# Design of Novel Cooling Systems Based on Metal Plates with Channels of Shapes Inspired by Nature
## 4 Results and Discussion
### Heat Transfer
#### 4.2.1 Transient Heat Transfer  
The cooling capacity of the different configurations was evaluated by solving a transient heat transfer problem with the finite differences numerical model. For the transient cooling study, no heat flux on the plate surface was considered, thus \(q_{\text{in}}\) in Equation (1) is zero. The initial temperature of the whole plate was fixed to 55 \({}^{\circ}\)C and heat was dissipated to ambient air and to water flowing through the channel. The temperature of both ambient air and water in the channel was set at 25 \({}^{\circ}\)C. Therefore, for a sufficiently long time, the whole plate would reach a final temperature of 25 \({}^{\circ}\)C.  
A total physical time of 60 s of the cooling process was studied for each configuration. Figure 5 shows the evolution of temperature across the plate for all configurations as time progresses during the first 15 s of the transient cooling process, showing snapshots of the temperature distribution in time intervals of 2 s. After 1 s of cooling, the temperature distribution is similar in all cases, with most of the plate at the initial temperature of 55 \({}^{\circ}\)C and a clearly visible channel shape at a lower temperature. However, as time progresses, heat is principally dissipated by forced convection to the water-cooling channel, and the regions close to the channel cool down by conduction. Remarkable differences can be observed in the temperature distribution of the plate for the different configurations studied as transient cooling evolves. The heat dissipation occurs from a large region of high temperature at the center of the plate for the square configuration, whereas heat dissipation mainly takes place from the corners closest to the inlet and outlet of the channel for the fork configuration, which are the points with the largest distance to the channel according to the geometrical analysis. In the case of the crateriform configuration, heat dissipation happens from the center and the four corners of the channel, while the dissipation behavior of the salverform channel plate is similar to that of the square configuration, showing a higher temperature zone at the plate center. Heat dissipation seems to be faster and more homogeneous for the cruciform channel, which is apparently the best option to \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Configuration** & **r\({}_{\text{max}}\) (cm)** & **r\({}_{\text{m}}\) (cm)** & **\(\sigma_{r}\) (cm)** \\ \hline Square & 5.34 & 1.26 & 1.17 \\ Fork & 5.34 & 1.14 & 0.98 \\ Crateriform & 4.93 & 1.22 & 1.07 \\ Salverform & 4.29 & 1.13 & 0.94 \\ Cruciform & 3.68 & 1.11 & 0.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristic values of the geometrical analysis of the different configurations. refrigerate a hot plate by circulating a cold fluid. Noticeably, the temperature distribution maps depicted in Figure 5 are similar to the contour plots of the distance to the channel included in Figure 4, which was an expected result since heat dissipation is affected by the distance to a sink point, i.e., the cooling channel. This confirms the usefulness of the simple geometrical analysis discussed in Section 4.1. Figure 5: Time evolution of the temperature distribution across the plates for each configuration during the first 15 s of a transient cooling process. The time evolution of the average and maximum temperature of each configuration can be found in Figure 6 for a time period of 60 s. As anticipated from the results of the geometrical analysis of the configurations, the evolution of the average temperature of all configurations is similar, obtaining slightly lower values for the cruciform shape of the channel, in agreement with the results of the geometrical study. In all cases, an inverse exponential approximation of the average temperature of the plate from the initial temperature to the ambient and cooling fluid temperature was obtained.