# Moisture uptake within a semi-finite region

 This benchmark test is taken from Skerget et al. It reproduces the normative benchmark described in [CEN2007].

## 1. Problem description

Proposed in Annex A of the norm BS EN 15026:2007 [CEN2007], this normative benchmark test is designed for the validation of heat and moisture tranfer software.

The test deals with thick single homogeneous material in equilibrium with a constant surrounding climate. The material is perfectly airtight. At a certain time the temperature and the relative humidity undergoes a step change. The objective is to calculate the moisture and temperature distribution after 7 days, 30 days and 365 days.

Geometry

The domain diemsions are not specified in [CEN2007] but it should be taken large enough to be represented as a semi-finite region. As in [Škerget2014], we consider a [$12\text{m}\times2\text{m}$] wall. Here is the description of the benchmark’s test geometry:

Figure 1. A semi-infinite homogeneous structure: geometry, boundary and initial conditions.
Boundary Conditions

We have $\Omega = [0,12]\times [0,2]$. We write $\partial\Omega=\partial\Omega_D\cup\partial\Omega_N$ where :

• $\partial\Omega_D$ is the left border of the domain on which Dirichlet condition are applied,

• $\partial\Omega_N$ is composed of the three other borders, on which homogenous Neunamm conditions stand.

Initial condition
• Initial condition: $\phi = 50\%$, $T=20^{\circ}C$

• After the step change: $\phi = 95\%$, $T=30^{\circ}C$

General data
$T_\mathrm{ref} = 293.15 K$
$\rho_\mathrm{w} = 1000 kg/m^3$
$M_\mathrm{w} = 0.018 kg/mol$
$L_v = 2500 kJ/kg$
$R_\mathrm{H_2O} = 462 J/\left(kg \cdot K\right)$
Material data
• Water retention curve

$w=\frac{146}{\left(1+\left(8 \times 10^{-8} p_\text{suc}\right)^{1.6}\right)^{0.375}}$
$p_\text{suc}=0.125 \times 10^{8} \left( \left(\frac{146}{w}\right)^{\frac{1}{0.375}}-1 \right)^{0.625}$
• Sorption isotherm

$w=\frac{146}{\left(1+\left(-8 \times 10^{-8} \cdot R_{\mathrm{H}_{2} \mathrm{O}} T_\mathrm{ref} \rho_{w} \ln (\phi)\right)^{1.6}\right)^{0.375}}$
$\phi=\exp \left(-\frac{1}{R_{\mathrm{H}_{2} \mathrm{O}} T_\mathrm{ref} \rho_{\mathrm{w}}} 0.125 \times 10^{8} \left( \left(\frac{146}{w}\right)^{\frac{1}{0.375}}-1 \right) ^{0.625}\right)$
• Vapour diffusion

$\delta_{\mathrm{p}}=\frac{M_w} {R T_\mathrm{ref}} \frac{26.1 \times 10^{-6}}{200} \frac{1-\frac{w}{146}}{0.503\left(1-\frac{w}{146}\right)^{2}+0.497}$

with $R = 8.314~J/\mathrm{mol} \cdot K$ is the universal gas constant.

 We use $\delta_p$ as it is given in [SKER20014]. The formula given in [CEN2007] leads to erronous solution.
• Liquid water permeability

\begin{array}{l} K= \exp \left( -39.2619+0.0704 \cdot(w-73)-1.7420 \times 10^{-4} \cdot(w-73)^{2}-2.7953 \times 10^{-6}\right. \\ \left.\cdot(w-73)^{3}-1.1566 \times 10^{-7} \cdot(w-73)^{4}+2.5969 \times 10^{-9} \cdot(w-73)^{5}\right) \end{array}
• Porosity equal maximum point of moisture storage function.

• Thermal conductivity

$k_\text{eff}=1.5+\frac{15.8}{1000} w$
• Heat capacity for dry material

$\rho_{s} c_{s}=1.824 \times 10^{6}$

The value of $C_{p,\mathrm{w}}$ is get from this page.

## 2. Results from [SKER2014]

Time and space discretization
• Two non-uniform non-symmetric meshes of $M = 100\times 2$ and $M = 200 \times2$ macro-elements were used, each with two aspect ratios $R_x1 = 40$ and $R_x1 = 80$ between the largest and the smallest boundary element, thus four different discretization models were considered.

• Two time-integration schemes were considered, e.g. constant and linear approximation of functions over each time-step.

• The time-dependent analysis was performed by running the simulation from the initial state with a time step value of $\Delta t = 0.01$ day, $\Delta t = 0.1$ day and $\Delta t = 1.0$ day, applying constant approximation of functions over each time step, and $\Delta t = 1.0$ day for the linear approximation.

Results with constant time interpolation

The following figures show the grid-dependence study when the computational mesh is refined from $100\times2$ to $200\times2$ with aspect ratios $R_x=40$ and $R_x=80$ for the constant time model. The moisture and temperature distributions were plotted for time increment $\Delta t1=1.0$ day at time instants $7$, $30$ and $365$ days.

Time step dependence for the time constant model was studied on the finest grid $200\times2$ $R_x=80$ and by varying the time step from $0.01$ to $1.0$ day. The computational results are shown the following figures:

Results with linear time interpolation

The following figures show the grid-dependence study when the computational mesh is refined from $100 \times2$ to $200\times2$ with aspect ratios $R_x=40$ and $R_x=80$ for the linear time model. The water content and temperature distributions are plotted for time increment $\Delta t=1.0$ day at the time instants $7$, $30$ and $365$ days.

## 3. Variationnal problem

Let $X = M = \left\{v\in H^1(\Omega)\,\middle|\,v|_{\partial\Omega_D}=0\right\}$.

From what was done here, the problem become :

At time $n+1$, the variationnal problem is : Find $(T^{n+1},\phi^{n+1})\in X\times M$ such as

$\begin{array}{l} a(T^{n+1},v) + b(\phi^{n+1},v) = f(v) & \forall v\in X\\ c(T^{n+1},q) + d(\phi^{n+1},q) = g(q) & \forall q\in M \end{array}$

with :

• $a(T,v) = \displaystyle\int_\Omega\left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^{n+1}}{\Delta t} v + \int_\Omega \left(k^n_{\mathrm{eff}} \nabla T^{n+1} \right) \cdot \nabla v$

• $b(T,v) = \displaystyle \int_\Omega L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} p^n_{\mathrm{sat}} \left(\nabla\phi^{n+1}\right) \cdot \nabla v + \int_\Omega \left(L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \right) \cdot \nabla v - \int_{\partial\Omega_N} \left(L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \cdot n \right) v$

• $c(T,q) = 0$

• $d(T,v) = \displaystyle\int_\Omega \left(\xi^n \frac{\phi^{n+1}}{\Delta t}\right)q + \int_\Omega\left(\xi^n D^n_{\mathrm{w}} \nabla \phi^{n+1}\right)\cdot\nabla q + \int_\Omega \left(\delta^n_{\mathrm{p}} (\nabla\phi^{n+1}) p^n_{\mathrm{sat}}\right)\nabla q + \int_\Omega \left(\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\right)\nabla q - \int_{\partial\Omega_N} \left(\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\cdot n\right)q$

• $f(v) = \displaystyle\int_\Omega\left( \left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^n}{\Delta t} + Q^{n+1} \right) v$

• $g(q) = \displaystyle\int_\Omega \left(\xi^n\frac{\phi^n}{\Delta t}+G^{n+1}\right) q$

This variationnal problem gives a discretized problem : find $(T_h,\phi_h)\in X_h\times M_h$ such as

$\begin{bmatrix} A & B \\ 0 & D \end{bmatrix}\begin{bmatrix} T^{n+1} \\ \phi^{n+1} \end{bmatrix} = \begin{bmatrix} F \\ G \end{bmatrix}$

Where $A,B,C$ are the matrix associated to the bilinear forms and $F,G$ are the vectors associated to the linear form on the bases of the spaces $X_h$ and $M_h$.

## 4. Command

The configurations files (cfg, json and geo) can be found in the repository github src/cases/moisture-uptake.

The command to execute this simulation is :

`mpirun -np 8 ./feelpp_hm_heat_moisture --config-file cases/moisture-uptake/moisture-uptake.cfg --mod-file "\\$cfgdir/moisture-uptake.json" --hm.export={604800,2592000,31536000}`

## 5. Resolution

The results of the temperature and the water content at $t=7$ days, $t=30$ days and $t=365$ days are given of the following figures. The minimal and maximal values from [CEN2007] are also plotted.

The simulation above was made with a time-step of 1 day. With smaller time-steps, the results stay the same for both temperature and water content, as shown above :

## References

• [CEN2007] EN 15026, Hygrothermal performance of building components and building elements - Assessment of moisture transfer by numerical simulation, CEN, 2007.

• [HAM2002] C.-E. Hagentoft, HAMSTAD – Final report: methodology of HAM-modeling, Report R-02:8, Gothenburg, Department of Building Physics, Chalmers University of Technology, 2002.

• [Kumaran1994] Kumaran, M. K., Mitalas, G. P. and Bomberg, M. T. (1994), 'Fundamentals of transport andstorage of moisture in building materials and components', Trechsel, H. R. (ed), Moisture control in buildings.

• [KUN1995] Künzel H, Simultaneous Heat and Moisture Transport in Building Components, PhD thesis, Fraunhofer Institute of Building Physics, 1995.

• [Kunzel2004] Kunzel, H.M., Holm, A., Zirkelbach, D. and Karagiozis, A.N., Simulation of indoor temperature and humidity conditions including hygrothermal interactions with building envelope Solar Energy 78 (2005) 554–561.

• [Mendes2019] Mendes, N., Chhay, M., Berger, J. and Dutykh, D., Numerical methods for diffusion phenomena in building physics, Springer Nature Switzerland AG 2019.

• [Neymark2002] Neymark, J. and Judkoff, R. International energy agency building simulation test and diagnostic method for heating, ventilation, and air-conditioning equioement models (HVAC BESTEST). Volume 1: Cases E100-E200 Technical Report NREL/TP-550-30152.

• [Škerget2014] Škerget, L. and Tadeu, A. BEM numerical simulation of coupled heat and moisture flow through a porous solid Engineering Analysis with Boundary Elements, 2014.

• [Straube2002] Straube J. F., Moisture in Buildings, ASHRAE Jornal, January 2002.