## 3 Results
### First Sample  
As mentioned hereinbefore, the starting point is the first cycle of the 2 h ramp for the 12.4%\({}_{w}\)-PCM mortar. The experimental heat flow-rates on the left and right sides of the sample for the given imposed temperature ramp are shown in Figure 5 with the corresponding numerical heat flow-rates obtained after the inversion process. One can see that agreement between the experimental measurements and the numerical computations is very good. Then, an obvious asymmetry is observed between each sides. This can be easily explained by the fact that, experimentally, the thermal contact is not perfect and is always different from one face to the other since their roughness is different. Consequently, huge variations can arise between these two sides. Yet, in spite of this asymmetry, the method is capable of reproducing the experimental behavior without any difficulties. Moreover, it is worth mentioning that the real sample also shows a non-symmetrical behavior between the melting and the crystallization steps, which is perfectly captured by the numerical calculation. This remark has to be further commented. First, this behavior is not due to the presence of supercooling, which would lead to a clear discontinuity during cooling, which is not observed experimentally. Added to this, it would be rather difficult for the results to represent such a phenomenon since it has not been considered during the modeling. In fact, this asymmetric feature is obtained only because the enthalpy function is governed by the relation (3), which shows a different behavior when going to the solid to liquid state and vice-versa. In other words, the slope of the function (and so its derivative) being non-constant, dynamics of the phase transition is different between melting and solidification. In this respect, this means that any enthalpy function having a symmetrical expression, as for instance with pure substances or solutions where a linear profile is assumed between the liquidus and the solidus, will no be capable to reproduce such a behavior. Once again, this is possible if and only if enthalpy function obeys Equation (3), as demonstrated in [68]. Eventually, the corresponding enthalpy function is depicted in Figure 6 and the values for the associated parameters, obtained after identification, are given hereafter:  
Figure 5: 12.4%\({}_{w}\)-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the first cycle of the 2 h heating/cooling ramp experiment. \[c_{S} =1094\,\mathrm{J}\,\mathrm{K}^{-1}\,\mathrm{kg}^{-1} \tag{7a}\] \[c_{L} =1094\,\mathrm{J}\,\mathrm{K}^{-1}\,\mathrm{kg}^{-1}\] (7b) \[T_{M} =25.57\,^{\circ}\mathrm{C}\] (7c) \[T_{M}^{\star} =27.18\,^{\circ}\mathrm{C}\] (7d) \[L_{M} =12,322\,\mathrm{J}\,\mathrm{kg}^{-1} \tag{7e}\]  
Here again, some discussion is needed. Indeed, in the inversion process, it is important to recall that an unique enthalpy function is determined. So, with a single expression, obeying a non-symmetrical formulation as discussed above, the behavior of the material is described for both the solid and the liquid state. Consequently, this obviates the usual formulations assuming one expression for the heating and another for the cooling, which is indisputably inconsistent with basic thermodynamic laws, as further discussed in [62]. After this study for the first cycle, the purpose is extended to the other cycles of the 2 h ramp experiment. Thus, the second and third cycles are used as original inputs in the inversion process, as well as the entire set of cycles. The corresponding results are furnished in Figure 7. In all cases, the numerical results obtained with the present procedure show a very good agreement with the experimental data. Once again, both the non-equal thermal contact resistances at the left and right sides and the asymmetrical behavior between the fusion and solidification are perfectly rendered by the computations.