## 4 Mathematical Model
### Thermal Transfer by Conduction on the Storage Wall  
The method used to model the thermal behavior of the latent storage wall was presented in several articles [63; 64; 65]; it consists of characterizing by the inverse method the parameters requiredto model the heat transfer of a mortar integrating a micro-encapsulated PCM during the phase change (solid \(<\)-\(>\) liquid). To establish the state equations of the composite material (mortar + PCM), the thermodynamic behavior of the PCM could be assimilated to that of a binary solution. Since the PCM is micro-encapsulated, the assumptions here stipulate no natural convection in the microcapsules given that their dimensions vary between 5 and 40 \(\upmu\)m and are uniformly distributed in the cement mortar matrix. The model implemented in the Dymola/Modelica software was based on the enthalpy derivative equation (Equation (16)) [66].  
\[\frac{\partial\rho h}{\partial t}=\lambda\frac{\partial T^{2}}{\partial^{2}x} \tag{16}\]  
This model depicts the unidirectional conduction of a homogeneous mono-layer material under transient conditions. The discretization is performed by assigning for each mesh two resistances and a capacity that varies with temperature (see Figure 8):  
\[C\frac{dT}{dt}=\phi_{i}-\phi_{i+1} \tag{17}\]  
The wall is then discretized into a 1D mesh, in which the heat balance can be calculated using the following equations (Equation (18)):  
\[\phi_{i}=\left(\frac{\lambda.A}{e_{mesh}/2}\right)dT_{i}\;\;\;\;and\;\;\;\phi_ {i+1}=\left(\frac{\lambda.A}{e_{mesh}/2}\right)dT_{i+1} \tag{18}\]  
Each mesh possesses the same thermal properties (\(\lambda\), \(c\), \(\rho\)) and thickness (\(\mathrm{e_{mesh}}\)). After a parametric study, a discretization of 30 meshes was selected. The thermophysical and dimensional properties introduced are those of the apparent composite material, which means that the composite material resulting from the mixture of cement mortar and PCM is considered to be a homogeneous isotropic material:  
\[\phi_{i}=\left(\frac{\lambda_{m\_pcm}.A_{m\_pcm}}{e_{mesh\_m\_pcm}/2}\right)dT _{i}\;\;\;and\;\;\;\phi_{i+1}=\left(\frac{\lambda_{m\_pcm}.A_{m\_pcm}}{e_{ mesh\_m\_pcm}/2}\right)dT_{i+1} \tag{19}\]  
It is necessary therefore to determine the enthalpy curve h(T). The enthalpy curve of a binary solution without a eutectic is of the form shown in Figure 9a. The formulation of this curve involves the end of the melting temperature (T\({}_{\mathrm{M}}\)), the melting temperature associated with the pure substance (T\({}_{\mathrm{A}}\)) of the PCM, the specific heats of the composite material when the PCM is in the solid state (\(\mathrm{c_{solid}}\)) and in the liquid state (\(\mathrm{c_{liquid}}\)), as well as the latent heat of phase change (\(\mathrm{L_{A}}\)) contained in the composite material.  
The heat stored in the storage wall integrating PCM (mortar PCM) represents the latent heat in considering both its melting and solidification. For this storage wall, \(\mathrm{c_{m\_pcm}}\) represents the derivative of  
Figure 8: Model of a phase change material mesh.  
enthalpy with respect to temperature T [63; 66] and is presented in Figure 9b. The following equations are used to determine the value of c\({}_{\text{m\_pcm}}\) once the material temperature is known.  
\[C=c_{m\_pcm}.A_{m\_pcm}.e_{m\_pcm}.\rho_{m\_pcm} \tag{20}\]  
\[c_{m\_pcm}=\frac{dh_{m\_pcm}}{dT}=\begin{cases}\left(\frac{T_{A}-T_{M}}{T_{A}-T }\right)c_{liquid}+\left(1-\frac{T_{A}-T_{M}}{T_{A}-T}\right)c_{solid}+L_{A} \frac{T_{A}-T_{M}}{\left(T_{A}-T\right)^{2}}&\text{if $T<T_{M}$}\\ c_{liquid}&\text{if $T\geq T_{M}$}\end{cases} \tag{21}\]