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### 2.3. Transport of water vapor

The treatment of the gas transport phenomenon is more complicated because two phases have to be considered: air and water vapor. However, the problem can be simplified by considering the following assumptions. As mentioned in the previous section, the development of Eq. (21) rests on the hypothesis that pressure is uniform over the gaseous phase. This implies that there is no bulk movement of air in the gaseous phase. Consequently, there will be no convective transport of water vapor within the material pore structure. Still, there can be movement of molecules in the gaseous phase as a result of their thermal agitation. The other assumption is that gravity does not have any significant effect on the behavior of the water vapor.

The continuity equation for water vapor component of a gaseous phase is the following [15]:

 (29)

The quantity ρV is the mass of water vapor per unit volume of gaseous phase, and vV is the velocity of water vapor. The water vapor will be in movement as a result of its thermal agitation. It is therefore a diffusive process. According to Daian [10], the water vapor flux is given as:

 ρVvV=−DgradρV (30)

where D is the self-diffusion coefficient of water vapor in the gaseous phase. By combining Eqs. (29) and (30), one gets:

 (31)

The bulk equation is calculated from the integration of Eq. (31) over the REV:

 (32)

This integral can be divided in two parts, which yields:

 (33)

According to the definition of the volumetric phase average (Eq. (7)), Eq. (33) can be written as:

 (34)

The average of the time derivative is given by:

 (35)

where SoGL is the surface of the liquid/gas interface, SoGS is the surface of the gas/solid interface, u is the velocity of the interface, nGL is a unit vector pointing outward the gaseous phase at the liquid/gas interface, and nGS is a unit vector pointing outward the gaseous phase at the gas/solid interface. Because it is assumed that the deformations of the solid matrix are negligible, the last integral on the right-hand side of Eq. (35) can be neglected, which leaves:

 (36)

The average of the divergence gives:

 (37)

The first integral on the right-hand side of Eq. (37) is neglected because there is no exchange of water vapor between the solid and the gaseous phases. Accordingly, Eq. (37) can be simplified as:

Furthermore, by assuming that the coefficient D is constant, Eq. (38) can be written as:

The average of the gradient is given by [12] and [13]:

 (40)

The quantity τ is referred to by Bachmat and Bear [12] as the tortuosity of the material. Conceptually, it is the ratio of the macroscopic system length to the shortest path length through the pore (liquid or gas) space. As such, it is a quantity that strictly equal to or less than one. The parameter is defined as =xxo, where x is a position vector within the REV and xo is the position vector of the center of the REV. The first integral on the right-hand side of Eq. (40) involves the solid/gas interface. Except for the very low water content conditions, there will be no direct contact between these two phases because water will be adsorbed on the surface of the solid. Accordingly, the integral can be neglected. It is assumed that the term (gradρV·nGL) in the second integral on the right-hand side of Eq. (40) varies very slightly over the surface SoGL. Under this assumption, it leaves an integral of a position vector times a scalar over a closed surface, which gives zero. Eq. (40) is thus simplified as:

 (41)

Replacing Eqs. (36), (39) and (41) into Eq. (34) gives:

 (42)

Substituting Eq. (30) in Eq. (42), one finds:

 (43)

Withaker [15] showed that the integral in Eq. (43) has the same value as the one in Eq. (18). It represents the rate of condensation per unit volume of the water vapor phase at the liquid/gas interface. Therefore, Eq. (43) can be written as:

 (44)

It is possible to express the gradient in Eq. (44) as a function of θL since ρV=f(θL) [10]. Applying the chain rule, it gives:

 (45)

The quantities preceding the gradient within the parenthesis can be lumped together to form a single vapor diffusion coefficient:

 DV=θGDτG (46)

Eq. (45) can be written succinctly:

 (47)

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