# Numerical Design and Laboratory Testing of Encapsulated PCM Panels for PCM-Air Heat Exchangers
## 2 Methods
### Data Reduction  
#### 2.2.1 Numerical Design  
The heat transfer between panel and air was calculated by the energy balance through the air crossing the control volume (Figure 3) and is given by:  
\[\dot{Q}_{air}=\dot{m}_{air}\mathrm{C}_{p(air)}(T_{o}-T_{i}) \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{4}{c}{**Experiment Pressure Drop (Pa)**} & \multicolumn{4}{c}{**Pressure Drop Simulation in 2D (Pa)**} & \multicolumn{4}{c}{
\begin{tabular}{c} **Difference between Experiment** \\ **and Simulation (Pa)** \\ \end{tabular} } \\ \hline Air flow (Re) & 1 TB & 2 TB & 3 TB & 1 TB & 2 TB & 3 TB & 1 TB & 2 TB & 3 TB \\
10089 & - & 10.30 & - & - & 11.07 & - & - & 0.77 & - \\
18736 & 15.38 & 28.58 & 40.98 & 19.80 & 37.77 & 52.69 & 4.42 & 9.19 & 11.71 \\
21600 & 20.20 & 43.47 & 51.24 & 25.51 & 45.92 & 67.29 & 5.32 & 2.44 & 16.05 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Experiment and simulation pressure drop and their difference.  
Figure 5: Thermal battery, section and top view of existing panel and diagram of the test rig.  
where: \(\dot{m}_{air}\) is the air mass flow rate in \(\frac{kg}{s}\); \(C_{p(air)}=1007\left[\frac{I}{kgK}\right]\); \(T_{i}=26\ ^{\circ}\)C; \(T_{o}\) is the air outlet temperature calculated by CFD, \(\dot{Q}_{air}\) is in Watts and it is assumed that there are no losses through the walls. The Nusselt number is given by:  
\[Nu=\frac{hD_{h}}{k_{air}} \tag{4}\]  
where \(D_{h}\) is the hydraulic diameter, \(k_{air}\) is the thermal air conductivity and \(h\) is the convective heat transfer coefficient.  
To evaluate \(h\), a control volume was applied (Figure 3) and all energy released by the air \((Q_{air})\) is absorbed through convection \((\dot{Q}_{c})\) by the panel (Equation (3)),  
\[\dot{Q}_{air}=\dot{Q}_{c} \tag{5}\] \[\dot{m}_{air}C_{p(air)}(T_{m,o}-T_{m,i})=hA_{panel}\Delta T\]  
where \(A_{panel}=L_{panel}\cdot W_{panel}\). For cases using constant surface temperature, the difference between hot and cold is given by a logarithmic mean temperature difference \(\Delta T_{LMDT}\)[25]:  
\[\Delta T_{LMDT}=\frac{(\Delta T_{o}-\Delta T_{i})}{\ln\left(\frac{\Delta T_{o }}{\Delta T_{i}}\right)}=\frac{(T_{s}-T_{o})-(T_{s}-T_{i})}{\ln\left[(T_{s}-T _{o})/(T_{s}-T_{i})\right]} \tag{6}\]  
where \(T_{m,o}\) is the outlet fluid temperature; \(T_{m,i}\) is the inlet fluid temperature and \(T_{s}\) is the surface temperature. Adding Equation (4) in Equation (3):  
\[h=\frac{\dot{m}_{air}C_{p(air)}(T_{o}-T_{i})}{(L_{panel}\cdot W_{panel})\left( \frac{(T_{s}-T_{s})-(T_{s}-T_{i})}{\ln\left[\frac{T_{s}-T_{s}}{T_{s}-T_{i}} \right]}\right)}\left[\frac{W}{m^{2}K}\right] \tag{7}\]  
The introduction of ridges requires more pumping power from the system. The thermal enhancement factor (\(\eta\)) [26] analyses the ratio of the convective heat transfer of the augmented surface over a smooth surface at a constant pumping power and is given by:  
\[\eta=\frac{\left(\frac{Nu}{Nu_{o}}\right)}{\left(\frac{\Delta p}{\Delta p_{o }}\right)^{1/3}} \tag{8}\]  
where \(Nu_{o}\) and \(\Delta p_{o}\) are the Nusselt number and the pressure drop in a smooth panel. Values higher than 1 suggest an increase in heat transfer or reduction in pressure drop when compared to a smooth panel, and values lower than 1 suggest the opposite.