# Study of the Hydrothermal Behavior of Wood Fiber Insulation Subjected to Non-Isothermal Loading
## 2 Materials and Methods
### Numerical Model  
The heat and moisture model developed was based on two conservation equations: energy and moisture (water in liquid and vapor form). The transfer potentials chosen are temperature \(T\) and vapor concentration \(\rho_{v}\). The model presented is based on the works of [PERSON] [23] and on works developed more recently available in [24; 25; 26; 27; 28; 29].  
The general equation describing the transfer of humidity in a porous medium is given by Equation (1), that results from the sum of conservation equations of liquid water and water vapor with diffusion and capillarity as transfer phenomena expressed by the laws of [PERSON] and [PERSON], respectively:  
\[\frac{\partial\rho^{m}}{\partial t}=-\vec{\
abla}\left(\int_{v}^{\text{-dif} }+\int_{I}^{\text{-cap}}\right) \tag{1}\]  
where \(\rho^{m}\) is the moisture content, \(\int_{v}^{\text{-dif}}\) is water vapor diffusion flux according to [PERSON]'s law, and \(\int_{I}^{\text{-cap}}\) is liquid flow according to [PERSON]'s law.  
For building applications, global air movements are neglected in the case of very low air velocities and low variations of temperature and pressure found in practice [28].  
The energy conservation equation (Equation (2)) was obtained from an enthalpy balance by explaining the enthalpies of different phases and elements composing the system. The hypothesis of local equilibrium was chosen with the consequence of equality of temperatures between the different phases and components:  
\[\rho_{s}c_{p}^{*}\frac{\partial T}{\partial t}+\left[\left(c_{p}^{\prime} \int_{I}^{\text{-cap}}+c_{p}^{\prime}\int_{v}^{\text{-dif}}\right)\vec{\
abla} \right]T=-\vec{\
abla}\cdot\int_{q}^{\text{-dif}}-L_{\text{vl}}\vec{\
abla} \cdot\int_{v}^{\text{-dif}} \tag{2}\]  
where \(T\) is the temperature, \(\rho_{s}\) is the dry density, \(\int_{q}^{\text{-dif}}\) is diffusion heat flux according to [PERSON]'s law, \(L_{\text{vl}}\) is the specific latent heat for water and \(c_{p}\) is the specific heat capacity. Exponents \(v\), \(l\), and * denote, respectively, vapor, liquid and effective state.  
Considering the moisture loading in the current study, mainly adsorption not exceeding 88% relative humidity (see Table 4), capillary mouvment in the over hygroscopic range and hysterisis were not taken into account.  
The coupled heat and moisture transfer model obtained takes the form of a series of partial differential equations (PDE) that govern all the transfers. These non-linear and coupled PDE are resolved simultaneously using the COMSOL Multiphysics software based on the Finite Elements methods. We chose this environment since it can be used for modeling and resolving a wide range of engineering and research problems, mainly problems in PDE form. It is particularly well-adapted to solving Multiphysics problems in which several phenomena are studied simultane  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**State Variable** & **Interior Air** & **Exterior Air** \\ \hline T (°C) & 19 & 5 \\ \hline RH (\%) & 55 & 88 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Experimental configurations studied.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Measurement** & **Domain** & **Uncertainty** \\ \hline Temperature (SHT75) & 5–45 \({}^{\circ}\)C & 0.17 \({}^{\circ}\)C \\ Relative humidity (SHT75) & 40–90\% & 2\% \\ Temperature (Pt100) & 5–45 \({}^{\circ}\)C & 0.15 \({}^{\circ}\)C \\ Temperature (thermocoouples) & 5–45 \({}^{\circ}\)C & 0.16 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table}
Table 3: Domain and measurement uncertainties.  
of coupled HAM transfers. The use of such tools is justified because they are capable of solving this type of problem and allow the implementation of reliable and well-defined input data and the choice of meshing. It comprises several solvers adapted to a series of given problems.