# Impact of Temperature on the Moisture Buffering Performance of Palm and Sunflower Concretes
## 2 Numerical Model  
### Mass Transfer Model in Single-Layer Wall  
In this article, we use a model based on the theory of [PERSON] (1957) [35]. The moisture transport within a simple layered wall can be described by one-dimensional diffusion using the moisture content in the material as a driving force [35]. The mass balance equation is written as  
\[\frac{\partial\theta}{\partial t}=\frac{\partial}{\partial x}\bigg{(}D_{\theta }\frac{\partial\theta}{\partial x}\bigg{)} \tag{1}\]  
where \(\theta\) is the moisture volumetric content in the material (m\({}^{3}\) of water/m\({}^{3}\) of material), and \(D_{\theta}\) is the diffusion coefficient of the moisture in the material (m\({}^{2}\)/s). The boundary conditions (in \(x=0\), outdoor environment, and \(x=L\), indoor environment) are  
\[-\rho_{l}\bigg{(}D_{\theta}\frac{\partial\theta}{\partial x} \bigg{)}\bigg{|}_{x=0,e} =h_{M,e}(\rho_{vv,a,e}-\rho_{vv,s,e}) \tag{2}\] \[-\rho_{l}\bigg{(}D_{\theta}\frac{\partial\theta}{\partial x} \bigg{)}\bigg{|}_{x=L,i} =h_{M,i}(\rho_{vv,s,i}-\rho_{vv,a,i}) \tag{3}\]  
where \(\rho_{vv,a,i}\) and \(\rho_{vv,a,e}\) are concentrations of water vapor in inside and outside air (kg/m\({}^{3}\)), \(\rho_{vv,s,i}\) and \(\rho_{vv,s,e}\) are concentrations of water vapor in the surface layers (inside and outside, (kg/m\({}^{3}\))) and \(h_{M,e}\) and \(h_{M,i}\) are coefficients of exchange of water vapor by convection (m/s) for the external and internal surfaces.  
The diffusion coefficient \(D_{\theta}\) is the total mass transport coefficient which is the sum of the liquid and vapor water phase:  
\[D_{\theta}=D_{\theta,\;v}+D_{\theta,J} \tag{4}\]  
In this article, we focus on the hygroscopic domain and the mass transport coefficient due to the liquid phase being negligible compared to that of water vapor (\(D_{\theta,\;l}\;\ll D_{\theta,\;v}\)), and therefore [36]  
\[D_{\theta}=D_{\theta,\;v}\;=\frac{\pi P_{\mathrm{v},s}(T)}{\rho_{l}}\;\frac{ \partial HR}{\partial\theta} \tag{5}\]  
The moisture sorption isotherm curve is a particularly important hygric property because it provides a true hygro-structural identity card of porous materials. It also impacts on the hygrothermal behavior of hygroscopic materials. Bio-based materials have high porosity and high sensitivity to moisture. These materials have the ability to adsorb and desorb water vapor, allowing for regulating the variation in the indoor relative humidity level in buildings [30]. When the humidity of the surrounding air increases, it results in an increase in mass. This mass gain is due to the phenomenon of physical adsorption. It allows water molecules to be fixed on surface pores in a reversible way. Likewise, a decrease in the humidity of the surrounding air results in an apparent loss of mass. This is the phenomenon of physical desorption.  
Many theories analyze the shape of the experimental sorption isotherm curves. [PERSON]'s model describes a monomolecular adsorption. The first adsorption layer for a porous middle is only verified at low humidity [37]. The Brunauer-Emmett-Teller model [38] is an extension of [PERSON] theory, and it was developed to analyze multilayer adsorption. Additionally, the Guggenheim-Anderson-de Boer model [39; 40], which was developed from two other models, will be used in this article to describe the sorption isotherm curves of the studied bio-based building materials. The GAB model [41] takes into account the heat of adsorption for all molecular layers and describes the dependence of the adsorption isotherm on the temperature. Using the GAB model has many advantages such as having a viable theoretical basis and giving a good description of the sorption behavior of bio-based concretes such as hemp lime concrete [42; 43]. The GAB model can be written as follows [41]:  
\[w=\frac{w_{m}\;CK\varphi}{(1-K\varphi)(1+KC\varphi-K\varphi)} \tag{6}\]  
where \(w_{m}\) is the water content (kg/kg). The parameters \(C\) and \(K\) are energy constants (J/mol) of the GAB model [40; 41].