At present，Stirling cryocoolers and Stirling type pulse tube cryocoolers have been commonly used in space applications in the past few years and the absence of a moving displacer in PTCs makes them have many potential advantages over Stirling cryocoolers for the cooling of infrared sensors ^［1-4］. Stirling cryocoolers and PTCs have long service life and low failure rates ^［5］. There are three different geometries that have been used in PTCs：in-line，coaxial and U-tube. The in-line arrangement is the most effective because the gas flow goes and returns without turbulence from the flow reversal at the cold head，but the disadvantage that the cold head is located between the two hot ends limits its development. U-tube type PTC uses a thin tube to connect the regenerator and pulse tube so that the gas cross-section flow and large dead volume produce large energy loss，which results in low efficiency. The most widely used arrangement in space applications is the coaxial arrangement because of its compact structure. The coaxial PTC places pulse tube inside of regenerator so that its cold head is on one side，which greatly improves its coupling efficiency with detector Dewar. However，helium gas flowing in and out of the cold head will have a 180 degrees transition，and there is a mismatch of temperature profiles in the regenerator and pulse tube that would lead to radial irreversible heat flow because of temperature distribution. Therefore，it is of great significance to analyze the thermal effect of coaxial PTC，especially to compare with the other two kinds of PTCs. A general analysis has been carried out for the experimental performance comparison between linear and U-type pulsed-tube cryocoolers by Tendolkar M V，et al. ^［6］. They pointed out that the performance of the linear PTC is apparently better than that of the U-type configuration respect to cooling down time，minimum temperature，and cooling power at the same pulse tube and regenerator dimensions. Although the effects of some different operating parameters were considered，the detailed thermodynamic behaviors of various components in PTC were not further discussed. In addition，linear and coaxial PTCs are widely used in space，especially coaxial PTCs. The coaxial PTC has a compact structure and is more convenient to use，while the linear structure is simple with high cooling efficiency. At present，the research on the difference between the two PTCs is relatively rare. Therefore，it is very valuable to carry out some comparative studies between the two PTCs.

This paper introduces a Stirling single-stage linear pulse tube cryocooler for cooling space infrared detectors. The linear chiller is based on the existing one coaxial pulse tube cryocooler transformation. The basic dimensions of the key components of both structures are the same. Through the establishment of one-dimensional numerical model，the energy loss and gas pressure ratio of linear and coaxial pulse tube system are compared and analyzed. The comparison between the experimental value and the simulated value is carried out to verify the accuracy of the model.

Figure 1 shows the schematics of two PTCs. In the in-line PTC shown in Fig. 1（a），the acoustic power flowing out of the compressor flows alternately in the aftercooler，regenerator，CHX，pulse tube，inertance tube and the reservoir. When the pulse tube is placed into the regenerator，the linear chiller becomes coaxial as shown in Fig. 1（b）. The volume of the two regenerators and the filling rate of the screen are basically the same. The gas flows through the regenerator and then enters the pulse tube reversely，resulting in energy losses. The main component parameters of the coaxial PTC（Case 1）and the in-line PTC（Case 2）are the same so that it is beneficial to carry out some comparative study. The main dimensions of the two PTCs are listed in Table 1.

The behavior of the PTC is predicted by a classical thermodynamic model and the mass and energy balance equations are applied to the control volumes of the cryocooler components. The working process of the pulse tube cryocooler is complicated due to the nature of unsteady，oscillating compressible gas flow. To trace the process，it is considered to be one dimensional，periodic and unsteady compressible flow. The following assumptions are introduced into the model as well^［7］：

1. The outer sidewall of the regenerator is thermally insulated.

2. The gas is ideal gas.

3. The entrance effects are neglected.

4. The fluid phase satisfies the no-slip condition on the fluid-solid interface.

5. The porosity is constant.

The general governing equations：mass equation，momentum equation，energy equation，gas equation of state for the solids in the regenerator and equation of state for ideal gas are as follows：

After the physical properties of working fluid in Stirling-type PTC are estimated according to these equations，the enthalpy and the expansion work at the cold side of pulse tube are calculated. Then，the losses transferred to the CHX are considered with the enthalpy and the expansion work，and the cooling capacity of Stirling-type PTC is finally predicted. Therefore，the accurate estimation of the physical property and losses is important to accurately predict the cooling capacity.

Figure 2 is the experimental bench of the linear and coaxial pulse tube cryocooler. The whole system includes single-stage pulse tube cryocooler，control power supply，measurement system，data acquisition system，vacuum pump and water-cooling system，etc. In order to reduce the radiation and convective heat transfer loss of the cold end heat exchanger. The whole cold finger is packed with aluminum foil and placed in a vacuum cover to maintain a high vacuum degree about 10^-4 Pa.

The compressor is cooled by air force. The hot end heat exchanger is connected with the water cooler through water pipes，and the temperature of hot end heat exchanger is accurately controlled by the water cooler. The experimental temperature is 300 K. The phase regulation parts of the cryocooler are the inertance tube and the gas storage，which is used to adjust and optimize the mass flow，pressure wave and the phase difference between them.

Figure 3 shows the behavior of the enthalpy at various locations. When the power flows alternately and reaches the steady state，its enthalpy value hardly changes until the helium fluid begins to flow through the aftercooler，which causes greater enthalpy drop because of bulk heat exchange by water-cooling. In the regenerator，the enthalpy of the isothermal heat transfer state almost does not change，while the enthalpy in the cold heat exchanger increases because of the transfer of cooling power. Under the same power，the enthalpy of the cold heat exchanger of the linear PTC is slightly higher than that of the coaxial PTC，and this phenomenon is reflected in the performance comparison between the two PTCs.

Figure 4 shows the comparison diagram of the energy losses in each part of the two PTCs. It is shown obviously that the main losses of the two PTCs are mainly concentrated in the regenerator，inertance tube，pulse tube，heat exchanger and other components. Under the same PV input power，the coaxial PTC has higher losses in its regenerator and pulse tube，and lower heat exchanger losses and heat dissipation losses in the inertance tube compared with in-line PTC. The main reason is that the optimized length of inertance tube in the former is smaller than that in the latter. The coaxial system cooling capacity is also significantly smaller than that of the in-line PTC because of the following reasons. On the one hand，the pulse tube is placed in the regenerator of the coaxial system. When the working gas flows alternately in the system，the direction of internal gas flow in the regenerator is opposite to that in the pulse tube，resulting in turbulence generation and increasing friction losses. On the other hand，the heat exchanger of the hot end of the coaxial PTC is placed inside. Thus，it is difficult to fully transfer the heat and the heat dissipation of the system is small，which makes the performance of the whole system worse. Moreover，the pressure ratio of working gas in coaxial PTC system is relatively small，as shown in Fig. 5.

Figure 5 shows the behavior of pressure ratio. With the operation of working gas in the system，its pressure ratio gradually decreases from left to right and its value is close to 1 in gas reservoir. The aftercooler，cold and hot end heat exchangers are all slit structures，and the pressure ratio changes very small when the working gas flows through the three heat exchangers. However，the pressure ratio changes smaller quickly in the inertance tube，which is thin and long. Under the same PV power，the pressure wave of the working gas in the coaxial PTC varies greatly. The pressure in the outlet of the compressor is relatively larger，and the internal pressure ratio of other parts is smaller in the coaxial PTC system. The pressure ratio is only 1.116 in the regenerator，which is smaller than the linear regenerator pressure ratio of 1.124. The small regenerator pressure ratio is not conducive to improve the performance of the PTC^［8］，so the cooling capacity is lower.

The design of PTC system should match the impedance of the linear compressor ^［9］. Fig. 6 shows the relationship between the electric-to-acoustic efficiency of the compressor and the real and imaginary parts of its impedance. It can be seen from the figure that the electric-to-acoustic efficiency of the coaxial PTC is slightly higher than that of the linear type no matter the cooling temperature is high or low. This phenomenon is also reflected in the experiment.

Figure 7 shows the relationship between the performance of the two PTCs with different frequencies. When the frequency increases，the input power required by the two PTCs decreases firstly and then increases. The optimum frequency is about 46 Hz. Higher frequency would accelerate the helium flow of working medium and increase heat exchange frequency，while the excessive flow of working medium is not conducive to the heat transfer between helium and the regenerator. The heat penetration depth is reduced and heat exchange is insufficient. Therefore，the regenerator loss and frictional resistance loss of the cold storage are increased，and the cooling efficiency of the PTC is reduced. The optimal operating frequency of the PTC mainly depends on the size of the inertance tube. The lengths of the inertance tubes of the two PTCs are obtained separately based on 46 Hz optimization，which also verifies the accuracy of the built model. In addition，a cooling capacity of 6 W is obtained，and the input electric power of the linear PTC is less than that of the coaxial type.

Figure 8 shows the cooling down characteristics of the two PTCs and that the PTCs work efficiently at 46 Hz with 30 bar average pressure. The temperatures of the two PTCs are close to the lowest points after running for about 20 minutes. The minimum cooling temperatures of the coaxial and linear PTCs are the no-load low-temperatures of 45.3 K and 50 K，respectively. The coaxial PTC could reach lower cooling temperature at the same input power because the pulse tube is placed in the regenerator of the coaxial system and precooled by the regenerator.

Figure 9 is a comparison diagram of the refrigeration performance of two PTCs under the same operating conditions between simulated and experimental values. The temperatures of cold and hot end are maintained at 90 K and 300 K，respectively. It can be seen from the figure that the refrigeration load significantly decreases as one moves from in-line（6.21 W @ 90 K）to coaxial type configuration（5.23 W @ 90 K）for a given charging pressure（30 bar）and an input power（100 Wac）. No matter a simulation value or an experimental value，the input PV power/electric power of the PTC value is almost linearly distributed over the cooling capacity. The ratio of the corresponding PV power to the input electric power is about 63%. The performance difference between the two PTCs is small when the cooling capacity is not large. However，the input PV power/electric power of the coaxial PTC is significantly higher than that of the linear PTC when the cooling capacity is large.

The coaxial PTC can provide 0 W or 8 W cooling power at a cold temperature of 90 K if the input electric power is kept at 35.5 W or 143 W，respectively. In addition，the in-line PTC can efficiently provide 0 W or 10 W cooling power at a cold temperature of 90 K if the input electric power is kept at 34 W or 149 W，respectively.

In-line and coaxial pulse tube cryocoolers are commonly used in space for cooling infrared detectors. Many published papers investigating this topic have been carried out by experimental results，but there is very little theoretical research. In this paper，the in-line and coaxial PTCs are compared by simulation and experiment. The results show that the in-line PTC has a relatively stronger cooling capacity，while the coaxial PTC can be cooled at a lower cooling temperature. Due to the changes in system parameters such as energy losses，enthalpy flow distribution and electric-to-acoustic efficiency in the system caused by the structural differences between the two PTCs，the in-line PTC has better cooling performance than the coaxial one under the same working conditions. In addition，by comparing the simulation value with the experimental value，the accuracy of the model is verified.