## 4 Results and Discussion
### Thermal Characteristics
At fixed \({X_{T}}^{*}=2\) and variable \({X_{L}}^{*}=1.25\), 1.5 and 2, the dimensionless solid temperature and dimensionless fluid temperature in the same position of various heat sinks have not varied with \({X_{L}}^{*}\), meaning \({X_{L}}^{*}\) is not a sensitive parameter to the overall temperature distribution. Figure 14 shows the Nusselt numbers of various rows of heat sink array at different \({X_{T}}^{*}\) and \({X_{L}}^{*}\) values of Samples 1-5 when Reynolds number \(Re=500\). The number of pin fins of Sample 3 heat sink is 9 \(\times\) 9, the porosity is 0.75. This heat sink array has a higher Nusselt number among all the heat sink arrays, and the Nusselt number of various heat sink arrays increases with decreasing \({X_{T}}^{*}\), but not varies with \({X_{L}}^{*}\). This result conforms to the dimensionless temperature contours in Figures 12 and 13. Figure 14: Nusselt number distributions along the main stream direction for various heat-sink arrays: (**a**) Sample 1; (**b**) Sample 2; (**c**) Sample 3; (**d**) Sample 4; and (**e**) Sample 5. Figure 15 shows the relationship of average Nusselt number to the porosity and number of pin fins of heat sink array when Reynolds number \(Re=500\). In order to clarify the influences of porosity and number of pin fins on the average Nusselt number, the heat transfer characteristics of heat sink arrays with higher porosity (Sample 6 and Sample 7, see Table 1) and heat sink array with more pin fins (Sample 8, see Table 1) were also simulated. Figure 15a indicates that when the porosity of heat sink with 9 \(\times\) 9 pin-fins increases from 0.358 to 0.556, the average Nusselt number changes slightly. When the porosity increases from 0.556 to 0.750, the average Nusselt number is increased significantly by 11.7% (for cases with \(X\!r^{*}\) = 1.25)-24.8% (for cases with \(X\!r^{*}\) = 2). However, when the porosity increases continuously from 0.750, the average Nusselt number decreases obviously. This is because the overall heat exchange area decreases with the pin diameter of pin-fin heat sink, which is adverse to convection heat transfer, and the upward thermal conduction capability of pin-fins declines, thus reducing the fin efficiency. As a result, the average Nusselt number decreases on the contrary. Figure 11b indicates that when the porosity is fixed at 0.75, increasing the number of pin fins continuously can increase the average Nusselt number. In order to keep the fixed porosity, the pin diameter must be reduced while the number of pin fins is increased; at this point, the overall heat transfer area still increases with the number of pin fins, which is advantageous to the convection heat transfer. According to empirical Equations (7) and (8) suggested by [PERSON] _et al._[24], the decrease in pin diameter increases the convection heat transfer coefficient. Therefore, the average Nusselt number increases with the number of pin fins. However, at the Reynolds number \(Re=500\), when the number of pin fins is large, the average Nusselt number increases with the number of pin fins slowly. The pressure drop of the heat sink array increases with the number of pin fins, thus preventing the air flow from entering the heat sink effectively for cooling. Figure 15: Average Nusselt numbers of pin-fin heat-sink array: (**a**) effect of heat-sink porosity and (**b**) effect of pin-fin numbers.