# Demonstration of Phase Change Thermal Energy Storage in Zinc Oxide Microencapsulated Sodium Nitrate
## 2 Materials and Methods
A third equation considers the caloric capacities of the tank components for the two loading m\({}_{1}\) and m\({}_{2}\), resulting in:  
\[\mathrm{Q_{m2}}\mathrm{Q_{m1}}\mathrm{=(m_{2}-m_{1})\;c_{p}\;dT} \tag{3}\]  
where dT is a small temperature change that occurred in dt\({}_{1}\), the same that occurred in dt\({}_{2}\) (temperature decay in experimental T1, immediately after the 8-9 h plateau interval--characteristic for latent storage); and c\({}_{p}\) is the specific heat of the storage material (microspapsules), determined by DSC just below the solidification offset temperature. For tank calculations, dT is monitored from the T1 curve; the difference between T1 and T2 affects \(\mathrm{\Delta T_{in\text{-out}}}\) estimation less than 8% in the relevant temperature interval. In the tank experiments, a T1 curve just below solidification (point indicated by the offset temperature in T4) is almost a straight line, therefore the dt (respectively, dT) can be quite large without affecting the integration of heat flows to Q\({}_{m1}\) and Q\({}_{m2}\) heats for Equation (3). **Thermal analysis using transient nonlinear FEM. **  
_Model and implementation._ The tank model was implemented in Microsoft Excel spreadsheets, with the FEM simulation step progressing on the columns of a simulation table. Visual representation (time course, temperature dependency, etc.) of input parameters and output results was available in charts. To estimate the heat transfer, the tank was modeled as a cylinder, then FEM was applied for 10 concentrical layers (cells), imposing T1 boundary conditions from the experimental dataset. Only conductive transfer was taken into consideration, as suggested by [PERSON] and [PERSON] (2008) [18] for sodium nitrate. Heat transfer during PCM solidification was calculated by the apparent capacity method, derived from the enthalpy method ([PERSON] et al., 2012; [PERSON] et al., 2020) [19; 20]. Thus, microparticle heat transfer during solidification was modeled based on conversion of the solidification heat into specific heat. Thermal conductivity and specific heats were then approximated on specific intervals by polynomial functions on temperature to best fit the experimental data. For a narrow T4 interval at the onset of solidification, a constant thermal conductivity value, 2\(\times\) higher than measured, was in place for T4 simulations, as it was the single major contributor to simulation-measurement overlap on that interval, probably due to some convective heat transfer by microcapsule deformation in the tank. Simulations start from the presumed steady-state equilibrium reached at the end of the charging phase. Thus, inside the tank, the radial symmetric temperature distribution was set to range linearly from maximal temperature (T4) to the lowest temperature (T1). After few cycles of initial calculation, the temperature differences between consecutive cycles converged in each calculation point, therefore reaching a more realistic radial temperature distribution than initially assumed. This calculated quasi-steady state was attained with time-space resolution of the FEM mesh (time step 0.25 s and layer thickness 10-2 m) that ensured agreement with the second principle of thermodynamics. _Model limitations and corrections._  
Symmetry assumptions. The radial symmetry of the tank is broken on partial tank filling because of the flat horizontal lining at the top of the storage material. In our model, this affects the cross-layer transfer in the material, introducing parallel heat transfer (microspulse-air in parallel with microcapsule-microcapsule) for a same FEM layer. To overcome this without increasing FEM mesh complexity, for adjacent layers we calculated the effective transfer area (representing only the layers contact within the microcapsule-material block), and use it to express the exchanged heat. Thus, radial heat transfer between consecutive layers occurred on local temperature difference and effective transfer area.