Models for Non-isothermal Steady-State Diffiision in Porous Buildirg Materials

Two modek for non-isothermal difusion of uater aapour in building m.aterials haae been deaeloped and their numcrical outputs compared wi,th a snndard isothermnl approximation.


I Introduction
The structure of most building materials of silicate origin consists of mutually interconnected microscopic pores. The diameters of the pores show wide dispersion in their values ranging from nanometers to millimeters. Since the mean free path of water vapour molecules at normal room conditions Iies below lnm, almost all pores are open for vapour diffusion. Howeveq this is not the only transport mechanism that enables moisture to enter a building material. At higher relative humidity (approx. above 50 %) intensive capillary condensation occurs on the surface of pores and the pores are subsequently filled with liquid water which can migrate into a material. Liquid transport in pores takes place both by surface diffusion and by capillary flow. Howeve4 the purpose of this paper is to deal with diffusion of the gas phase only, i.e., of water vapour, without the liquid transport. This simplification is commonly applied, e.g., within the Glaser condensation model It], widely used in thermal building technology.
A common practice in thermal building technology is to calculate the vapour diffusion flux and pressure profile p(x) inside building envelopes within an isothermal model, although the envelopes normally experience non-isothermal conditions throughout the year. The temperature of the envelope within the isothermal model is represented by a one-year average for a given climate region (e.g., l0 "C for Central Europe). Such an approximation has been incorporated into the national thermal standards ofvarious countries t2l, t3l and has been in service for many years. Although this procedure might provide satisfactory results for one-year assessment of envelopes, it is clear that that the results for a shorter period can show severe deviations from the experimental data.
2 Non-isothermal diffusion Let us suppose that the building envelope through which the diffusion flux goes is represented by a plain brick wall of thickness d = 44 cm. The wall separates a heated room with a usual environment (surface temperature and relative humidity: ?n,=293.15 K, er=60 7o RH) from an outdoor space (Tz=255.15 K, Q:= 84 % RH). The atmospheric pressure of dry air is usually considercd to be approximately the same on both sides (1"=98066.5 Paused in [a]). The atmosPheric pressure p consists of partial pressures of water vapor p* and dry air p^F =f*+f^, !*<F^, F"Fu' (l) Similar relations also hold for mass concentrahsns aswell c=cw+c^, cw<ca, c=ca Frck's first law for non-isothermal diffusion assumes the following form [12] 9* = )*(9* + 8u) -cD*^(T)Y1* , ,* =T = where g* and q^ are diffiLrsion fluxes of water vaPor and air, respectively, and D*u(T) is the temperature-dependent di{fusivity. The diffusion flow should fulfil the condition of continuous flow d\l s lt! =YO (5) which is ri.r.t ,!lo.,a law. Assuming unidirectional steady--state diffusion @yl0t=0) along the x-axis and a negligibly small dif;hrsion flow (g" -+0) of heary air molecules (Nr+Or+... ) as compared with lighter H2O molecules @* > q), Fick's nro equations can be rewritten as follows ,. = -,D*u(T) d/* , lL =, < l, Ywl-1,(l+r)dx' q* df rf* =o -4w =conSt.
Inserting the second boundary condition (8) into (11) we obtain the diffusion flux z =181 which, together with ( I I ), gives the vapor profile 1* ( x) Thking into account the difinim through an immobil;izcd air lnyer (DIAL model), i.e., q^ -+ 0 (r -+ 0), relations (12) Since 11.< I and y2*< l, further simplification can  (14) -(17) hold within the framework of rhe DIAL approximation, when the air layer embedded in a porous material is only slightly perturbed by the diffusion of water vapor molecules that possess smaller mass and much lower concentration than those of air. It seems to be natural that not only the pressure of dry air remains constant but also the concentrations of dry and wet air vary only slightly across the wall. This can be nicely illustrated when the concentration ratios are calculated for a particular case, e.9., for the internal and external conditions defined at the beginning ofthe previous Section ofner/router =16.23, ,lnntt/r:"ttt = 0870, ,innerr/router = 0gg3 From ( I 8) it can be seen that the variations of air concentrations ca, c across the wall are negligibly small and, thus, the profiles c^(x), c(x) may be considered as approximately horizontal, i.e., constant for common climatic conditions. This does not hold at all for water vapor whose concentration varies considerably and, thus, its profile shows clear func- If these relations are included into Fick's Eq.(6), we can obtain a more simplified transport equation Ar(Ta\ qa=-D*u(T):tr, l-)*=l' (zo) r x0, c !const.
An analogous transport equation holds for non-isothermal diffusion of a gas in a solid compact body, so present approximation might be termed as Diffusion through a 'Rigid' Air Layer (DRAL model). At first sight this approximation might seem rather unrealistic, but the final account of all physical factors and their approximate befuruior leads to such a conclusion. Eq. (20) is quite analogous to those presented in technical literature for cases when the total concentration is constant (see e.g. Eq. 16.2-3 in [7]). The assumPtion c = const.
does not necessarily mean the condition for the isothermal state. Nearly constant total concentration c can be expected not only with non-isothermal diffusion of a gas in a solid cotnpact body which does not contain any air pores but also in solid materials containing closed pores (cavities) filled with air.
The diffirsion flux in such materials goes either through the airless solid structures whose concentration (density) is almost unaffected by temperature and through voids with a constant air content (constant concentration), provided the walls of the voids are hardly penetrable for heary air molecules in conrast to lighter water molecules. Foam building materials such as foam polyethylene approach this type of material. It seems probable that the DRAL model might be more applicable to such materials. However, it is necessary to stress that if a strong non-isothermal state causes essential variations in the total concentration profile, i.e., c=f(x), the DRAL model will fail to determine a realistic diffusion flux. Briefly, the transport equation (20)    By means of relations (30) the non-isorhermal diffirsion flux g" expressed within the DRAI approximation can be easily calculated.

11
-12 At first sight it is obvious that thep*(x) and c,(x) profiles are not linear. Nevertheless, for usual temperature and partial pressure differences between outdoor and indoor spaces in the Central European climatic region the graphs ofp*(x) and c*(x) will closely follow linear behaviou4 as can be easily verified.

IM-TDR and IM-TIR models
Glaser's standard condensation model [l] is based on isothermal diffusion, i.e., the temperature of a wall is considered to be a constant Tand equal to the mean value of the surface temperatures t T1+T2 /t" = -: ., Fick's equations for diffusion in an isothermal structure can be obtained from (21) and (22)  As can be seen, the pressure profile p(x) is a linear function of x in conrast to non-isothermal profiles (15)   year, all the models seem to be applicable within civil engineering practice.
Nevertheless, if strongly non-isothermal conditions are established (AT>40 K), it is necessary to distinguish carefully behveen computational models. While the IMjIIR scheme is not applicable at all under such conditions, the applicability of the remaining three models will depend on the material structure. For materials with macroscopic open pores filled with air (such as silicate building materials or mineral wool) the most convenient models seem to be DIAL and IM:TDR, the first of which should be given priority in practical calculations. The applicability of the DRAL model under strongly non-isothermal conditions is more problcmatic since it requires a constant total concentration profile to be established, which is fulfilled only with special materials.