Theory for Moisture Transport in Air
In moist air, the transport of moisture is done by convection and binary diffusion of vapor in air.
Two formulations are available, depending on the magnitude of moist air density variations with moisture content.
In both formulations, the moisture is supposed to be constituted only of vapor when considering moist air. In other terms the liquid concentration is null, except on boundaries where condensation may take place.
Diluted Species Formulation
In the diluted species formulation, a transport equation is written in terms of vapor concentration. It accounts for convection, binary diffusion in air, and source terms. Writing it in a conservative form:
Since the molar mass is constant, it can be taken out of the derivatives. Then, to express the nonconservative form, the spatial derivative is distributed:
The continuity equation for moist air is:
With the diluted species formulation, it is supposed that the spatial and temporal variations of the vapor concentration are small enough not to induce significant variations of the moist air density, that is, the continuity equation reduces to:
In other terms, this formulation supposes that the flow of moist air is incompressible.
The incompressible hypothesis can be used to simplify the nonconservative form of the moisture transport equation. Additionally, in the diluted species formulation, the vapor diffusive flux is expressed as a function of the vapor concentration. The Moisture Transport in Air Interface solves for the following nonconservative equation, in which the moisture content variation is expressed through the transport of vapor concentration, cv, by convection and binary diffusion in air:
(4-137)
with the following material properties, fields, and source:
•
 Μv (SI unit: kg/mol) is the molar mass of water vapor.
•
  ϕw (dimensionless) is the relative humidity.
•
 csat (SI unit: mol/m3) is the vapor saturation concentration.
•
 D (SI unit: m2/s) is the vapor diffusion coefficient in air.
•
 u (SI unit: m/s) is the air velocity field.
•
 gw (SI unit: kg/(m2⋅s)) is the vapor diffusive flux.
•
 G (SI unit: kg/(m3⋅s)) is the moisture source (or sink).
Concentrated Species Formulation
In the concentrated species formulation, a transport equation is written in terms of vapor mass fraction. It accounts for convection, binary diffusion in air, and source terms. Writing it in a conservative form:
The following relation between vapor concentration cv and vapor mass fraction ωv always stands:
where ρg is the moist air density (SI unit: kg/m3). Therefore, we obtain the same equation in conservative form than the diluted species formulation:
The only difference is that the vapor diffusive flux gw is written as a function of the vapor mass fraction ωv instead of the vapor concentration cv.
When the spatial and temporal variations of the vapor concentration induce significant variations of the moist air density, that is, when the moist air flow cannot be supposed incompressible, the velocity divergence term cannot be neglected. The Moisture Transport in Air Interface solves for the following nonconservative equation, in which the moisture content variation is expressed through the transport of vapor concentration, cv, by convection, binary diffusion in air, and includes compressibility effects:
(4-138)
By neglecting the spatial and temporal variations of ρg in the equations above, the equation of the diluted species formulation, Equation 4-137, is obtained.