## 3 Results and Discussion  
### Overall Heat Transfer Performance for Different Flow Channels  
The core volume goodness factor (\(\eta_{g}h_{std}a\)) and \(E_{std}a\) were proposed by [PERSON] and [PERSON] [29], and they were used to comprehensively evaluate the flow channels of three different structures, including equal, trapezoidal and circular-arc cross-sections, as shown in Figure 4.  
\[\eta_{g}h_{std}a=\frac{c_{p}\mu}{Pr^{2/3}}\frac{4\sigma}{d_{h}^{2}}jRe \tag{16}\]  
\[E_{std}a=\frac{\mu^{3}}{2g_{c}\rho^{2}}\frac{4\sigma}{d_{h}^{4}}fRe \tag{17}\]  
The significance of using the core volume goodness factor is that it can compare the overall heat transfer performance for different flow channels and different heat transfer areas. \(\eta_{g}h_{std}a\) is energy consumption per unit volume while the temperature changes by 1 K in a heat exchanger, and \(E_{std}a\) is the friction power consumed by the unit volume of the heat exchanger. \(E\) is the fluid pumping power per unit surface area, \(g_{c}\) is the proportionality constant in [PERSON]'s second law of motion, \(\eta_{g}\) is the extended surface efficiency on one  
Figure 4: Comparison of core volume goodness factor of three flow channels.  
fluid side of the extended surface heat exchanger and \(\sigma\) is the ratio of the free flow area to the frontal area. The subscript \"_std_\" is a benchmark and follows the ARI standard [30,31]. For constant \(E_{std}\alpha\), a surface having a high plot of \(\eta_{o}h_{std}\alpha\) vs. \(E_{std}\alpha\) is characterized as the best from the viewpoint of the heat exchanger volume; that is, under the same \"\(E_{std}\alpha\)\", a larger value of \(\eta_{o}h_{std}\alpha\) means that only a small heat exchanger volume is required at the same capacity of heat transmission. During the experiment, the Reynolds number (_Re_) on the air side ranged from 1200 to 5100.  
Figure 4 shows the variation curve of the \"\(\eta_{o}h_{std}\alpha\)\" value for three different-shaped flow channels. In the figure, the slope angle of the trapezoidal cross-section flow channel is 40\({}^{\circ}\), which is because the corresponding overheat transfer performance of this angle is best. It can be seen from the figure that the \"\(\eta_{o}h_{std}\alpha\)\" values of the three structures increase with the increase in \"\(E_{std}\alpha\)\". Under the same \"\(E_{std}\alpha\)\", the circular-arc-shaped flow channel has the best overall heat transfer performance, while the equal cross-section flow channel has the worst overall heat transfer performance. The sorting of the overall heat transfer performance for the three channels is circular-arc cross-section-shaped, trapezoidal cross-section-shaped and equal cross-section-shaped. The value of \(\eta_{o}h_{std}\alpha\) in the circular-arc-shaped flow channel is about 9-26.2% larger than that in the equal cross-section flow channel. In fact, when \(E_{std}\alpha\) is maximum, the value of the \(\eta_{o}h_{std}\alpha\) of the circular-arc-shaped flow channel is about 26.2% larger than that in the equal-section flow channel. The \(\eta_{o}h_{std}\alpha\) of the circular-arc flow channel is 3.6-11.8% better than that of the trapezoidal flow channel. This indicates that the curved flow channel has better overall heat transfer performance than that of the straight flow channel. It is true that the volume goodness factor is pretty much the same as the trapezoidal case with only a small percentage enhancement at a low Reynolds number, as shown in Figure 4. However, at high Reynolds numbers, the overall heat transfer performance of the trapezoidal flow channel was about 20% higher than that of the equal cross-section flow channel; that is to say, the trapezoidal case is mainly used at a high Reynolds number.  
Performance of Heat Transfer and Flow Characteristics in Different Circular-Arc-Shaped Flow Channels