Analytical verification: drying out of a layer

This benchmark test is taken from [Škerget2014]. It reproduces the second benchmark of HAMSTAD project.

1. Problem description

A homogeneous layer is analyzed under isothermal conditions in one dimension. The layer is initially in moisture balance with the ambient air, having constant relative humidity. At time zero there is a sudden change in the relative humidity of the surrounding air. The structure is perfectly airtight. The benchmark studies the drying process of the layer. Results should be displayed for the cross-sectional distribution of moisture content, \(w(x) \, (kg/m^3)\) at \(100\), \(300\) and \(1000\) hours.

1.1. Geometry

We consider the same geometry as in [Škerget2014]: \(H=0.2 \mathrm{m}\) thick and \(L=0.8 \mathrm{m}\).

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

1.2. Boundary conditions

The surounding temperature is constant: \(T_e = T_i = 20^{\circ}C\).

The top (exterior) and bottom (interior) surfaces of the structure are exposed to \(\varphi_e = 45\%\) and \(\varphi_i = 65\%\), respectively.

Homogeneous Neunmann conditions stand on the lesft and right sides of the structure.

1.3. Initial conditions

The initial hygrothermal conditions of the structure are: \(T_0 = 20^{\circ} C\) and \( \varphi = 95 \%\).

1.4. Material properties

  • Sorption isotherm:

\[w = \frac{116}{\left[1- \frac{1}{0.118} \ln(\varphi)\right]^{0.869}} kg/m^3\]
\[\varphi = \exp \left(0.118 \cdot \left(1 - \left( \frac{116}{w} \right)^{\left(\frac{1}{0.869}\right)} \right) \right)\]
  • Vapor diffusion

\[\delta_p = 1.0 \times 10^{-15} s \quad \mathrm{(constant)}\]
  • Moisture diffusivity

\[D_w = 6 \times 10^{-10} m^2/s \quad \mathrm{(constant)}\]
  • Thermal conductivity

\[k = 0.15 W/(mK) \quad \mathrm{(constant)}\]
  • Specific hest capacity

\[c_s = 800 J/(kgK)\]
  • Density

\[\rho_s = 525 kg/m^3\]

2. Analytical solution

Under isothermal conditions, the 1D mass transfer equation reads:

\[\frac{\partial w}{\partial t} = D_w \frac{\partial ^2 w}{\partial y^2}\]

its analytical solution is given by:

\[w(y,t) = w_i + \frac{w_e - w_i}{H} y + \sum_{n=1}^{\infty} B_n \sin\left( \frac{n \pi y}{H} \right) \exp\left( - D_w \left( \frac{n \pi }{H} \right)^2 t \right)\]


\[B_n = 2 \left[ \frac{w_0 - w_i}{n \pi} \left( 1 - \cos(n \pi) \right) + \frac{w_e - w_i}{n \pi} \cos(n \pi) \right]\]


  • \(w_0 = w\left( \varphi_0 \right) = 84.8 ~ kg/m^3\), \(\varphi_0 = \varphi \left( y, t = 0 \right)\)

  • \(w_i = w\left( \varphi_i \right) = 19.5 ~ kg/m^3\), \(\varphi_i = \varphi \left( y = 0, t \right)\)

  • \(w_e = w\left( \varphi_e \right) = 30.5 ~ kg/m^3\), \(\varphi_e = \varphi \left( y = H, t \right)\)

  • \(D_w = 6 \cdot 10^{-10} \quad m^2/s\)

The following figure shows the moisture content profile after 100h, 300h and 1000h obtained with the analytial solution.

The following figure shows the moisture content variation at \(y= \frac{H}{2}\) obtained with the analytial solution.

3. Results from [Škerget2014]

3.1. Time and Space Discretization

Different meshes of \(M=1 \times 4, M=20 \times 10, M=20 \times 20\) and \(M=20 \times 40\) are considered.

The time dependent analysis is performed by running the simulation from the initial state with a time step value of \(\Delta t=1 \mathrm{h}=3600\mathrm{s}\).

The aim is to calculate the moisture distribution after \(t=100,300\) and \(1000 h\).

3.2. Model

At constant moisture diffusivity, mass transfer is controlled according to equation (8) under the following boundary conditions of the first:

\[\begin{array}{l} w=\overline{w\left(\varphi_{i}\right)} \text { on } y=0 \quad \text { and } \quad 0 \leq x \leq L \text { for } t>0 \\ w=\overline{w\left(\varphi_{e}\right)} \text { on } y=H \quad \text { and } \quad 0 \leq x \leq L \text { for } t>0 \end{array}\]


\[\begin{array}{l} \displaystyle \frac{\partial w}{\partial n}=0 \text{ on } x=0 \quad \text{ and } \quad 0 \leq y \leq H \text{ for } t>0 \\ \displaystyle \frac{\partial w}{\partial n}=0 \text{ on } x=L \quad \text{ and } \quad 0 \leq y \leq H \text{ for } t>0 \end{array}\]

3.3. Results

Fig. 2 shows the initial moisture content and moisture distribution across the structure at \(t=100,300\) and 1000 h for four different discretization models.

Figure 2. Moisture content \(W\) profiles at \(t=100,300, 1000 \mathrm{h}\) from [Škerget2014]

Fig. 3 shows the moisture content variation \(\left(w(t)\right)\) in the middle of the structure \(\left(y=\frac{H}{2}\right)\) for four different discretization models.

moisture time
Figure 3. The moisture content variation \(\left(w(t)\right)\) in the middle of the structure \(\left(y=\frac{H}{2}\right)\) from [Škerget2014]

3.4. Simulation

To launch the simulation, use this line of code, with those cases files :

mpirun -np 4 ./feelpp_hm_heat_moisture --config-file cases/drying-layer/drying-layer.cfg --mod-file "\$cfgdir/drying-layer.json" --hm.export={360000,1080000,3600000}

3.5. Resolution

The results at \(t=100\)h, \(t=300\)h, \(1000\)h obtained with Feel++ are given on this figure :

The following figure shows the moisture content variation at \(y= \frac{H}{2}\) obtained with the analytial solution.

We can see that for both measures, the results obtained with the application correspond to the analytical solution. It tells us that the model and the code of the application is quite correct !


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