## 4 Preliminary Thermal Analysis as a Proof of Feasibility  
Bubbling is started once the desired starting temperature is reached. The thermal response of Xylitol during bubbling is recorded by two thermocouples: one close to the tube injecting air and the other one placed on the opposite side. The air is injected at room temperature (20 \({}^{\circ}\)C).  
Figure 8: Xylitol crystallization pictures: (**a**) small air bubbles are dragged by the liquid circulation cell then white solid particles are formed by crystallization of Xylitol on the air bubbles surface. Finally, numerous solid particles can be observed in suspension; (**b**) Pictures of 6 particles extracted from the beaker and measured using a caliper (dimensions are given in millimeters); (**c**) magnified pictures of the cross-sections of particles cut in half.  
Figure 7: Top view pictures of bubbles reaching the free-surface of the liquid at different times of the crystallization process. The bubbles burst and generate small daughters.  
A typical thermal response of high undercooled Xylitol during bubbling is in Figure 9 where three different periods can be clearly distinguished:  
* _The induction period_, from the beginning of bubbling to the beginning of crystallization. In a perfectly adiabatic reactor, the temperature of the melt should be constant during this period. In practice, it slightly decreases due to thermal losses. Thermal losses take place both, by conduction through the reactor wall and by convection due to air bubbling at ambient temperature.
* _The recalescence period_, where Xylitol is crystallizing and its temperature raises. In a perfectly thermally insulated system, Xylitol should reach the melting point and the crystallization process should stop at that moment. However, due to thermal losses, the maximum temperature reached by Xylitol is inferior to its melting point and the crystallization can progress beyond this point. We consider in the following that the recalescence period ends when Xylitol reaches its maximum temperature.
* _The end-period_, where crystallization (if any) is driven by heat extraction due to thermal losses.  
From the recorded temperature data, the evolution of the solid fraction over time as well as the total crystallization rate can be estimated. It has been shown in [19] that the solid fraction (\(f_{s}\)) can be estimated the following equation:  
\[\frac{df_{s}}{dt}=\left(\frac{c_{p}}{L_{m}}\right)\,\left[\frac{dT}{dt}+k(T-T_ {\infty})\right] \tag{1}\]  
where \(c_{p}\) is the specific heat of Xylitol, \(L_{m}\) is the latent heat and \(k\) the thermal exchange coefficient. Once, \(c_{p}\), \(L_{m}\) and \(k\) parameters are known, the crystallization rate (\(df_{s}/dt\)) can be determined.  
An example of the estimated time evolution of the crystallization rate as well the solid fraction corresponding to temperature data recorded by both thermocouples are depicted in Figure 9.  
It can be seen that the crystallization progresses slowly at the beginning, then accelerates up to a maximum crystallization rate, and the rhythm of crystal production reduces afterwards. At the end of the recalescence period (\(t\approx 2500s\) in Figure 9), the crystallization rate is such that power delivered by solidification equals the thermal losses.  
These results confirm the possibility of using Xylitol as phase change material for seasonal thermal energy storage despite its high viscosity and by using a simple, non-intrusive and thus cost-effective bubbling technique.  
Figure 9: Time evolution of the solid fraction (a) and time behavior of the crystallization rate (b) calculated from the temperature data obtained using both thermocouples.  
Further works will consist in studying the bubbling performances for thermal storage applications. Quantitative analyses will be performed to estimate the thermal properties influencing the discharge process and to evaluate the reachable performances.