# Influence of the Water-Cooled Heat Exchanger on the Performance of a Pulse Tube Refrigerator
## 1 Introduction  
Owing to their high reliability, high efficiency, and compact size, pulse tube refrigerators have attracted much attention in the past decades [1]. Many of them have been developed to cool devices such as infrared detectors, HTS filters, and germanium detectors. Their cooling power is often less than ten watts. Recent advances in high-temperature superconductivity (HTS) devices, small gas liquefaction, and cryogenic storage tanks, have spurred the demand for high-capacity refrigerators. The cooling power required by such applications varies from hundreds to thousands of watts, and the corresponding cooling temperature varies from 80 to 120 K [2; 3; 4]. The pulse tube refrigerator was considered to be one of the best candidates, so the development and use of high-capacity pulse tube refrigerators boomed. At present, the cooling power of these refrigerators can exceed 1 kW at a temperature of 80 K, and a relative Carnot efficiency of 22% has been achieved [5; 6; 7; 8; 9]. Although this efficiency is quite high, some studies have indicated that the room temperature heat exchanger could be further improved [10; 11; 12]. The water-cooled heat exchanger (WCHX) is one of the core components in a pulse-tube cooler [13; 14]. The function of the heat exchanger is to transfer time-averaged heat from the gas to the water, which acts as an external heat sink. It must provide good thermal contact between the two flowing streams, while causing a minimal pressure drop in either stream. There are three typical configurations for water-cooled heat exchangers employed in pulse tube refrigerators. The first is the finned-tube type introduced in References [15; 16]. In this type, fins of different lengths are arranged in a round case at regular intervals. Thermal contact between the fins and tubes acting as channels for the water is ensured by soldering. Although this is a very appropriate high-capacity design in theory, it is technically difficult. The water-cooled heat exchanger presented in References [17; 18; 19], shown inFigure 1b, can also be considered a fin-tube type. The fins are machined by wire electrical discharge machining (WEDM), and the water only flows around the circumference of the case. Meanwhile, the temperature difference on the fins caused by heat conductance will increase with the size of the exchanger [8; 20]. It is much easier to machine, but more expensive. The third configuration is the shell-tube type, shown in Figure 1c. As is known, the heat transfer coefficient at the water side is much higher than that at the gas side [21], so the heat exchange at the gas side should be strengthened. With the present machining process capability, it is impossible to produce such fins in a pipe with a small hydraulic diameter. If the hydraulic diameter is less than 1 mm, the flow area of the gas is often less than 15% of the cross-sectional area of the heat exchanger. To overcome this drawback of the shell-tube heat exchanger, Hu introduced another configuration, as shown in Figure 1d [22]. Small tubes acting as fins are welded inside the larger tubes, to strengthen the heat transfer at the gas side. Because the fins are not long, a uniform temperature distribution on the heat exchanger can be ensured. The hydraulic diameter and flow area can be easily adjusted by changing the parameters of the small tubes. The effectiveness of this configuration was confirmed in Reference [19]. Generally, a smaller hydraulic diameter and high flow velocity can improve heat transfer, but also result in greater flow loss. A good design achieves a reasonable compromise between these variables, in order to obtain the best performance of the refrigerator. The pressure drop can be calculated based on thermoacoustic theory. Meanwhile, the heat transfer is hard to evaluate because the empirical correlation for steady flow cannot be directly applied. A well accepted heat transfer correlation for oscillating flow is still absent, although much investigation has been carried out in this  
Figure 1: Four configurations of WHCXs.