Theory for Moisture Transport in Porous Media
In a porous material, moisture may be present under vapor and liquid state, and moisture transport occurs through convection and diffusion, liquid water being fixed on the pores walls mainly in two ways:
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Adsorption: the vapor is fixed under molecular films on the pores walls. Equilibrium vapor pressure is reached in the pores under thermodynamic conditions that are significantly different than those necessary in a free medium. Thus saturation conditions are changed, and the relative humidity is significantly changed as well, becoming a relevant variable for the characterization of moisture content.
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Capillarity: when the porous matrix is wetted by liquid water, there is a pressure difference between the liquid and gas (moist air) phases, due to the curvature of the wetting interfaces in each pore. The pressure difference, called capillary pressure, becomes the driving potential for a further filling of the pores by the water. Note that in some applications, the capillary pressure is replaced by the suction.
The predominance of adsorption or capillarity depends both on the amount of liquid water present in the porous medium, and on the size and shape of the pores, and should be defined in function of the material and process under study. Two moisture regions can be distinguished:
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Hygroscopic region: adsorption is the predominant phenomenon, and the equilibrium moisture content is better characterized from the relative humidity, through the specification of a sorption curve. For relative humidity values below 0.95, the slope of the sorption curve is relatively small, and the moisture content increases in a nearly linear way with relative humidity. Building materials are most of the time in this moisture region. Above 0.95, a very small variation of relative humidity induces a large variation of the moisture content, and the capillary pressure is best suited to characterize the moisture content.
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Capillary water region: capillarity is the predominant phenomenon, and the equilibrium moisture content is better characterized from the capillary pressure, through the specification of a retention curve (the van Genuchten and Brooks & Corey retention models for example).
The relative humidity and capillary pressure are called state variables: they describe the thermodynamic state of water in the pore space under two phases in equilibrium. At equilibrium, they are related through Kelvin’s law, which is a relation of thermodynamical origin. As long as equilibrium is satisfied, the state variables are continuous between two media (at the interface between two porous media with different porosities, or between a porous and a free medium), and Kelvin’s law is valid on both sides of the interface. In opposition, the liquid saturation or the moisture content depend on the porous matrix.
Note that capillary transport of liquid water should be considered in both regions.
Considering a porous medium made of a solid porous matrix of density ρs (SI unit: kg/m3) which pores are filled with a fluid of density ρf (SI unit: kg/m3). Noting its porosity εp (dimensionless), the density of the whole medium is
As mentioned above, the fluid itself is made of liquid phase, and a gas phase. The liquid saturation sl (dimensionless), defines the proportion of the pores that is filled with liquid. The fluid density can thus be expressed in terms of liquid density ρl and gas density ρg:
The liquid phase is pure water, however, the gas phase is moist air, that is a mixture of dry air and water vapor. Thus, in turn, its density can be written in terms of dry air density ρa and water vapor density ρv with the help of the vapor mass fraction ωv (dimensionless):
With and .
The moisture transport equations in porous media ensures the conservation of water, that is the total moisture content:
The mass conservation equation for the liquid water and water vapor account for convection, diffusion, evaporation or condensation, that is an exchange of mass between both phases, and possible mass sources for the vapor Gv (SI unit: kg/(m3⋅s)) and liquid Gl (SI unit: kg/(m3⋅s)) phases:
Several notations are introduced:
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glc (SI unit: kg/(m2⋅s)) is the liquid water capillary flux, equal to εpsljl.
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gw (SI unit: kg/(m2⋅s)) is the moisture diffusive flux, equal to εp(1 − sl)jv.
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ul (SI unit: m/s) is the liquid water velocity, equal to , that should be interpreted as the Darcy velocity, that is, the volume flow rate per unit cross sectional area.
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uv (SI unit: m/s) is the moist air velocity, in the sense of the Darcy velocity, equal to .
The water mass conservation equations can be written:
(4-140)
(4-141)
Different strategies exist to solve for these equations. For example, one can assume that, in the porous medium, the liquid and vapor phases are in equilibrium. In that case since the amount of liquid and vapor are linked through the sorption isotherm relation, one can solve for a unique equation for the total water content. The theory for this hypothesis is detailed in the Equilibrium Moisture Transport section below. Otherwise, the Nonequilibrium Moisture Transport section describes the theory for modeling moisture transport in porous media with a vapor mass fraction and a liquid water saturation that do not verify the equilibrium conditions.