## 2 Modeling
### Performance Evaluation Method  
To carry out an efficient analysis and comparison of the PCTCS performance under varying operational conditions, we assessed its function based on several key performance indicators. These included the energy efficiency ratio of the pump (EERp), the total pressure drop (Pdt), the maximum temperature difference between cold and heat sources (Tdt), the average temperature disparity between the first and second CPUs (Td\({}_{\rm cpu}\)), the temperature variances of the first and second CPUs (S\({}_{\rm cpu1}\)\({}^{2}\), S\({}_{\rm cpu2}\)\({}^{2}\)), and the heat transfer efficiency of the condenser (\(\eta_{\rm con}\)). These indicators are pivotal in evaluating the system's performance. For instance, Tdt, Td\({}_{\rm cpu}\), and S\({}_{\rm cpu}\)\({}^{2}\) are critical for analyzing the system's temperature performance, while EERp and Ppump are significant for assessing its energy efficiency. Specifically, EERp is calculated as the total heat dissipation divided by the power utilized by the pump, providing a measure of energy efficiency. In contrast, \(\eta_{\rm con}\) represents the heat transfer efficiency of the condenser, determined by the ratio of the actual temperature difference between the cold and hot ends to the maximum possible temperature difference in the heat exchanger. This thorough approach allows for a comprehensive understanding of the PCTCS under different operating conditions.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Refrigerant** & **Pump Power (W)** & \multicolumn{2}{c}{**Total Pressure Drop**} & \multicolumn{2}{c}{**Average CPU**} \\
**Flow** & & & & **(kPa)** & & **Temperature (\({}^{\circ}\)C)** \\ \cline{2-7}
**(m\({}^{3}\)/h)** & **Exp.** & **Sim.** & **Exp.** & **Sim.** & **Exp.** & **Sim.** \\ \hline
0.12 & 36 & 33 & 57.8 & 54.7 & 62.3 & 60.3 \\
0.15 & 42 & 40 & 75.0 & 75.6 & 62.5 & 61.1 \\
0.18 & 51 & 48 & 101.5 & 99.1 & 62.6 & 62.3 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of the results between the simulation and experiment of different refrigerant flows.  
Figure 7: Comparison of the results between the simulation and experiment of different CPU load and cooling water temperatures.  
The parameters were calculated as Equations (24)-(29):  
\[\mathrm{EERp}=\frac{Q_{\mathrm{con}}}{P\mathrm{pump}}=\frac{Q_{\mathrm{eva}}+P \mathrm{pump}}{P\mathrm{pump}} \tag{24}\]  
\[\mathrm{Pdt}=P\mathrm{dpump}=P_{2}-P_{1} \tag{25}\]  
\[\mathrm{Tdt}=\mathrm{max}(\overline{T}_{\mathrm{cpu1}},\overline{T}_{\mathrm{ cpu2}})-\mathrm{T}_{\mathrm{win}} \tag{26}\]  
\[\mathrm{Td}_{\mathrm{cpu}}=\overline{T}_{\mathrm{cpu1}}-\overline{T}_{\mathrm{ cpu2}} \tag{27}\]  
\[\mathrm{\eta}_{\mathrm{con}}=\frac{\mathrm{max}[(\mathrm{T}_{\mathrm{wout}}- \mathrm{T}_{\mathrm{win}})_{*}(\mathrm{T}_{7}-\mathrm{T}_{8})]}{\mathrm{T}_{7}- \mathrm{T}_{\mathrm{win}}} \tag{28}\]  
Both \(\mathrm{S}_{\mathrm{cpu1}}\) and \(\mathrm{S}_{\mathrm{cpu1}}\) were computed using Equation (29):  
\[\mathrm{S}_{\mathrm{cpu2}}=\frac{\sum\limits_{i=1}^{\mathrm{num}}\left( \overline{T}_{\mathrm{cpu}}-\mathrm{T}_{\mathrm{cpu,i}}\right)^{2}}{\mathrm{num}} \tag{29}\]  
Here, num signifies the number of cells in the condenser, and i refers to the cell's serial number.