# Influence of the Water-Cooled Heat Exchanger on the Performance of a Pulse Tube Refrigerator
## 3 Experimental Results  
### Influence of Different Configurations  
To study the effect of the structure of the water cooler on the heat transfer performance, the performance of WCHX 1, WCHX 2, and WCHX 4, were compared. Figure 4 presents the dependence of the overall relative Carnot efficiency \(\eta\) on the input acoustic power.  
\[\eta=\frac{Q(T_{h}-T_{c})}{W_{a}T_{c}} \tag{2}\]  
where \(Q\) is the cooling power, \(T_{c}\) is the cold head temperature, and \(T_{h}\) is the cooling water temperature. A better cooling performance was achieved when WCHX 4 was used. When the input acoustic power was about 1500 W, the system performance was similar for WCHX 1 and WCHX 2; with a larger input acoustic power, the performance with WCHX 2 was better than that with WCHX 1; however, there was still a gap with that observed with WCHX 4. This was apparent in high-power conditions, where WCHX 4 had a great advantage compared with the other two WCHXs. Taking an input acoustic power of 4000 W as an example, the WCHX 4 system achieved a relative Carnot efficiency of 20.6% and cooling power of 319 W, whereas the system with WCHX 1 only achieved an efficiency of 18% and cooling power of 270 W, and the system with WCHX 2 only achieved an efficiency of 18.7% and cooling power of 281 W.  
Owing to the aforementioned technical difficulty, the gas temperature was measured at the inlet of the WCHXs, as shown in Figure 5. At 1270 W input acoustic power, there was no large difference in gas temperature among the three WCHX designs: 35 \({}^{\circ}\)C with WCHX 1, 24 \({}^{\circ}\)C with WCHX 2,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**WHCX** & **Porosity** & \
\begin{tabular}{c} **Gas-solid Heat** \\ **Transfer Area** \\ \end{tabular} &
\begin{tabular}{c} **Hydraulic** \\ **Diameter** \\ \end{tabular} & **Length** & **Remarks** \\ \hline
1 & 26.2\% & 0.38 m\({}^{2}\) & 0.8 mm & 64 mm & - \\
2 & 23.8\% & 0.11 m\({}^{2}\) & 2.44 mm & 64 mm & - \\
3 & 25.1\% & 0.261 m\({}^{2}\) & 1.35 mm & 64 mm & - \\
4 & 24.5\% & 0.386 m\({}^{2}\) & 0.76 mm & 64 mm & - \\
5 & 24.5 and 23 \({}^{\circ}\)C with WCHX 4. The gas temperature increased with the input acoustic power, especially in the system with the plated-fin WCHX. When the input acoustic power was 5000 W, the gas temperature was as high as 76 \({}^{\circ}\)C with WCHX 1, but only 33 \({}^{\circ}\)C with WCHX 4. This implies that WCHX 4 was better cooled.  
What puzzled us is that the gas temperature obtained with WCHX 2 was very close to that with WCHX 4 (Figure 5), but its efficiency was closer to that of WCHX 1 (Figure 4). The temperature distribution along the regenerator shown in Figure 6 may provide some explanations. The \(x\)-axis begins at the warm end of the regenerator and ends at its cold end. The data points are the temperatures measured by thermometers \(T_{1}\) to \(T_{4}\), and the straight lines are their fitted results. At an input acoustic power of 1270 W, the temperature at the warm end for WCHXs 1 and 2 were very close, and were 10 K higher than that for WCHX 4. This means that with WCHX 4, the temperature difference between the two ends of the regenerator was less, and thus, more cooling power was obtained. At an input acoustic power of 5000 W, the temperature at the warm end observed with WCHX 1 was about 10 K higher than that with WCHX 2, and about 24 K higher than that with WCHX 4. As a result, more effort was required to pump heat from the cold head to the ambient end of the regenerator. Figures 4-6 indicated that the more effective parameter to evaluate the performance of the WCHXs in a pulse tube refrigerator, is the hot-end temperature of the regenerator, instead of the inlet gas temperature of the WCHXs.