# Design of Novel Cooling Systems Based on Metal Plates with Channels of Shapes Inspired by Nature
## 4 Results and Discussion
### Heat Transfer
#### 4.2.2 Steady State Heat Transfer
Similar values of the maximum temperature of the plate under steady state conditions were reached for the crateriform and the salverform configurations due to the higher concentration of points, with higher distance to the channel for the salverform configuration, close to the plate center, than for the crateriform configurations, for which these distant points to the channel are distributed at the center and the four corners of the plate. Once more, the lowest values of the maximum temperature of the plate attained for steady state cooling were obtained for the cruciform configuration, confirming again the higher capability of this channel shape to cool down the plate and the applicability of the simple geometrical analysis proposed to predict the optimal shape for heat transfer purposes. Considering the linear increase in the maximum temperature of the plate with the heat flux supplied, i.e., \({q_{\text{in}}}^{\prime\prime}={q_{\text{in}}}/{L^{2}}\), the dimensionless temperature proposed by [PERSON] et al. [42], \(\overline{T}\), was used to quantify the effect of the shape of the channel on the performance of the cooling plate. The dimensionless temperature \(\overline{T}\) is defined as follows:  
\[\overline{T}=\frac{{T_{\text{max}}}-{T_{\text{w}}}}{{q_{\text{in}}^{\prime \prime}s}/k} \tag{22}\]  
where \({T_{\text{max}}}\) is the maximum temperature of the plate, \({T_{\text{w}}}\) is the temperature of the cooling water, \({q_{\text{in}}}^{\prime\prime}\) is the input heat flux, \(s\) is the plate thickness, and \(k\) is the thermal conductivity of the solid plate. The results for the dimensionless temperature can be found in Table 2. A similar value of \(\overline{T}\) was obtained for the standard configurations, i.e., square and fork shapes of the channel. However, the value of the dimensionless temperature was lower for all the novel configurations proposed. The average dimensionless temperature obtained for the common configurations can be reduced by 11.3, 9.8, and 21.1% using the crateriform, salverform, and cruciform configurations, respectively. Therefore, a significant reduction in the dimensionless temperature can be attained by using the novel configurations, especially for a cruciform shape of the cooling channel. Figure 9: (**a**) Mean and (**b**) maximum temperature of the plate as a function of the thermal power input for all configurations. From a practical point of view, it is also important to evaluate the difference between the maximum and the minimum temperature of the plate as a measure of the uniformity of the temperature distribution in the plate. The maximum variation in temperature of the plate, defined as the difference between the maximum temperature \(T_{\text{max}}\) and the minimum temperature \(T_{\text{min}}\), is plotted versus the thermal power input in Figure 10a for all the configurations studied. The maximum temperature variation obtained for the square and fork shape channels collapse, increasing linearly with the thermal power input. A substantial reduction in the maximum variation in temperature can be reached using the proposed novel shapes for the channel. Slightly higher values for the maximum variation in temperature in the plate were obtained for the crateriform compared to the salverform configuration, while the cruciform channel resulted again in the lowest temperature variations among all configurations analyzed. The crateriform and salverform configurations reduced the maximum temperature variation of the standard configurations, i.e., the square and fork configuration, by roughly 15%, whereas a reduction of approximately 28% could be attained by the cruciform configuration. These results agree with the maximum distances obtained from the geometrical analysis of the configurations and the main tendencies could be again directly inferred from the simple geometrical study carried out in Section 4.1.