# Optimal Design of Wearable Micro Thermoelectric Generator Working in a Height-Confined Space
## 3 Results and Discussion  
### Geometry Design of Finned Heat Sink  
Based on the relevant formula regarding convective heat transfer of the fin, as detailed in Section 2.3, it can be inferred that the temperature difference between the fin base and the ambient surroundings (\(\theta=T_{\text{base}}-T_{\text{m}}\)) is a key factor that influences the heat dissipation capacity of the system, as illustrated in Figure 2a. To investigate the impact of various geometric parameters on \(\theta\), we have established models with different TE leg widths (\(w\)), leg heights (\(l\)), and the number of leg pairs (\(N\)). As shown in Figure 2b, when the number of thermoelectric legs remains constant, the temperature difference \(\theta\) increases with the widening of \(w\), while it decreases with the heightening of \(l\). Additionally, when the number of TE leg pairs increases from 2 to 16, the temperature difference \(\theta\) demonstrates a notable increase. Within the range of geometric dimensions of the corresponding device, \(\theta\) ranges from approximately 3 to 8 K.  
Figure 2: (**a**) Detailed drawing of corresponding parameters in TEG; (**b**) effect of TEG geometry parameter (leg width \(w\), leg height \(l\), number of leg pairs) on the temperature difference \(\theta\) between fin base temperature and ambient temperature.  
We conducted further analysis on the effect of the temperature difference \(\theta\) on the heat dissipation capacity of the heat sink, as well as the equivalent convective heat transfer coefficient \(h_{\text{eff}}\) and convective heat transfer coefficient per unit area \(h_{\text{a}}\), respectively (Figures 3a, S3 and S4). In order to normalize the results, we used the equivalent convective heat transfer coefficient \(h_{\text{eff}}\) as an indicator of the heat dissipation capacity of heat sinks with different geometric parameters. The difference of corresponding surface area between \(h_{\text{eff}}\) and \(h_{\text{a}}\) is represented by different colors in Figure 3a. We found that the \(h_{\text{eff}}\) of heat sinks with different fin numbers (\(n\)) (Figure 3b) and fin heights (\(H\)) (Figure 3c) are basically the same when \(\theta\) increases from 4 K to 7 K. Therefore, the influence of the geometric size of the device on the heat dissipation capacity of the heat sink is negligible. In addition, we investigated the impact of the number of fins and fin heights on \(h_{\text{eff}}\) (Figure 3d,e). As presented in Figure 3d, with the increase of the number of fins, the \(h_{\text{eff}}\) initially increases to a certain value and then decreases sharply. The number of fins with a sudden drop decreases as the fin height increases. Moreover, Figure 3e shows that \(h_{\text{eff}}\) initially increased slowly with increasing fin height, sharply decreased at a certain point, and then increased slowly again. The above related laws are in line with the related theories of heat transfer. The fin height of the fin structure parameters of the fin radiator can effectively affect the natural convection heat transfer area, and the number of fins can effectively affect the air flow between the fins during natural convection heat dissipation [33].  
Finally, we have conducted a comprehensive study on the impact of geometric parameters of fins on the temperature difference (\(\theta\)) and heat dissipation capacity of a heat sink, as presented in our results. We have identified the optimal combination of geometric parameters of fins for achieving the highest heat dissipation capacity of the heat sink. Specifically, we have sorted out the optimal number of fins for the highest equivalent convective heat transfer coefficient (\(h_{\text{eff}}\)) corresponding to different fin heights, as illustrated in Figure 3f. Our findings reveal that the optimal number of fins gradually decreases as the fin height increases. These results provide valuable insights into the design and optimization of heat sinks and the impact of geometric parameters on the heat dissipation capacity.  
Figure 3: _Cont._