# Influence of Water Addition on the Latent Heat Degradation of Sodium Acetate Trihydrate
## 2 Materials and Methods
### Data Analysis  
The thermal behavior of the material during a full cycle (heating, cooling, and constant temperature steps), using the previously presented experimental bench, is described in Figure 2. Whereas all the surfaces are covered by sensors, only the record of the left and right fluxmeters (\(140\times 210\) mm\({}^{2}\)) is given. Indeed, the measurements of the four other surfaces are not considered significant as they represent about 4% of the total heat flux recorded.  
Here, the analysis was performed between two points (1 st and 2 nd point) on both sides of a temperature variation (heating or cooling). These two points were chosen at the end of the thermal plateau in order to have a stabilized heat flux. Hence, a baseline below 0.01 W could be obtained. The value of latent heat could then be obtained from an integration of the heat flux between these two points, after subtracting the sensible heat.  
This sensible heat was stored in the container, made of PMMA, whose specific heat capacity \(C_{P,\,\,\text{b}rick}\) has been determined by blank tests to be approximately 1.45 kJ-kg\({}^{-1.}\)\({}^{\circ}\)C\({}^{-1}\). The sensible heat was also stored in the solid \(C_{P,S}\) and liquid \(C_{P,L}\) SAT's specific heats. They can be calculated using linear approximations, given in Equation (1), based on the data from [PERSON] et al. [25]. These equations were considered valid for a mass fraction \(w_{SA}\) from 54.3 to 60.3 wt.% and a temperature between 27 and 58 \({}^{\circ}\)C for solid and between 27 and 77 \({}^{\circ}\)C for liquid (valid also in supercooled state), with an error of 5% maximum.  
\[C_{P}=A+B\ T+(C+D\ T)\ w_{SA} \tag{1}\]  
where \(T\) is the temperature (\({}^{\circ}\)C). The empirical parameters \(A\), \(B\), \(C\), and \(D\) are given in Table 1 for both solid and liquid specific heats.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(A\) (J\(\cdot\)kg\({}^{-1.}\)\({}^{\circ}\)C\({}^{-1}\)) & \(B\) (J\(\cdot\)kg\({}^{-1.}\)\({}^{\circ}\)C\({}^{-2}\)) & \(C\) (J\(\cdot\)kg\({}^{-1.}\)\({}^{\circ}\)C\({}^{-1}\)) & \(D\) (J\(\cdot\)kg\({}^{-1.}\)\({}^{\circ}\)C\({}^{-2}\)) \\ \hline \(C_{P,S}\) & 3367 & \(-10.74\) & \(-2412\) & 24.67 \\ \(C_{P,L}\) & 4091 & 5.8 & \(-2274\) & \(-2.52\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Empiric parameters for solid and liquid specific heat determination.  
Figure 2: Description of a full thermal cycle.  
These two specific heats allow for the calculation of the latent heat. This evaluation is based on the subtraction of the sensible energies to the global heat flux measured on the different surfaces, as shown in Equation (2).  
\[L=\int_{S}\int_{t_{1}}^{t_{2}}\phi\ dt\ dS\ -\int_{20}^{70}m\ C_{P,\ brick}dT-\int_{20}^{T_{melt}}m\ C_{P,sol}\ dT-\int_{T_{melt}}^{70}m\ C_{P, liq}\ dT \tag{2}\]  
where \(t_{1}\) and \(t_{2}\) are the time for the 1 st and 2 nd point (Figure 2), and \(T_{melt}\) is the melting temperature (58 \({}^{\circ}\)C). Equation (2) is focused on the melting of the material because it was assumed that the energy released during solidification was influenced by the possible presence of a non-repeatable supercooling degree. This equation could be compared with an empirical relation from [PERSON] et al. [25], given in Equation (3) that is valid for a mass fraction of sodium acetate \(w_{SA}\) from 47.5 to 60.3 wt.% with an error of 5% maximum.  
\[L=-462+11.8\ w_{SA} \tag{3}\]  
Moreover, the determination of the crystallization temperatures was necessary for a correct analysis of the material's solidification and its impact on the next melting. For each cycle (as presented in Figure 2), the onset temperature was determined in three steps. (1) The onset time was determined by the intersection of 2 tangents represented by black dash lines in Figure 3; (2) the crystallization temperature was then identified by a blue arrow; and (3) its value was read on the left axis.  
This procedure was then applied to the study of the two samples (SAT60% and SAT58%).