Water vapor transport to material surfaces-Simplified analytical expressions for non-linear material properties

The water vapour transfer between the indoor air and material surfaces is of importance for the moisture balance of the room. It can also be important for the moisture content and durability of the material surface layer such as artefacts in churches and historical buildings. For most building materials the penetration depth due to short time fluctuations, such as diurnal ones, is very limited. For these cases the assumption of semi-infinite analysis gives accurate results even for a rather thin material layer. In the paper, the moisture profile and surface moisture uptake are modelled in detail for isothermal cases and strongly non-linear material properties for the sorption isotherm and vapor permeability. An approximative formula is given for a quite accurate estimate of the moisture up take for a demonstration case with a strongly non-linear material. 1 Problem formulation Hygroscopic organic materials, such as wood, are particularly susceptive to changes in the ambient climate. With an increase in RH, wood will adsorb moisture from the ambient air and swell. With a decrease in RH it will desorb moisture and shrink. In a fluctuating climate, constantly moving moisture gradients will develop from the surface and inwards. If the changes in RH are significant, or frequent enough, permanent deformation or damage may occur. This problem coupled to wooden artefacts in churches is analyzed in (Melin et al. 2018). In that article, thin samples of wood with uniform temperature that is changing in time is analyzed. In the report and (Heirstraeten, 2019) the deformation due to changing RH is analyzed further. This paper is a continuation of the work. The uptake and release of moisture from the surfaces facing the interior climate in rooms is also an area of interest. This moisture interaction between the room and the surfaces can, especially for fast changes, influence the peak values of the moisture content of the indoor air. It is shown (Hagentoft, 2001) that the penetration depth of the propagating moisture change in a material after step change at the surface is very small. This makes it appropriate * Corresponding author: carl-eric.hagentoft@chalmers.se , 0 (2019) https://doi.org/10.1051/matecconf /201928 MATEC Web of Conferences 282 CESBP 2019 0 02 2020 20 2 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). to study semi-infinite domains and still get the same moisture profile in the surface of the material as for materials with a thickness of say 5 times the penetration depth. In this article both the moisture profile in semi-infinite materials with strongly nonlinear properties and the moisture uptake at the surface is investigated for step-changes in the ambient climate during isothermal conditions. In the study the effect of surface resistances and hysteresis is neglected. 2 The mathematical problem Using the humidity by volume v (kg/m 3 ) the moisture flow, g (kg/m/s), in the material is given by: ( ) v g x       (1) Here, the vapour permeability is given by ( )   (m 2 /s), and it depends on the relative humidity  (-). The moisture balance equation reads: ( ) w g x t       (2) Here, w (kg/m 3 ), is the moisture content in the material. It depends on the relative humidity. The slope of the sorption isotherm is introduced: ( ) w       (3) The relative humidity is defined from:


Problem formulation
Hygroscopic organic materials, such as wood, are particularly susceptive to changes in the ambient climate. With an increase in RH, wood will adsorb moisture from the ambient air and swell. With a decrease in RH it will desorb moisture and shrink. In a fluctuating climate, constantly moving moisture gradients will develop from the surface and inwards. If the changes in RH are significant, or frequent enough, permanent deformation or damage may occur. This problem coupled to wooden artefacts in churches is analyzed in (Melin et al. 2018). In that article, thin samples of wood with uniform temperature that is changing in time is analyzed. In the report and (Heirstraeten, 2019) the deformation due to changing RH is analyzed further. This paper is a continuation of the work.
The uptake and release of moisture from the surfaces facing the interior climate in rooms is also an area of interest. This moisture interaction between the room and the surfaces can, especially for fast changes, influence the peak values of the moisture content of the indoor air.
It is shown (Hagentoft, 2001) that the penetration depth of the propagating moisture change in a material after step change at the surface is very small. This makes it appropriate to study semi-infinite domains and still get the same moisture profile in the surface of the material as for materials with a thickness of say 5 times the penetration depth.
In this article both the moisture profile in semi-infinite materials with strongly nonlinear properties and the moisture uptake at the surface is investigated for step-changes in the ambient climate during isothermal conditions. In the study the effect of surface resistances and hysteresis is neglected.

The mathematical problem
Using the humidity by volume v (kg/m 3 ) the moisture flow, g (kg/m 2 /s), in the material is given by: Here, the vapour permeability is given by ()  (m 2 /s), and it depends on the relative humidity  (-).
The moisture balance equation reads: Here, w (kg/m 3 ), is the moisture content in the material. It depends on the relative humidity. The slope of the sorption isotherm is introduced: The relative humidity is defined from: Here, T (˚C) will be the constant temperature, and v s (kg/m 3 ) is the humidity by volume at saturation. A function for the moisture diffusivity a (m 2 /s) is also introduced: Combining the moisture balance equation (2) with (1) and (3-5): In order to find a dimensionless solution, the following transformation is introduced: We get: Here, both f and g are dimensionless functions depending on the RH only. The boundary condition at x=0 and the initial conditions are transformed according to: By combining (1), with (4) and (7,9), the moisture flow m (kg/m 2 /s) in to the material at the surface reads: The accumulated moisture uptake is: For materials with f=0 and g=1, e.g. materials with constant vapor permeability and constant slope of the sorption curve, we have the following equation to be solved: The solution is well known: We have: When the ODE-solver is used it requires value for: The value 2 (0) y is not known directly. However, it is found by finding the solution that meets the requirement (10), i.e.   1 i y   , starting with an initial guess using (15). The Matlab code, using the function fzero, is given in the appendix below.

Example
A generic material with an assumed same breaking point for both the vapor permeability and the sorption isotherm at  is used in the example: For f and g we get: The following case will be used (approximately wood); 0  =100 kg/m 3 , is analyzed in detail. The boundary and initial RH are: In the calculation later, for the moisture flow, the temperature T=20 ˚C is assumed. Figure 1 shows the vapor permeability and the sorption isotherm:     Table 1 gives the calculated moisture flow into the material (11-12) and the slope of the relative humidity at the surface.
Further investigations will show how the formulas can be further simplified for estimating moisture fluxes more generally.

Conclusions
A methodology is described by which the moisture profile inside a semi-infinite domain of a strongly nonlinear material can be calculated. The necessary code, here described in Matlab, is short, efficient and accurate. The formulation is based on one non-dimensional parameter that contains both time and space. Formulas for the moisture flow rate and the accumulated moisture uptake is given. A formula is outlined that gives a quite accurate estimation of the interval of the moisture uptake.