# Numerical Investigation of Latent Thermal Storage in a Compact Heat Exchanger Using Mini-Channels
## 2 Materials and Methods
### Simulation Model
For the PCM domain, the element sized varied in the cross-sectional plane and in length, and the effect on melt fraction was studied. As shown in Figure 4b, the simulated melt fraction changes slightly with refining the cross-sectional PCM mesh from 1 mm to 0.5 mm. Further reducing the cross-sectional element size in the PCM domain down to 0.3 mm showed an insignificant change in comparison to the element 0.5 mm. Figure 4c shows the influence of decreasing the element size from 3 to 1 mm in the axial direction for the PCM domain, making almost no influence on the melt fraction results. Furthermore, the variation in time-step displays its insignificant effect on the simulations, as shown in Figure 4d. In the end, considering the insignificant changes in the hydrodynamic air flow and melting simulations, the following element sizes were chosen. The final grid elements size, perpendicular to the air flow, were set to 0.4 mm, 0.15 mm and 0.1 mm for the PCM, aluminum and air zones, respectively, as shown in Figure 5, and the grid element size in the axial direction was set to 2 mm. The time step was fixed to 0.1 s in the simulations. In the used meshes, the minimum orthogonal mesh quality in the final grid and the maximum skewness factors were 0.5 and 0.8, respectively. The simulations with the chosen grid based on the performed mesh independency analysis were performed for the specified configurations in Table 2 with inlet HTF temperature of 30 \({}^{\circ}\)C and 15 \({}^{\circ}\)C for melting and solidifications, respectively. The initial domain temperature was set to 15 \({}^{\circ}\)C and 30\({}^{\circ}\)C. The total flow rate of the HTF entering the cuboid was varied within a range of 7-24 L/s. The minimum flow rate, corresponding to a room of 20 m\({}^{2}\), was chosen based on the Swedish regulation of maintaining a minimum flow rate of 0.35 L/m\({}^{2}\)-s in the ventilation systems of residential buildings. A parametric study of flow rate with fixed values of 7, 12, 16, 20 and 24 L/s was carried out. As the number of channels was varied, given a constant total air flow rate, the inlet velocity of air flow per channel was calculated and used as the inlet boundary conditions. With the simulated range of flow rate and the varying number of channels from 60 to 100 for channel-1 and 72 to 96 for channel-2, respectively, the Reynolds numbers were within ranges of 250-1440 and 430-1970. The total transient rate of heat transfer is calculated with Equation (8). \[\dot{\mathrm{Q}}(\mathrm{t})= N\rho_{\mathrm{air}}V_{\mathrm{air}_{\mathrm{in}}}A_{\mathrm{c}}\;c_{ \mathrm{Pair}}\mathrm{abs}\big{[}\mathrm{T}_{\mathrm{air}_{\mathrm{out}}}( \mathrm{t})-\mathrm{T}_{\mathrm{air}_{\mathrm{in}}}\big{]} \tag{8}\]  
where N is the number of channels, \(\rho_{\mathrm{air}}V_{\mathrm{HTF}_{\mathrm{in}}}A_{\mathrm{c}}\) is the air mass flow rate per channel, \(c_{\mathrm{Pair}}\) is the air specific heat and \(\mathrm{T}_{\mathrm{air}_{\mathrm{out}}}(\mathrm{t})-\mathrm{T}_{\mathrm{air}_ {\mathrm{in}}}\) is the transient air temperature difference of the outlet and the inlet. The total UA value is calculated as a ratio of the total rate of transferred heat to the logarithmic mean temperature difference (LMTD) between the HTF and the PCM, using Equations (9) and (10). \[\mathrm{UA}\;=\;\frac{\dot{\mathrm{Q}}(\mathrm{t})}{\mathrm{LMTD}\;(\mathrm{t})} \tag{9}\]  
\[\mathrm{LMTD}\;(\mathrm{t})\;=\;\frac{\big{[}\mathrm{T}_{\mathrm{air}_{\mathrm{ in}}}-\mathrm{T}_{\mathrm{s/m}}\big{]}-\big{[}\mathrm{T}_{\mathrm{air}_{ \mathrm{out}}}(\mathrm{t})-\mathrm{T}_{\mathrm{s/m}}\big{]}}{\ln\frac{\mathrm{ T}_{\mathrm{air}_{\mathrm{in}}}-\mathrm{T}_{\mathrm{s/m}}}{\mathrm{T}_{\mathrm{air}_{ \mathrm{out}}}(\mathrm{t})-\mathrm{T}_{\mathrm{s/m}}}} \tag{10}\]  
The air heat transfer coefficient is calculated as the ratio of the total heat flux to the logarithmic mean temperature difference between the HTF temperature and the channel wall temperature at the inlet and outlet, as specified in Equations (11) and (12).