## 4 Results and Discussion
### Air-Cooling BTMS with the AFHS of Gradient Porosity Designs
This is mainly because the porosity near the heating surface is large with the PDGP, and the solid skeleton volume in the porous medium area is small, which makes the heat conduction between the aluminium foam and the heated surface deteriorate, and ultimately leads to a weakening in the heat transfer effect. Figure 13b shows the \(T_{d}\) of the battery with the AFHS with gradient porosity designed in the \(Z\)-axis direction. The \(T_{d}\) of the battery with AFHS of PIGP was almost the same as that with the equivalent homogeneous porosity AFHS. However, using the AFHS of PDGP leads to \(T_{d}\) being slightly less than that at the equivalent porosity. Figure 11: Pressure drop with AFHS of gradient porosity designs in the \(Z\)-axis direction. Figure 12: Temperature distributions of battery: (**a**) using AFHS with porosity-increasing gradient pattern m = 1, (**b**) using AFHS with porosity-decreasing gradient pattern m = 1. Figure 14a shows the temperature control coefficient \(\alpha\) versus pumping power \(\Omega\) with AFHS with gradient porosity designed in the \(Z\)-axis direction. At low porosity level, the \(\alpha\) was slightly greater over the entire pumping power range. Within a constant porosity level, there was no obvious difference in the \(\alpha\). However, it can be seen from the enlarged figure, when the \(\alpha\) with the homogeneous porosity AFHS was equal to that with AFHS of PIGP, the \(\Omega\) required for AFHS of PIGP was smaller. When the same \(\Omega\) was required for AFHS of PIGP and PDGP, the AFHS of PIGP had a lower \(T_{av}\), resulting in the AFHS of PIGP having a larger \(\alpha\) than that of PDGP. At low pumping power (\(\Omega<0.2\) W), the AFHS of PIGP with \(m=0.1\) had the highest \(\alpha\). Figure 14b shows the temperature uniformity coefficient \(\beta\) versus pumping power \(\Omega\) with the AFHS with gradient porosity designed in the \(Z\)-axis direction. As the mass flow rate increased, the \(T_{d}\) of the battery with AFHS of different gradients became closer, so the maximum value of \(\beta\) was almost the same in the whole pumping power range. Since the \(T_{d}\) of the battery with AFHS of PDGP was smaller than that of PIGP under the same equivalent porosity level, it led to a slightly larger \(\beta\) with PDGP. Figure 15 shows FOM versus mass flow rate with the AFHS with gradient porosity designed in the \(Z\)-axis direction. Compared with the homogeneous porosity AFHS, the AFHS with gradient porosity designed in the \(Z\)-axis direction (perpendicular to the flow direction) reduces the pressure drop. Therefore, with little difference in heat transfer performance, the AFHS of PIGP required less pumping power and resulted in a larger FOM. Meanwhile, the AFHS of PDGP had a small convective heat transfer coefficient  
Figure 14: (**a**) Temperature control coefficient \(\alpha\), (**b**) temperature uniformity coefficient \(\beta\) (with AFHS with gradient porosity designed in the \(Z\)-axis direction). Figure 13: (**a**) Average temperature of battery, (**b**) temperature difference of battery (with AFHS with gradient porosity designed in the \(Z\)-axis direction). leading to a smaller FOM. Considering the heat convection and energy consumption, an AFHS with PIGP in the \(Z\)-axis direction is superior to those with homogeneous porosity.