Jerzy Wyrwał*, Andrzej Marynowicz
Technical University of Opole, Faculty of Civil Engineering, Katowicka 48, 45-061 Opole, Poland
Simultaneous one-dimensional heat and vapour transfer with condensation in a porous wall is analytically investigated. Spatially-steady state distribution of accumulated moisture, less than the critical content, is described. Closed-form analytical expressions for the temperature, condensation rate and moisture content are obtained. Presented model requires material properties which are relatively simple and easy to determine. The results of the paper are illustrated with an example of multilayer building wall under climatic conditions.
Key words: Heat; Vapour; Condensation; Porous material; Critical moisture content
Simultaneous heat and vapour transfer with condensation in porous materials is of practical importance in applications in the civil engineering. The transport of vapour across the building walls and its possible condensation increases the thermal conductivity of the building porous materials and may cause structural damage.
The theoretical and experimental study of the heat and moisture transfer in building walls has been the target a lot of important research work. We can refer to the works of Andersson , Budaiwi et al. , De Freitas et al. , Kiessl , Kohonen , Künzel and Kiessl , as well as Pedersen . However, very little work exists on the particular subject of vapour condensation in a porous materials.
Water-vapour condensation within a porous wall has been observed, particularly when the wall is exposed to large temperature differences and high humidity environments. This phenomenon was first rigorously studied by Ogniewicz and Tien  where the coupling between temperature and concentration of condensing vapour was taken into account. In reference by Motekef and El-Masri , one-dimensional transport of heat and mass with phase change in a porous slab was studied, and analytical solutions for the cases of immobile and mobile condensate were obtained. Recently Shapiro and Motakef  proposed an analytical solution of a large class of transient problems, and compared their results with experimental data.
The present paper uses the approach of Motakef and El-Masri  for the problem of one-dimensional flow of heat and diffusion of vapour in a porous wall. It is assumed that the moisture content in wet zone is lower than the critical one and, as a consequence of this, the liquid water is practically immobile. This leads to very simple relations describing the temperature and moisture profiles as well as the condensation rate in the wet zone. The major advantage of the proposed solution is its relative ease in determining the moisture accumulation due to vapour condensation for different materials in different conditions.
2. Formulation of the problem
Consider one-dimensional transfer of heat and vapour in a homogeneous porous wall (Fig. 1), the boundaries of which are exposed to two different environments: an indoor environment with temperature Ti and humidity φi, and an outdoor environment with temperature Te and humidity φe. Under winter conditions the indoor temperature is higher than the outdoor one, and vapour diffuses towards the colder boundary.
For a constant pressure system, the concentration of saturation vapour is a unique function of temperature. Therefore, the vapour saturation-concentration curve in the wall is defined by the temperature distribution. Depending on the values of the prescribed boundary conditions, the vapour concentration profile may touch the saturation concentration curve in the wall. The diffusing vapour would then undergo phase change and condense in some region of the wall. With the relative humidity at the boundaries less than 100%, condensation occurs over the wet zone, separated from the boundaries by two dry zones as illustrated in Fig. 1.
The condensation of vapour in the wet zone can be considered to be simultaneously a vapour sink, water source and heat source. Hence, three processes of vapour diffusion, vapour condensation and heat conduction are coupled through the condensation rate. The vapour concentration, moisture content and temperature profiles in the wet zone are obtained by the simultaneous solution of the three coupled conservative equations for heat, vapour and liquid water.
At moisture content less than the critical one, condensate is in a pendular state (Harmathy ) and does not exhibit any tendency to migration. Beyond the critical moisture content, as the pendular drops coalesce and the capillary pores are wetted, condensate is propelled by surface tension forces from region of its higher content to the drier regions.
In the absence of moisture migration in the wet zone, the system of differential equations for heat and vapour transfer may be written as
x co-ordinate [m],
λ coefficient of heat conduction [W/(m∙K)],
T temperature [K],
L latent heat of condensation [J/kg],
R condensation rate [kg/(m3ˇs)],
l width of the condensation zone [m],
ρa air density [kg/m3],
ε porosity of the material [m3/ m3],
D vapour diffusion coefficient in the air [m2/s],
Csat concentration of saturation vapour [kg/kg].
Equation (1) is subjected to the following boundary conditions
T0 temperature of interior surface of the wet zone [K],
Tl temperature of exterior surface of the wet zone [K].
The problem of calculation of the location of wet zone, and the temperatures of its surfaces has been solved by Motakef and El-Masri .
In considered process first occurs a relatively short initial transient stage in which the temperature and vapour concentration fields are developing within the porous slab. During this phase a very small quantity of liquid water is accumulated in the porous material (Fig. 2). The initial transient stage is of little significance due to its relatively short duration and, therefore, is not studied here. Condensation of vapour is defined here as the accumulation of liquid water beyond the phase described above.
Beyond the initial transient time, the temperature and vapour concentration profiles remain invariant with time, vapour condenses continuously in the wet zone, the condensate accumulates with time, and for moisture content less than critical one the transport of water in liquid phase within the capillaries can be ignored. Therefore, conservation equation for liquid water simplifies to (Ogniewicz and Tien )
ρw liquid density [kg/m3],
W moisture content [m3/m3],
t time [s],
Wcr critical moisture content [m3/m3],
with the initial condition
W0 initial moisture distribution in the wet zone [m3/m3].
The critical moisture content for selected building materials contains Table 1.
Motakef and El-Masri  defined the solution which satisfies the above conditions as the first spatially-steady regime. In such regime the temperature and vapour concentration profiles are at steady-state. There is no condensate motion. The moisture content in the wet zone increases linearly with time, and the location of the wet region is spatially fixed and determined by the continuity of heat and vapour fluxes at the wet-dry boundaries. When the local value of moisture content reaches its critical level, the capillary forces lead to the migration of condensate into the dry regions and the subsequent expansion of the wet zone.