## 3 Results and Discussions
### Effects from Blockage on Radiation Heat Flux in Practical Battery Packs  
In practical battery packs, depending on the separation distance d, thermal radiation from a trigger cell to another cell in the pack may be blocked by adjacent cells. For example, as shown by the schematic plot in Figure 12, radiation flux from O1 to O2 is blocked by the presence of O3, in that any photon emitted beyond point A on the surface of O1 is unable to directly reach the surface of O2. In general, the interest is in determining the extent of radiative exchange that will occur between any pair of cells in a battery pack, how this is influenced by the presence of other cells, and how the cells are arranged in the pack. Here, we considered two specific packing arrangements: equal-spacing homogenous or orthogonal. Following [PERSON]'s classical cross string theory [37], the view factor for arbitrary 2D geometry can be evaluated as:  
\[F_{12}=\frac{\sum\limits_{\mbox{$\mathit{crossed\;strings}$}}-\sum\limits_{ \mbox{$\mathit{uncrossed\;strings}$}}}{2\times\mbox{$\mathit{source\;string}$}} \tag{6}\]  
For the case of homogeneous arrangement of cells, we can express the view factor between Cells 1 and 2, F12 as:  
\[\begin{array}{l}F_{12}=\frac{2\times\left(\left.\left(\bar  
the other limit with gradually increased cell separations, the effect from blockage gradually vanishes as expected, and hence Equations (5) and (7) reduce to the same results.  
Instead of being homogenously arranged, a more widely adopted practical real geometry of battery pack is the orthogonal configuration, as shown in Figure 14. Compared to the homogeneous arrangement, this configuration has reduced energy density for a given packing volume but provides more convenience for sequential and parallel electrical interconnection, as well as the provisioning of cooling. In this configuration, assuming the hot radiation source is the center Cell 5, the radiation heat flux from Cell 5 to any one of its immediate neighbors 2, 4, 6, and 8 is free of blockage, and, using Equation (5), the view factor may be written as  
\[F_{52}=F_{54}=F_{56}=F_{58}=\frac{\sqrt{h^{2}-4}-h+2 arcsin\left(\frac{2}{h} \right)}{2\pi} \tag{8}\]  
Figure 14: A common orthogonal arrangement of cylindrical battery packs with radius R and cell center distance H, h = H/R. O1 to O9 denote the center locations of each cell. Figure 13: Comparison of view factors with and without the blockage between adjacent cells in equally spaced homogenous array. Radiative heat flux from the center Cell 5 to any one of the other Cells 1, 3, 7, and 9 can potentially be blocked, depending on the average separation among the cells. Due to symmetry, one can easily recognize that these view factors are equal:  
\[F_{51}=F_{53}=F_{57}=F_{59} \tag{9}\]  
Given the definition of view factor, the sum of view factor from Cell 5 to all reachable surfaces must be unity with sufficient small cell gaps. In other words, no radiation emission from Cell 5 is able to escape from all the cells around it. We hence can obtain:  
\[F_{51}=F_{53}=F_{57}=F_{59}=\frac{1}{4}-\frac{\sqrt{h^{2}-4}-h+2\arcsin\left( \frac{2}{h}\right)}{2\pi} \tag{10}\]  
With these view factors analytically obtained, the radiation heat flux from the center Cell 5 to all its adjacent cells with and without blockage can be easily evaluated. A comparison of the view factor in Equations (8) and (10) is presented in Figure 15, with Cells 2, 4, 6, and 8 as the closest cells relative to the trigger Cell 5, and 1, 3, 7, and 9 being next-closest cells. It is very interesting to see that there is a threshold cell separation distance h-2.7, within which the closest cells receive higher radiation while beyond which the corner cells receive higher radiation. For 18650 cell arrays, this limiting aspect ratio of h = 2.7 corresponds to a cell gap of 6.3 mm. In the other limit of h = 2, where cells directly touch with their closest neighbors, the radiation heat flux on the closest neighboring cell is more than two times higher than that received by the corner ones.