducible saturation threshold, the drying rate is very slow andthere are not imposed temperature differences.So, in practice, a model accounting

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Figure 2. Water saturation curves at various times (hours) for a brick sample, resulting froma drying simulation based on the full model.

Figure 3. Water saturation curves at various times (hours) for a brick sample, resulting froma drying simulation based on the isothermal model.

for vapor diffusion alone is acceptable only for phenomena taking place in a porousmedia having a very low water content(i.e., final stages of drying).

5.1. numerical example

As an example of the difference between the fully coupled model and the isothermal(purely diffusive) one, some numerical simulations have been carried out about theone-dimensional drying process of a brick sample 10 cm thick. For the details of thenumerical implementation see [29] and [47–48].

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Figure 4. Average saturation of the brick sample versus time as obtained from a dryingsimulation with the full model and with the isothermal model.

Figure 5. Average temperature of the brick sample versus time as obtained from a dryingsimulation with the full model.

The brick sample (S. Marco brick) was assumed to have a total porosityϕ = 46%,an absolute permeabilityK equal to 2.0 × 10−15 m2, a dry thermal conductivityλequal to 0.495 W m−1K−1 and the pore size distribution shown on Figure 1. Thediffusion of water vapor was assumed to take place in the part of the pores notfilled with liquid and a tortuosity factorτ = 0.5 was considered to take in accountthe microstructure of brick. The effective diffusivity was then set equal toDeff =τ(1 − S)D0 whereD0 is the diffusivity of water vapor in air.

The sample has been discretized with a mesh of 40 nine noded lagrangian ele-ments and the evolution of the transient drying process has been simulated for 672 h(28 days) using a variable time step (ranging from 5 s to 1800 s depending upon the

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stage of the drying process). The sample was assumed to have an initial water satu-rationS = 0.92. The upper surface was considered as exposed to surrounding airhaving a relative humidity equal to 50% and the lower surface as impermeable andadiabatic. The simulation based on the full model has been performed assuming aninitial temperature of the sample equal toT = 293.15 K (20◦C) and a temperatureof the surrounding air equal toT a = 303.15 K (30◦C). The corresponding boundaryconditions for the upper surface were a fixed gas pressurePg = 101325 Pa (atmo-spheric pressure), a convective boundary condition for the surface temperature witha surface heat exchange coefficient equal to 10 W m−2 K−1 and a similar boundarycondition for the mass flux (water vapor) with the surface mass exchange coefficientfor the water vapor set equal to 0.00547 m s−1.

The water saturation distribution inside the sample resulting from this simulationat various times is shown in Figure 2. Another simulation has been done based onthe isothermal (diffusive only) model, i.e., on Equation (27) without the energy con-servation constraint (28), using the same boundary conditions as above for the massflux and evaluating the properties of water and air at a (fixed) temperature equal to298.15 K(25◦C). In Figure 3 the results of the isothermal simulation are presented.It is immediately evident that the simulated drying process in such case evolves ata faster pace. This can be seen also in Figure 4, where the average water saturationinside the sample during the process resulting from the full model simulation is com-pared with the one resulting from the isothermal model. Obviously, the differencestend to disappear in the final stages.

In Figure 5, the average temperature of the brick sample resulting from the fullmodel simulation is shown: the heat flux necessary for the evaporation of the water isobtained from the surrounding air and for this reason the temperature of the samplemust be lower than that of the air. In the initial phase of the process the brick sampletemperature stays close to the wet bulb temperature of the surrounding air and onlyof the drying the temperature difference between the sample and the air becomessmaller (because in the final stage also the heat flux needed for evaporation becomessmaller, i.e., the drying rate tends to be controlled by the vapor diffusion process).

6. Conclusions

• Although the starting point is the same, Clapeyron’s Equation and Kelvin’sEquation are obtained letting the equilibrium evolve through different pathsand are not the same thing. Clapeyron’s Equation accounts for the relation-ship between temperature variation and pressure variation for the equilibriumevolution of two phases separated by a flat surface. Kelvin Equation accountsfor the relationship between vapor tension of a liquid and vapor pressure of itsvapor when the interface between liquid and gas phase is a meniscus (and canbe extended to other bounding mechanisms). When needed, they can be usedtogether (Whitaker [10]), to evaluate the modification of equilibrium that takesplace inside the porous media when, in addition to the enthalpy of vaporization,

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other enthalpy variations must be taken in account because of the bound stateof the water (capillarity/adsorption) as in Equation (25).

• Both thermodynamic relationships and conservation equations are needed. Itmust be stressed that the Clapeyron’s Equation and the Kelvin’s Equation arenot conservation equations (no rate of change can be derived from them), butonly relationships between thermodynamic coordinates at (local) equilibrium.On the contrary, the energy conservation equation grants the conservation ofenergy (first law of thermodynamics) throughout a nonequilibrium process (let-ting to analyze the time evolution of the system). They can (peacefully) coexistbecause it is supposed that the conservation equations are valid on a macroscopicscale, but the gradients of the thermodynamic coordinates (appearing in conser-vation equations) are small enough that, locally, the system can be consideredin equilibrium.

• Neglecting liquid transfer and heat transfer can be a rather crude approximationwhen modeling moisture transfer and phase change in a porous media. Thisis acceptable when both the water content and the evaporation rate are verysmall (final stages of so-called ‘isothermal drying’), but using a purely diffusiveapproach outside this range requires to reintroduce the neglected aspects throughthe use of complex and nonlinear diffusion coefficients. Moreover, neglectingthe energy conservation equation can lead to formulations that violate (locallyor generally) the first law of thermodynamics.

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some considerations on modeling heat and mass transfer in porous media

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