## 4 Numerical Simulation and Result Analysis
### Effect of Microchannel Roughness on Heat Dissipation Capability  
The double-layer reflow microchannel heat sink in this paper was fabricated by laser 3D printing technology, and the roughness of the inner wall of the pipe was unavoidable. The roughness may increase the heat exchange area between the heat sink and the cooling mass and may also change the flow state of the cooling mass near the wall. Therefore, it is necessary to study the effect of roughness on the heat dissipation capacity of the microchannel heat sink. Roughness is involved in the solution of NS equation in the form of wall function. When roughness is considered, the logarithmic distribution law of wall function is:  
\[u^{+}=\frac{u_{p}}{u^{*}}=\frac{1}{k}In\bigg{(}E\frac{y_{p}u^{*}}{\
u}\bigg{)} -\Delta B \tag{7}\]  
where \(\Delta B\) is a function of roughness. A dimensionless number \(K_{S}^{+}=\frac{K_{s}u^{*}}{\
u}\) is introduced for roughness, where \(K_{s}\) is roughness (mm):  
Figure 10: Variation curve of inner pressure of double-layer microchannel heat sink tube with coolant flow under different hydraulic diameters.  
Figure 9: Variation curve of thermal resistance of double-layer microchannel heat sink with coolant flow under different hydraulic diameters.  
(1) 0 \(<K_{S}^{+}=\frac{K_{S}u^{*}}{\
u}<\) 2.25, \(\Delta B=0\), the viscous sublayer covers the whole roughness peak, (2) 2.25 \(<K_{S}^{+}=\frac{K_{S}u^{*}}{\
u}<\) 90,  
\[\Delta B=\frac{1}{k}In\left[\frac{K_{S}^{+}-2.25}{87.75}+C_{s}K_{s}\right] \times sin\big{[}0.4258 In\big{(}K_{S}^{+}-0.811\big{)}\big{]} \tag{8}\]  
(3) \(K_{S}^{+}=\frac{K_{S}u^{*}}{\
u}>\) 90,  
\[\Delta B=\frac{1}{k}In\big{[}1+C_{s}K_{s}\big{]} \tag{9}\]  
peak is almost completely exposed in the turbulent region, where \(C_{s}\) is the roughness constant, which is generally 0.5.  
Figure 11 shows the influence curve of the inner wall roughness of the microchannel on the thermal resistance of the chip. It can be seen that when the channel wall roughness was 0-20 \(\upmu\)m, the thermal resistance of the chip did not change significantly and only decreased from 0.258 \({}^{\circ}\)C/W to 0.256 \({}^{\circ}\)C/W. However, when the microchannel wall roughness exceeded 20 \(\upmu\)m, the curve dropped slowly from 0.256 \({}^{\circ}\)C/W at 20 \(\upmu\)m to 0.234 \({}^{\circ}\)C/W at 90 \(\upmu\)m with a decrease of 8%. The result shows that the increase in channel wall roughness could reduce the thermal resistance of the chip. The reason might be when the roughness was small, the heat exchange area and the flow state of the cooling mass did not change significantly compared with the smooth wall surface. With the increase of micro-channel wall roughness, on the one hand, the heat transfer area between the cooling medium and heat sink is increased, on the other hand, the degree of disturbance and turbulence of the cooling medium is increased, and the heat dissipation effect was enhanced. However, the roughness continued to increase, which means that the surface grain was less constrained, and the surface was prone to plastic deformation [20], which is easy to cause pipeline blockage caused by erosion and corrosion of the cooling medium, affecting the service life of the microchannel heat sink.  
According to the above analysis, the optimal design of the three microchannel radiator structures is double-layer backflow, and the optimal parameters are as follows: Hydraulic diameter of the microchannel is 0.5 mm, flow rate of the cooling medium is 24 L/h, inner wall roughness is 0-20 \(\upmu\)m, and the thermal resistance of the chip is 0.256 \({}^{\circ}\)C/W. The temperature cloud diagram is shown in Figure 12.  
Figure 11: Relationship of thermal resistance versus pipe roughness.