## 2 Modeling
### Mathematical Models and Correlations
#### 2.2.2 Mini-Channel Evaporator Model  
Figure 4 displays both the external photograph and internal structure diagram of the mini-channel evaporator utilized in this study. The mini-channel evaporator is a pivotal component in the PCTCS. Its internal channel is designed as a straight channel. Fabrication of the evaporator was accomplished using a blend of precision machining techniques. The process was initiated by independently processing the base and cover plates. The base plate underwent a milling procedure,  
Figure 4: The exterior photo and internal structure diagram of the mini-channel evaporator. during which mini channels with a width of 1 mm were carefully carved. Following this, both elements were integrated through vacuum brazing, forming the finished structure of the evaporator. After the machining process, we introduced a surface finishing phase aimed at reducing any surface roughness that might have been introduced during the previous stages. * Pressure drop for single-phase and two-phase flow  
In conventional data center arrangements, servers are typically situated horizontally within their racks, which subsequently dictates the horizontal orientation of the CPUs housed within these servers. Correspondingly, the mini-channel evaporator within these servers also maintains a horizontal orientation. However, it is noteworthy that real-world installation procedures could yield slight deviations in tilt. These deviations, although existent, are typically negligible. Therefore, in the modeling, we disregard their potential influence, presuming the evaporator's channels to be horizontally aligned. Given this horizontal alignment of the mini-channel evaporator's flow channels and the fact they undergo phase change, we have chosen to disregard the gravitational pressure drop. However, the accelerated pressure drop is considered. The frictional pressure drop for single-phase and two-phase flow in the mini-channel evaporator is calculated utilizing Equations (3)-(5) and (7), while the accelerational pressure drop is determined using Equation (11):  
\[\mathrm{Pda}=\mathrm{G}^{2}.\left\{\left[\frac{\mathrm{x}^{2}}{\alpha\cdot \mathrm{p}_{\mathrm{g}}}+\frac{\left(1-\mathrm{x}\right)^{2}}{\left(1-\alpha \right)\cdot\mathrm{p}_{1}}\right]_{\mathrm{out}}-\left[\frac{\mathrm{x}^{2} }{\alpha\cdot\mathrm{p}_{\mathrm{g}}}+\frac{\left(1-\mathrm{x}\right)^{2}}{ \left(1-\alpha\right)\cdot\mathrm{p}_{1}}\right]_{\mathrm{in}}\right\} \tag{11}\]  
In the equation, \(\mathrm{G}\) symbolizes the mass flux of the refrigerant, and the calculation of \(\alpha\) is derived from Equation (10). * Heat transfer model  
The CPU's surface and the mini-channel evaporator's bottom surface were tightly affixed and coated with thermally conductive silicone grease, facilitating the heat transfer from the CPU to the refrigerant in the evaporator. Specifically, we utilized DOWSIL TC-5026 thermal grease, a product of DOW, to ensure efficient thermal conduction. The boundary condition for heat transfer is treated as a constant heat flux. The heat transfer process is computed using Equation (12):  
\[\mathrm{Q}_{\mathrm{eva}}=\frac{\left(\overline{\mathrm{T}}_{\mathrm{r}}- \mathrm{T}_{\mathrm{cpu}}\right)}{\frac{1}{\mathrm{h}\alpha_{\mathrm{eva}} \cdot\mathrm{A}_{\mathrm{tol}}}+\frac{\mathrm{t}_{\mathrm{co}}}{\mathrm{A}_{ \mathrm{wall}}\cdot\mathrm{A}_{\mathrm{co}}}+\frac{\mathrm{t}_{\mathrm{g}}}{ \mathrm{A}_{\mathrm{wall}}\cdot\mathrm{A}_{\mathrm{g}}}} \tag{12}\]  
where \(\overline{\mathrm{T}}\) is the arithmetic mean temperature of the refrigerant, \(\mathrm{T}_{\mathrm{cpu}}\) is the CPU temperature, \(\mathrm{t}_{\mathrm{co}}\) is the bottom plate's thickness, \(\mathrm{t}_{\mathrm{g}}\) is the thermal grease's thickness, \(\mathrm{\lambda}_{\mathrm{co}}\) is the bottom plate's thermal conductivity, \(\mathrm{\lambda}_{\mathrm{g}}\) is the thermal conductivity of the thermal grease, \(\mathrm{A}_{\mathrm{tol}}\) is the effective heat transfer area, and \(\mathrm{A}_{\mathrm{wall}}\) is the area of the evaporator's bottom surface.