## 2 Method
### Numerical Method  
Given the small size of the PCM inclusions and their homogeneous distribution into the solid cement mortar (\(cm\)) matrix, we model the sample as a homogeneous medium, characterized by macroscopic thermo-physical properties. According to the same principle, the latent heat of the inclusions is homogenized and allows to define an apparent latent heat of the homogeneous material. These assumptions also make it possible to consider only conduction transfers [68].  
Thus, the physical model writes:  
\[\rho\frac{\mathrm{d}h}{\mathrm{d}t}=\overrightarrow{\
abla}\cdot(\lambda \overrightarrow{\
abla}T) \tag{1}\]  
where all involved parameters are apparent ones.  
Closure of (1) is then done by defining the initial and boundary conditions:  
\[T =T^{0} \tag{2a}\] \[\varphi_{L}^{\mathrm{num}} =\frac{T_{p,L}-T_{L}}{R_{L}}\] (2b) \[\varphi_{R}^{\mathrm{num}} =\frac{T_{p,R}-T_{R}}{R_{R}} \tag{2c}\]  
where \(T_{p}\) is the temperature of the exchange plate and \(R\) the corresponding thermal resistances for the left and right side respectively, as shown in Figure 3b.  
Finally, the enthalpy function \(h(T)\) is the one to be identified. Furthermore, the properties of the PCM are not constant, since they depend on its physical state (liquid or solid): consequently, the variation of the thermo-physical properties of the composite material depends on the liquid  
Figure 3: Detailed description of the experimental set-up. (**a**) Schematic view; (**b**) Detailed view (with boundary conditions).  
fraction \(\chi\) of the PCM and involves its solid and liquid properties.  
Moreover, the following assumptions are made:  
1. The PCM being micro-encapsulated, the thermal expansion during the phase transition can be neglected, which implies that solid and liquid densities are equal. _n.b._ in such a case, there is no difference between the mass and volume fractions, which are therefore both denoted by \(\chi\).
2. The thermodynamic behavior of the PCM, i.e., its enthalpy function, obeys a binary solution model [68]: \[\mathrm{d}\chi =\frac{T_{M}^{\star}-T_{M}}{\left(T_{M}^{\star}-T\right)^{2}} \,\mathrm{d}T\] (3a) \[\frac{\mathrm{d}h}{\mathrm{d}T} =\frac{T_{M}^{\star}-T_{M}}{\left(T_{M}^{\star}-T\right)^{2}}L_{M} (T)+c_{s}\left(1-\frac{T_{M}^{\star}-T_{M}}{T_{M}^{\star}-T}\right)+c_{l}\frac{ T_{M}^{\star}-T_{M}}{T_{M}^{\star}-T}\] (3b) Here, the involved latent heat for the composite mortar depends on the mass fraction \(Y\) of the PCM included in this latter one: \[L_{M}=YL_{M}^{\star}\] (4)
3. The thermal conductivity is governed by a classic average [72; 73; 74]: \[\lambda=\chi\lambda_{l}+(1-\chi)\lambda_{s}\] (5) where indices \(s\) and \(l\) stand for the homogeneous material, that is to say to the initial cement mortar with the liquid or solid contribution of the PCM.  
Lastly, the system of partial differential equations described by Equations (1) and (2) is solved numerically using a finite-volume method, based on a first-order scheme for the flux calculations and on an Euler integration. The mesh used contains 100 cells, which corresponds to a space step of 0.4 mm, and the time step is fixed at 0.1 s. These parameters have been set after a convergence study, presented in Figure 4, which has shown that the solution was not sensible to further refinement.