## 3 Results and Discussions
### Analytical Modeling of the Effective Thermal Conductivity
#### 3.4.1 Effective Thermal Conductivity of Porous Mortar
Considering the thermal conductivity of solid/fluid phase (\(k_{J\mathrm{Ms}}\), \(k_{J}\)) and the morphological parameter f\({}_{\mathrm{s}}\) of the solid phase (with f\({}_{\mathrm{s}}\) = 1 in the case of fiber particles), the effective thermal conductivity of the crushed fibers is determined using the following expression:  
\[k_{J\mathrm{M},eff}=\phi_{\mathrm{inter}}k_{f}+(1-\phi_{\mathrm{inter}}) \times\left(\frac{2-f_{\mathrm{s}}}{3}\right)k_{J\mathrm{Ms}} \tag{13}\]  
Where  
* \(k_{f}\) is the thermal conductivity of the fluid phase (air)
* \(k_{J\mathrm{Ms}}\) is the local thermal conductivity of Juncus maritimus fibers
* \(\phi_{\mathrm{inter}}\) is the porosity between the particles
* \(k_{J\mathrm{M},eff}\) is the effective thermal conductivity of the packed bed of the crushed fibers  
The local thermal conductivity of Juncus maritimus particles estimated from Glicksman model (Equation (13)) and crushed fibers packed beds effective thermal conductivity measurement is  
\[k_{J\mathrm{Ms}}=\frac{3}{2-f_{s}}\frac{k_{J\mathrm{M},eff}-\phi_{\mathrm{ inter}}k_{f}}{1-\phi_{\mathrm{inter}}}=0.719\;\mathrm{W/m\cdot K} \tag{14}\]  
A recent model, proposed by [PERSON] et al. [22] is also used to identify the local thermal conductivity of Juncus maritimus fibers. This model is based on the Self-consistent scheme, it was previously applied to model the thermal conductivity of hemp insulation material. \[k_{J\mathrm{Ms}}=\frac{3k_{J\mathrm{M},eff}}{1+\frac{\phi_{\mathrm{inter}}}{ 1-\phi_{\mathrm{inter}}}\times\frac{k_{f}-k_{J\mathrm{M},eff}}{k_{f}+2k_{J \mathrm{M},eff}}}-2k_{J\mathrm{M},eff} \tag{15}\]  
The local thermal conductivity of Juncus maritimus fibers estimated from Equation (15) and crushed fibers packed beds effective thermal conductivity measurement is \(k_{J\mathrm{Ms}}\) = 0.8 W/m\(\cdot\)K. The relative deviation between the estimated values using Equation (13) or (15) does not exceed 6%. It can be concluded that models are in good agreement. A mean value \(k_{J\mathrm{Ms}}\) = 0.76 W/m\(\cdot\)K can be considered in the next. It is important to note that radiation heat transfer inside the porous material could contribute to the total heat transfer measured due to the high porosity (70%). Thus, we can retain that the local thermal conductivity of Juncus maritimus fibers identified could be overestimated using this method. Radiation contribution is often neglected in previous works dealing with local solid thermal conductivity determination for hemp insulation materials and other bio based materials [8,22]. However, it can be noted that for traditional porous insulating media used in building application \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fibers Loading (wt \%)** & **Porosity (\%)** & \(K_{e}\) **(W/m\(\cdot\)K)** \\ \hline
0 & 0.27 & 2.868 \\
2 & 0.39 & 0.399 \\
5 & 0.43 & 0.239 \\
7 & 0.47 & 0.211 \\
10 & 0.52 & 0.182 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Thermal conductivity of porous mortar for different porosity. (such as polystyrene foam at ambient temperature), radiation contribution can reach 35% of the total effective thermal conductivity. 4.3 Effective Thermal Conductivity of the Composite Samples/Experimental and Analytical Results Analytical Models  
3D-uCT image of composite samples (Figure 5c,d) show, heterogeneous medium with fibers randomly distributed in the mortar matrix. An estimate of the effective thermal conductivity of composite samples can obtain by the Geometric mean model [31]. This model uses the estimated thermal conductivities of each phase and their volume fractions. It can be noted that this model has been used to estimate the thermophysical properties of date palm fibers/gypsum composite [35]. The geometric mean model assumes random distribution of two components of the materials as a function of their volume fractions.