1. Reference problem#
1.1. Geometry#
The enclosure is modelled on a height \(\text{Lint}=\text{Lext}=1m\)
Outer radius of the enclosure: \(\mathit{Rext}=\mathrm{23,5}m\)
Inner radius of the enclosure: \(\mathrm{Rint}=\mathrm{22,5}m\)
1.2. Material properties#
For thermal calculation:
Thermal diffusion coefficient: \(\lambda =80J/(\mathit{h.cm.}°C)=2.22W/(\mathit{m.}°C)\)
Volume heat: \(\rho {C}_{p}=2.4{10}^{6}J/{m}^{3}/°C\)
For drying calculation:
In the drying equation:
\(\frac{\mathrm{dC}}{\mathrm{dt}}-\text{div}\left[D(C,T)\mathrm{grad}C\right]=0\)
the diffusion coefficient \(D\) will be of the form recommended by Granger [bib1], [bib2]:
\(D(C,T)=A\mathrm{exp}(\mathrm{BC})\frac{T}{{T}_{0}}\mathrm{exp}\left[-\frac{{Q}_{S}}{R}(\frac{1}{T}-\frac{1}{{T}_{0}})\right]\)
\(A=3.8{10}^{-13}{m}^{2}/s\)
\(B=0.05\)
\({T}_{0}=0°C\)
\(\frac{{Q}_{S}}{R}=4700{K}^{-1}\)
1.3. Boundary conditions and loads#
For thermal calculation:
A heat exchange is imposed on the inner and outer walls of the enclosure wall (mesh groups \({l}_{\text{int}}\) and \({l}_{\text{ext}}\)).
For the first five years, the outside temperature is \(15°C\) on each wall:
\({T}_{\text{int}}={T}_{\text{ext}}=15°C\)
Starting in the fifth year, the internal outside temperature changes to \(35°C\):
\({T}_{\text{int}}=35°C\) and \({T}_{\text{ext}}=15°C\)
The inner wall is considered to be unventilated and its exchange coefficient is \({h}_{\text{int}}=4\frac{W}{\mathit{m².}°C}\).
The outer wall is considered to be ventilated and its exchange coefficient is \({h}_{\text{ext}}=6\frac{W}{\mathit{m².}°C}\). (page 136 in [bib1]).
In practice, since the drying time scales are much greater than those of thermal drying, it can be considered that the thermal balance is almost immediate. The calculation is carried out using linear thermal methods.
For drying calculation:
The boundary conditions are expressed in terms of the normal flow of moisture on the inner and outer walls of the enclosure (mesh groups \({l}_{\text{int}}\) and \({l}_{\text{ext}}\)). We use option FLUX_NLde the AFFE_CHAR_THER_F operator. In general, in a drying calculation, the normal flow is expressed as a function of the initial concentration \({C}_{0}\), the 50% humidity concentration \({C}_{50}\) and the external concentration \({C}_{\mathit{ext}}\), in the form:
\(w=-D\left(C,T\right)\frac{\partial C}{\partial n}=\frac{0.5\beta }{{\left({C}_{0}-{C}_{50}\right)}^{2}}\left[C-\left(2{C}_{0}-{C}_{\mathit{ext}}\right)\right]\left(C-{C}_{\mathit{ext}}\right)\)
with
\(\beta ={\mathrm{3.41557.10}}^{–6}\frac{l}{\mathit{m².s}}\) (page 181 in [bib1])
The data used in the case of the Flamanville enclosure are as follows:
\({C}_{0}=105.7l/{m}^{3}\) and \({C}_{50}=57.5l/{m}^{3}\) (page 194 in [bib1])
On the inside:
On the outside:
\({C}_{\mathit{ext}}=69.1l/{m}^{3}\)
The drying calculation is carried out in non-linear heat, by chaining with the linear thermal calculation.
1.4. Initial conditions#
The initial conditions consist of the initial temperature, which is taken to be \(15°C\), and the initial water concentration, which is \({C}_{0}=105.7l/{m}^{3}\)