1. Reference problem#
The aim here is to model the drying of a bar with water exchange conditions at the end and which can be assimilated to a thermal problem under certain conditions. The modeling will be done in 3D, 2D (Plane Deformation) and axysimetry.
1.1. A and D geometry#
Coordinates of the points (\(m\)):
A |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
B |
0, 2 |
0 |
0 |
0 |
0 |
0, 01 |
0 |
|
E |
0 |
0.01 |
0.01 |
0.01 |
0.01 |
0.01 |
||
F |
0.2 |
0.01 |
0.01 |
0.01 |
H |
0 |
0.01 |
1.2. Geometry B, C, E, and F#
Coordinates of the points (\(m\)):
A |
0 |
0 |
0 |
C |
0, 2 |
0, 01 |
B |
0, 2 |
0 |
0 |
0 |
0 |
0, 01 |
1.3. Material properties#
We only give here the properties on which the solution depends, knowing that the command file contains other material data (elasticity modules, heats, porosity…) which ultimately play no role in the solution of the problem being treated.
Liquid water |
Density (\(\mathit{kg}\mathrm{.}{m}^{-3}\)) Dynamic viscosity of liquid water (\(\mathit{Pa.s}\)) |
\({10}^{3}\) \({\mathrm{\mu }}_{w}=8E-4\) |
Steam |
Molar mass (\({\mathit{kg.mol}}^{\mathrm{-}1}\)) Gas viscosity (\(\mathit{kg}\mathrm{.}{m}^{-1}\mathrm{.}{s}^{-1}\)) |
\(\mathrm{0,001}\) \(\mathrm{5,0}\times {10}^{-7}\) |
Gas |
Molar mass (\(\mathit{kg}\mathrm{.}{\mathit{mol}}^{-1}\)) Gas viscosity (\(\mathit{kg}\mathrm{.}{m}^{-1}\mathrm{.}{s}^{-1}\)) |
\(\mathrm{0,001}\) \(\mathrm{5,0}\times {10}^{-7}\) |
Dissolved air |
Henry’s constant (\(\mathit{Pa}\mathrm{.}{m}^{3.}{\mathit{mol}}^{-1}\)) |
\(\mathrm{1,30719}\times {10}^{5}\) |
Homogenized coefficients |
Biot coefficient |
\(1\) |
Fick diffusion coefficient of liquid media |
\(0\) |
|
Fick diffusion coefficient of the gaseous medium |
\(0\) |
|
Intrinsic Permeability \(K\) (\({m}^{2}\)) |
\(\mathrm{1,0}\times {10}^{-18}\) |
|
Relative water permeability \({\mathit{kr}}_{w}(S)\) |
|
|
Relative gas permeability \({\mathit{kr}}_{g}(S)\) |
|
|
Sorption isotherm |
\(S({p}_{c})=1-{p}_{c}\times 1E-7\) |
|
Porosity |
\(\mathrm{0,085}\) |
1.4. Initial conditions#
The vapour pressure is negligible. It is initially set to \({p}_{\mathit{vp}}=\mathrm{921,6}\) throughout the domain.
The initial gas pressure \({p}_{\mathit{gz}}(x\mathrm{,0})=10000\mathit{Pa}\) and the capillary pressure over the entire domain are set to \({p}_{c}(x\mathrm{,0})=10000\mathit{Pa}\).
This corresponds to a saturation of \(\mathrm{0,999}\).
The initial temperature is 20°C. The mechanical stresses break down into effective stresses, \(\overline{\overline{\mathrm{\sigma }}}\text{'}\) and hydraulic stresses \({\overline{\overline{\mathrm{\sigma }}}}_{p}\):
Initially, the hydraulic and effective constraints are zero.
The other generalized constraints [voir U2.04.05] are initialized to zero.
1.5. Boundary conditions#
We impose:
zero travel in all directions;
a temperature, \(T\) zero;
a water exchange condition on the right edge ([BC] in 2D or face [BCFG] in 3D.
For models a, b and c, the exchange will focus on pressures and water flow \(q_{w}^{ext}\) will then be written:
\(q_{w}^{ext}.n={\mathrm{\lambda }}^{H}\nabla ({p}_{c}\mathrm{.}n)={h}^{H}({p}_{\mathit{ext}}-{p}_{c})\) with \({\mathrm{\lambda }}^{H}=\frac{K\times {\mathit{kr}}_{w}}{{\mathrm{\mu }}_{w}}\) (in \(\mathit{Pa}\mathrm{.}s\)), \({h}^{H}=\mathrm{1,0}\times {10}^{-14}\) and \({p}_{\mathit{ext}}=\mathrm{1,5}\times {10}^{8}\mathit{Pa}\);
For the models of, c and e, the exchange will focus on densities (or relative humidities):
\(q_{w}^{ext}.n= \alpha.\rho_{vp}.\left( exp\left[ \frac{Pc.M_{H2O}}{\rho_{l}.R.T}\right] -{HR}^{ext} \right)\)
with
\(\rho_{vp} = \frac{Pv_{sat}.M_{H2O}}{\rho_{l}.R.T}\) with \(M_{H2O}\) the molar mass of water, \(\rho_{l}\) its density, R the constant of ideal gases and \({HR}^{ext}\), \(Pv_{sat}\) and \(\alpha\) respectively the external humidity, the saturation vapor pressure and the exchange coefficient entered by the users. Here \({HR}^{ext}=0.5\); \(Pv_{sat}=2341 Pa\) and \(\alpha=0.001 m/s\)
a condition of zero hydraulic flow on all other edges.