## 4 Results and Discussion
### Air-Cooling BTMS with the AFHS of Gradient Porosity Designs  
Compared with the situation without AFHS, using homogeneous AFHS could remarkably enhance the thermal performance, but pressure drop and temperature differences were even greater. Therefore, the gradient design in aluminium foam was considered to reduce flow resistance and improve temperature uniformity. Considering that the pore density has little effect on the heat transfer characteristics, the gradient porosity designs were adopted in subsequent analyses, and the pore density was all 10 PPI. Figure 8: (**a**) Temperature control coefficient \(\alpha\), (**b**) temperature uniformity coefficient \(\beta\). Figure 9: FOM versus mass flow rate. The gradient of porosity is controlled by the following equation:  
\[ \begin{split}\varepsilon&=\varepsilon_{\text{min}}+( \varepsilon_{\text{max}}-\varepsilon_{\text{min}})(\frac{x}{T})^{m}\\ \varepsilon&=\varepsilon_{\text{max}}-(\varepsilon_{ \text{max}}-\varepsilon_{\text{min}})(\frac{x}{T})^{m}\end{split}  \tag{17}\]  
where \(\varepsilon_{\text{max}}\) and \(\varepsilon_{\text{min}}\) are the maximum and minimum porosity, respectively, in the current study \(\varepsilon_{\text{max}}\) = 0.92 and \(\varepsilon_{\text{min}}\) = 0.68; \(m\) represents the parameter controlling the porosity; \(x/l\) is the normalised distance. The variation of the porosity gradient of aluminium foam for increasing and decreasing patterns is shown in Figure 10. For the porosity-increasing gradient pattern (PIGP), m = 0.1, 1, and 10 correspond to the equivalent homogeneous porosity levels of 0.9, 0.8 and 0.7, respectively, while for the porosity-decreasing gradient pattern (PDGP), m = 0.1, 1, and 10 correspond to the equivalent homogeneous porosity levels of 0.7, 0.8 and 0.9, respectively. The gradient porosity can be realised by the user-defined function UDF in Ansys-fluent, and the change of coefficient of viscous resistance terms and inertial resistance terms caused by the change of gradient porosity can be calculated by the Ergun formula [37]. The porosity is set to change perpendicular to the flow direction (\(Z\)-axis direction) and along the flow direction (\(X\)-axis direction), and \(C_{1}\) and \(C_{2}\) change accordingly with porosity. 3.1 AFHS of Gradient Porosity Designs in the \(Z\)-Axis Direction (Perpendicular to the Flow Direction)  
Figure 11 shows the pressure drop \(P_{d}\) of AFHS with gradient porosity designs in the \(Z\)-axis direction (perpendicular to the flow direction). As can be seen, there was no difference in \(P_{d}\) between using PIDP and PDGP at the same equivalent porosity, indicating that the two patterns have the same influence on flow performance in the \(Z\)-axis direction. Moreover, the \(P_{d}\) of the two patterns was lower than that of the homogeneous porosity AFHS at the same equivalent porosity, which indicates that the gradient porosity designs in the \(Z\)-axis direction can improve the flow performance. The reason for this phenomenon is that when the porosity change direction is perpendicular to the flow direction, the air tends to flow to the part with large porosity at a higher speed, so the overall flow resistance decreases and the pressure drop decreases. Figure 10: Variation of porosity gradient versus normalised distance: (**a**) porosity-increasing gradient pattern, (**b**) porosity-decreasing gradient pattern. Figure 12 shows the battery temperature distributions with AFHS in the two gradient patterns at \(q_{m}=5\) g\({}^{-1}\), but the difference between the two cases is not obvious. Figure 13a shows the average temperature \(T_{av}\) of the battery with the AFHS of gradient porosity designs in the \(Z\)-axis direction. We can see that the \(T_{av}\) of the battery with AFHS of PIGP was almost the same as that with equivalent homogeneous porosity AFHS, while the \(T_{av}\) of the battery with AFHS of PDGP was higher than that with equivalent homogeneous porosity AFHS.