## 3 Results and Discussions
### Analytical Modeling of the Effective Thermal Conductivity
#### 3.4.1 Effective Thermal Conductivity of Porous Mortar
Considering the mortar matrix as solid phase and the Juncus maritimus particles as dispersed phase, the effective thermal conductivity is calculated using the following expression:  
\[k_{eff}=k_{JMs}{}^{\varphi_{v,m}}\times k_{e}{}^{(\phi+\varphi_{v,m})} \tag{16}\]  
The Schuetz and Glicksman (Equation (17)) and [PERSON] analytical models (Equation (18)) are applied to composites samples constituted of porous mortar and Juncus maritimus fibers. \[k_{eff}=\frac{\varphi_{v,JM}}{3}k_{JMs}+(\phi+\varphi_{v,m})k_{e} \tag{17}\]  
\[2k_{eff}{}^{2}+k_{eff}\big{(}k_{e}(1-3(\phi+\varphi_{v,m}))-k_{JMs}(2-3(\phi+ \varphi_{v,m}))\big{)}-k_{JMs}k_{e}=0 \tag{18}\]  
Experimental identification of local Juncus maritimus thermal conductivity from composite effective conductivity  
Experimental results are compared with previous theoretical models using the identified local thermal conductivity, \(k_{JMs}\) = 0.76 W/m\(\cdot\)K, (obtained from crushed fibers effective thermal conductivity measurements and analytical model results) (Figure 7). It can be observed that there are non-negligible deviations between theoretical results. The [PERSON], Geometric mean and Nguyen models overestimate experimental results of the effective thermal conductivity. These results tend to show that identified value of the local thermal conductivity, \(k_{JMs}\), is overestimated. These results could be explained by the non-negligible radiation heat transfer contribution in the crushed fibers sample which is taken into account in the \(k_{JMs}\) value inducing overestimation. Alternatively, the JM local thermal conductivity of Juncus maritimus fibers was identified using analytical models and experimental measurements of thermal conductivity of composite samples for different Juncus maritimus volume fraction. The [PERSON], Geometric mean and Nguyen models are applied to composite samples constituted of mortar matrix and dispersed fibers. The estimated values of JM local thermal conductivity by these latter models are 0.472 \(\pm\) 0.09, 0.125 \(\pm\) 0.028 and 0.130 \(\pm\) 0.033 W/m\(\cdot\)K, respectively. As expected the value are inferior to \(k_{JMs}\) = 0.76 W/m\(\cdot\)K (obtained in Section 3.4.2). The relative deviations between the estimated values of local thermal conductivity of JM crushed fibers and of the JM fibers embedded in composite samples estimate by Equations (16)-(18) are 83.55%, 37.8%, and 83%, respectively. It seems that the Juncus maritimus local thermal conductivity estimated by Glicksman model is the most appropriate. The reduction is too large for the both other models and seems not realistic. Thus, the value \(k_{JMs}\) is estimated to be 0.472 W/m\(\cdot\)K.  
Using the thermal conductivity of porous mortar values, \(\mathrm{k_{e}}\), (Table 4) and the estimated local thermal conductivity of crushed Juncus maritimus fibers \(k_{JMs}\) =0.472 \(\pm\) 0.09 W/m\(\cdot\)K, both Glicksman model and the effective thermal conductivity experimental data of composite samples are plotted in Figure 8. It is observed a good agreement between results for the different volume fraction. Figure 8: Comparison of JM/mortar composites effective thermal conductivity experimental results with Glicksman model calculated using the estimated \(k_{JMs}\) = 0.472 W/m\(\cdot\)K.  
Figure 7: Experimental results compared with theoretical models calculated with \(k_{JMs}\) = 0.76 W/m\(\cdot\)K.