## 3 Results and Discussions
### Analytical Modeling of the Effective Thermal Conductivity
#### 3.4.1 Effective Thermal Conductivity of Porous Mortar  
The 3D-uCT images (Figure 5a,b) reveal that the reference mortar is constituted of two phases: the continuous solid phase: sand and cement mixture, and the dispersed phase: air spherical pores randomly distributed in mortar matrix. The Hadley model [34] that combines the Maxwell upper bound with an adjustable function f\({}_{0}\)(\(\phi\)), is applied to the medium made of the fluid phase \"f\" surrounded by a solid matrix \"s\". The thermal conductivity, \(k_{s}\), of the solid phase of reference mortar (without fiber) is identified using the following Hadley expression (Equation (10)) and reference mortar effective thermal conductivity measured experimentally:  
\[\frac{k_{e}}{k_{f}}=(1-\alpha_{0})\frac{\phi f_{0}+\frac{k_{s}}{k_{f}}(1-\phi f _{0})}{1-\phi(1-f_{0})+\phi(1-f_{0})\frac{k_{s}}{k_{f}}}+\alpha_{0}\frac{2 \left(\frac{k_{s}}{k_{f}}\right)^{2}(1-\phi)+(1+2\phi)\frac{k_{s}}{k_{f}}}{(2 +\phi)\frac{k_{s}}{k_{f}}+1-\phi} \tag{10}\]  
Where \(\phi\) is the porosity of samples, k\({}_{f}\) is the air thermal conductivity and the coefficients f\({}_{0}\) and \(\alpha_{0}\) are given by:  
\[f_{0}=0.8+0.1\phi \tag{11}\]  
\[\left\{ \begin{array}{ll}\log\alpha_{0}=-4.898\phi&0\leq\phi\leq 0.0827\\ \log\alpha_{0}=-0.405-3.154(\phi-0.0827)&0.0827\leq\phi\leq 0.298\\ \log\alpha_{0}=-1.084-6.778(\phi-0.298)&0.298\leq\phi\leq 0.58\end{array} \right. \tag{12}\]  
Using the measured effective thermal conductivity of the reference mortar \(ke=k_{\text{eff}}\) = 0.521 W/m\(\cdot\)K obtained \(\phi=0.27\) and \(k_{f}\) = 0.026 W/m\(\cdot\)K, the inversion of Equation (10) leads to determine the solid thermal conductivity of dense mortar: \(k_{s}\) = 2.868 W/m\(\cdot\)K.  
Then, knowing \(k_{s}\), this model is used to calculate the effective thermal conductivity of porous mortar, \(k_{e}\), for different porosities. Results are presented in Table 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **K\({}_{\text{eff}}\) (W/m\(\cdot\)K)** & **\(\alpha\) (\(\times\)10\({}^{-7}\) m\({}^{2}\)/s)** & **\(\rho_{\text{c}}\) (Kg/m\({}^{3}\))** & **References** \\ \hline Reference mortar & 0.521 & 2.6 & 1933 & [This work] \\ JM/mortar (10\%) & 0.171 & 1.623 & 1081.96 & [This work] \\ Wood/mortar (30\%) & 0.24 & 1.5 & 1495 & [31] \\ DPF mesh/mortar (51\%) & 0.243 & 1.5 & 1217 & [13] \\ Cork/mortar (10\%) & 0.96 & - & 2100 & [32] \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the thermal properties of Juncus maritimus/mortar composites and some building materials used for thermal insulation. DPF: date palm mesh. 4.2 Estimation of the Local Juncus Maritimus Thermal Conductivity from Crushed Fibers Effective Conductivity  
This section focuses on the thermal conductivity determination of the local Juncus maritimus fibers. This property is identified by fitting theoretical results and experimental measurements of the effective thermal conductivity of the crushed fibers packed beds. The Schuetz and Glicksman analytical model [21], is applicable to porous media containing randomly oriented fibers.