Performance improvement of a pulse tube cryocooler with a single compressor through cascade utilization of cold energy

Abstract

The high-frequency pulse tube cryocooler (HPTC) has been attracting increasing and widespread attention in the field of cryogenic technology because of its compact structure, low vibration, and reliable operation. The gas-coupled HPTC, driven by a single compressor, is currently the simplest and most compact structure. For HPTCs operating below 20 K, in order to obtain the mW cooling capacity, hundreds or even thousands of watts of electrical power are consumed, where radiation heat leakage accounts for a large proportion of their cooling capacity. In this paper, based on SAGE10, a HPTC heat radiation calculation model was first established to study the effects of radiation heat leakage on apparent performance parameters (such as temperature and cooling capacity), and internal parameters (such as enthalpy flow and gas distribution) of the gas-coupled HPTC. An active thermal insulation method of cascade utilization of the cold energy of the system was proposed for the gas-coupled HPTC. Numerical simulations indicate that the reduction of external radiation heat leakage cannot only directly increase the net cooling power, but also decrease the internal gross losses and increase the mass and acoustic power in the lower-temperature section, which further enhances the refrigeration performance. The numerical calculation results were verified by experiments, and the test results showed that the no-load temperature of the developed cryocooler prototype decreased from 15.1 K to 6.4 K, and the relative Carnot efficiency at 15.5 K increased from 0.029% to 0.996% when substituting the proposed active method for the traditional passive method with multi-layer thermal insulation materials.

Graphical abstract

Keywords

radiation heat leakage / active thermal insulation / cascade utilization / cold energy / performance improvement / cryocooler

1 Introduction

The development of low-power-consumption and high-efficient energy conversion machines is an important concern in the field of engineering [1–3]. Refrigerators, which provide the mandatory low-temperature environment for many advanced scientific instruments, especially in the fields of space detection, low-temperature superconductivity, low-temperature biology, and low-temperature medicine, are crucial for the proper functioning of various systems [4–6]. There has been an increasing demand for small cryocoolers due to ease of operation compared with cryogen [7,8]. The high-frequency pulse tube cryocooler (HPTC) without any moving parts in the cold part has attracted increasing and widespread attention due to its low vibration, low noise, reliable operation, and compact structure [9]. However, as the operating frequency increases, the irreversible heat transfer loss in the system significantly increases, and other factors, including real gas effects, low matrix heat capacity, and difficulties of phase-shifting with small acoustic power, pose a challenge to using HPTCs to achieve a very low temperature [10,11].

One way to obtain a lower cooling temperature is to employ additional HPTCs or cryogen as the precooling stage. National Institute of Standards and Technology (NIST), simultaneously using two HPTCs as the pre-cool stage of the third stage, obtained a temperature of 6.4 K by using helium 4 or a temperature of 5.4 K by using helium 3 with a total input power of 730 W [8]. Zhejiang University, adopting a thermal-coupled three-stage structure, achieved a no-load temperature of 4.97 K with a total input power of 950 W by using helium 4 [12]. After continuous optimization, the temperature was reduced to 4.26 K by using helium 4 and 4.03 K by using helium 3 [13,14]. Chen et al. also developed a thermal-coupled HPTC and obtained a temperature of 3.4 K with a total input power of 385 W, which is the lowest temperature record obtained by HPTCs using helium 4 [15]. Although the thermal-coupled cryocooler can obtain a lower cooling temperature, the system requires at least two compressors, often with a more complicated structure. Besides, the thermal-coupled multi-stage structures inevitably require thermal bridges, in which the heat transfer loss is inevitable, and the cooling efficiency of the system is also reduced. Compared to the thermal-coupled structure, the gas-coupled HPTC usually requires only one compressor, without additional thermal bridge connection, and the heat transfer is directly performed through the working gas [16]. Therefore, the gas-coupled HPTC has a more compact structure and higher-potential cooling efficiency.

In addition to optimizing the cooling process of a HPTC, another way to improve the cooling performance is to reduce the loss in a certain component of the cryocooler. Qiu et al. reduced the entropy production in a pulse tube by additionally pre-cooling the pulse tube, so that the cooling temperature was reduced from 26.60 K to 18.02 K, and the relative Carnot efficiency at 30 K increased from 1.4% to 4.1% [17]. Dai et al. also pre-cooled a pulse tube to reduce its exergy loss, so that the temperature was reduced from 18.60 K to 15.35 K, and the relative Carnot efficiency at 20 K based on acoustic power increased from 1% to 2.7% [18]. Besides, optimizing different filling methods of regenerators also contributes to improving the cooling performance. Dang et al. reduced the no-load temperature from 7.80 K to 6.82 K and increased the 10 K cooling capacity from 91 mW to 112 mW by simultaneously filling different types of rare-earth materials in the lower-temperature regenerator [19]. Optimizing individual components usually avoids changing the existing cooling process of a HPTC, thus improving cooling performance without destroying the original compactness and complexity of the system.

At present, the efficiency of HPTCs operating in the temperature range from liquid hydrogen to liquid helium is still very low. To obtain the mW cooling capacity, hundreds or even thousands of watts of electrical power are consumed [20–22]. It should be noted that as the temperature decreases, the proportion of radiation heat leakage accounting for the refrigeration power increases. Therefore, the performance can be improved by reducing the radiation heat leakage. Currently, there are two methods to reduce the radiation heat leakage of HPTCs: one is to wrap multi-layer thermal insulation materials, reducing the radiation heat leakage by reducing the surface emissivity of the cold part [23,24]. This method avoids affecting the complexity of the system, but the reduction of radiation heat leakage is limited; another is to employ a radiation shield, reducing the radiation heat leakage by lowering the surrounding temperature [8,12–14]. This method requires an additional cold source and is often employed in the thermal-coupled structure, where the pre-cooling cryocooler, or the cryogen additionally outputs a portion of the cooling power to cool the radiation shield. In these structures, the shield temperature is directly determined by the pre-cooling cryocooler or the cryogen, and the cooling of the radiation shield increases the thermal load of the pre-cooling cryocooler, thereby increasing the power consumption of the whole system. For the HPTC driven by a single compressor, there is no additional pre-cooling stage like the above thermal-coupled structure, and the radiation shields can only be cooled by the outputting cold energy at the central position of the regenerator [20,21] or the cold head of the gas-coupled hotter stage [15,25,26]. However, for the single-stage structure or the gas-coupled structure, the additional output of the cold energy of the system affects the energy flow distribution inside the refrigerator. Especially for the gas-coupled structure, it affects the distribution of internal mass flow, enthalpy flow, acoustic power flow and phase of the hotter stage, adversely affecting the refrigeration performance of the colder stage. Presently, the research on the improvement mechanism of the refrigeration performance after employing the radiation shield cooled by the cold energy of the system in the gas-coupled process is not clear, which is the purpose of this paper.

This paper designed and optimized a novel cooling process, a gas-coupled multi-bypass HPTC with only one compressor, which is similar to a single-stage HPTC judging from the outside appearance. One or more radiation shields, which were cooled by the transition section between the hot end and the cold head of the developed cryocooler, were employed to reduce the radiation heat leakage between the cryocooler and ambient. Besides, detailed theoretical analysis and numerical modeling were first conducted. In addition, the effect of cooling a radiation shield at different positions of the developed cryocooler model on radiation heat leakage and different internal losses in the lower-temperature section was analyzed. Moreover, the traditional passive thermal insulation method by using multi-layer insulation and the active thermal insulation through cascade utilization of cold energy at different positions simultaneously to cool more radiation shields were also simulated. For different insulation solutions, the reasonable inherent loss compositions and values were given detailly. Furthermore, experiments were implemented to verify relevant conclusions. The following first revealed various losses of each component and deduced the expression of the actual net refrigeration power of the HPTC.

2 Thermodynamic analysis

2.1 Actual losses of each component of HPTC

Taking a single-stage HPTC as an analysis example, a typical structure is often composed of a linear compressor, a duct, an aftercooler, a regenerator, a cold-end heat exchanger, a pulse tube, a hot-end heat exchanger, an inertance tube, and a gas reservoir [27].

The actual losses in a regenerator mainly include the pressure amplitude reduction loss caused by the gas viscosity, the solid and gas heat conduction losses resulted from the axial temperature gradient, and the incomplete heat exchange loss generated by the gas temperature fluctuation. The pressure reduction loss weakens the working ability of the gas, the heat conduction loss directly reduces the net cooling power, and the heat exchange loss produces the axial enthalpy flow in the regenerator [10,27]. Additionally, the external wall of the regenerator also exchanges radiation heat with the surrounding environment. Although this leakage will not directly reduce the net refrigeration power, it changes the temperature distribution in the regenerator, thereby increasing other internal losses. Generally, these losses can be calculated as follows.

For pressure amplitude reduction loss:

(1){\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{lost,}}{}_{\mathrm{reg}},\mathrm{where}\mathrm{\hspace{0.17em}}\langle \stackrel{·}{S_{\mathrm{irr}}}\rangle \mathrm{=}-{\displaystyle \underset{\mathrm{d}t}{\propto }{\displaystyle \underset{\mathrm{d}x}{\int }{\displaystyle \frac{uAF}{T}}}}\mathrm{=}{\displaystyle \frac{R}{\tau P_{\mathrm{m}}}}{\displaystyle \underset{\mathrm{0}}{\overset{\tau }{\int }}\stackrel{·}{m}}\mathrm{\Delta }{P}_{\mathrm{1}}\mathrm{d}t.

For heat conduction loss in the regenerator wall:

(2){\stackrel{·}{Q}}_{\mathrm{cond,reg,w}},\mathrm{where}\mathrm{\hspace{0.17em}}\langle \stackrel{·}{S_{\mathrm{irr}}}\rangle =-{\displaystyle \underset{\mathrm{d}t}{\propto }{\displaystyle \underset{\mathrm{d}v}{\int }{\displaystyle \frac{\mathit{q}\cdot \nabla T}{T^{\mathrm{2}}}}}}={\displaystyle \frac{\mathrm{1}}{\tau }}{\displaystyle \underset{\mathrm{0}}{\overset{\tau }{\int }}{\displaystyle \underset{v}{\overset{\infty }{\int }}{\displaystyle \frac{A_{\mathrm{w}}{k}_{\mathrm{w},i}{(T_{\mathrm{w},i}-T_{\mathrm{w},i+\mathrm{1}})}^{\mathrm{2}}}{T_{\mathrm{w},i}{T}_{\mathrm{w},i+\mathrm{1}}\mathrm{\Delta }{x}_i}}\mathrm{d}v\mathrm{d}t}}.

For heat conduction loss in the regenerator matrix:

(3){\stackrel{·}{Q}}_{\mathrm{cond,reg,r}},\mathrm{where}\mathrm{\hspace{0.17em}}\langle \stackrel{·}{S_{\mathrm{irr}}}\rangle =-{\displaystyle \underset{\mathrm{d}t}{\propto }{\displaystyle \underset{\mathrm{d}v}{\int }{\displaystyle \frac{\mathit{q}\cdot \nabla T}{T^{\mathrm{2}}}}}}={\displaystyle \frac{\mathrm{1}}{\tau }}{\displaystyle \underset{\mathrm{0}}{\overset{\tau }{\int }}{\displaystyle \underset{v}{\overset{\infty }{\int }}{\displaystyle \frac{(\mathrm{1}-\phi )A_{\mathrm{reg}}{k}_{\mathrm{r},i}{(T_{\mathrm{r},i}-T_{\mathrm{r},i+\mathrm{1}})}^{\mathrm{2}}}{T_{\mathrm{r},i}{T}_{\mathrm{r},i+\mathrm{1}}\mathrm{\Delta }{x}_i}}\mathrm{d}v\mathrm{d}t}}.

For heat conduction loss in the gas:

(4){\stackrel{·}{Q}}_{\mathrm{cond,reg,g}},\mathrm{where}\mathrm{\hspace{0.17em}}\langle \stackrel{·}{S_{\mathrm{irr}}}\rangle =-{\displaystyle \underset{\mathrm{d}t}{\propto }{\displaystyle \underset{\mathrm{d}v}{\int }{\displaystyle \frac{\mathit{q}\cdot \nabla T}{T^{\mathrm{2}}}}}}={\displaystyle \frac{\mathrm{1}}{\tau }}{\displaystyle \underset{\mathrm{0}}{\overset{\tau }{\int }}{\displaystyle \underset{v}{\overset{\infty }{\int }}{\displaystyle \frac{\phi A_{\mathrm{reg}}{k}_{\mathrm{g},i}{(T_{\mathrm{g},i}-T_{\mathrm{g},i+\mathrm{1}})}^{\mathrm{2}}}{T_{\mathrm{g},i}{T}_{\mathrm{g},i+\mathrm{1}}\mathrm{\Delta }{x}_i}}\mathrm{d}v\mathrm{d}t}}.

The heat exchange loss can be represented by the average enthalpy flow:

(5){\langle \stackrel{·}{H}\rangle }_{\mathrm{reg}}\mathrm{=}{\displaystyle \underset{\mathrm{d}t}{\propto }{\displaystyle \underset{\mathrm{d}x}{\int }uA(\rho e+P)}}/L\mathrm{=}{\displaystyle \frac{\mathrm{1}}{\tau L}}{\displaystyle \underset{\mathrm{0}}{\overset{\tau }{\int }}{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\stackrel{·}{m}(R/(\gamma -\mathrm{1})T_{\mathrm{g},i}+u_i{}^{\mathrm{2}}/\mathrm{2}+ZR{T}_{\mathrm{g},i})\mathrm{d}x\mathrm{d}t}}.

where P_m is the average pressure; ΔP₁ represents the pressure amplitude reduction, A_w and A_reg are the cross-section area of the regenerator wall and the regenerator, respectively; k_w, k_r, and k_g are the thermal conductivities of the regenerator wall, the matrix, and the working gas; \phi represents the regenerator porosity; L is the effective length of the regenerator; and Z is the compressibility factor [27,28].

In an actual pulse tube, the viscosity, heat dissipation, and disturbance of the working gas make the interaction between the gas boundary layer and the internal wall deviate from perfect adiabatic behavior. Such losses as those due to turbulence and acoustic streaming within the pulse tube are extremely challenging to the model. An empirical factor can be used to integrally calculate gross losses (including heat transfer loss and pressure reduction loss) in a pulse tube [27], as expressed in Eq. (6). Radiation heat leakage is also generated between the external wall of the pulse tube and the surroundings, which also increases these internal losses. Additionally, the large temperature gradient causes the axial heat conduction of the pipe wall and the working gas. They can be represented individually as

{\stackrel{·}{Q}}_{\mathrm{cond,pt,w}},\mathrm{\hspace{0.17em}}{\stackrel{·}{Q}}_{\mathrm{cond,pt,g}}

(6){\epsilon }_{\mathrm{pt}}={\displaystyle \frac{{\langle \stackrel{·}{H}\rangle }_{\mathrm{pt}}}{{\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{pt}}}}=\mathrm{1}-{\displaystyle \frac{{\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle }_{\mathrm{pt}}}{{\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{pt}}}}.

In an actual cold-end heat exchanger, the temperature difference between the working gas and the wall causes irreversible heat transfer loss. The gas viscosity causes pressure reduction loss when the working gas is flowing through the component. The partial inhomogeneous temperature causes conduction loss. Especially, the cold-end heat exchanger also exchanges radiation heat with the surrounding environment, which directly reduces the net refrigeration power. The total loss and radiation heat leakage from the cold-end heat exchanger are represented separately as

{\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle }_{\mathrm{ch}},\mathrm{\hspace{0.17em}}{\stackrel{·}{Q}}_{\mathrm{rad,ch}}.

Besides, the gross losses of an actual linear compressor, duct, and aftercooler are expressed respectively as {\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle }_{\mathrm{co}}, {\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle }_{\mathrm{du}},{\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle }_{\mathrm{ac}}. Moreover, the unavoidable empty volume in each component will also affect the cooling efficiency of the HPTC. However, the entropy increase caused by the empty volume alone is zero. Only when the pressure amplitude reduction or the temperature fluctuation is also considered, can the empty volume cause irreversible entropy production [29]. Some losses also occur at the boundary between an isothermal element and an adiabatic element, such as the pulse tube and the cold-end heat exchanger [27]. These losses caused by multi-factor interactions are high-order values, which are not specially analyzed in this paper.

2.2 Actual net cooling capacity of the HPTC

The theoretical gross refrigeration power with an ideal working gas of the HPTC is given by Eq. (7).

(7){\langle \stackrel{·}{Q_{\mathrm{c}}}\rangle }_{\mathrm{rev}}={\langle \stackrel{·}{H}\rangle }_{\mathrm{pt}}={\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{pt}}={\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{reg,c}}={\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{reg,h}}\cdot {\displaystyle \frac{T_{\mathrm{c}}}{T_{\mathrm{h}}}}=\langle \stackrel{·}{W_{\mathrm{e}}}\rangle \cdot {\displaystyle \frac{T_{\mathrm{c}}}{T_{\mathrm{h}}}},

where the subscripts pt, reg, c, and h represent the pulse tube, the regenerator, the cold end, and the hot end, respectively [30].

After considering various losses of each component, combining with Eq. (7), the actual net cooling power with a real working gas of the HPTC can be calculated [10,27–30], as demonstrated in Eqs. (8) and (9).

(8)\begin{array}{l}\langle \stackrel{·}{Q_{\mathrm{c}}}\rangle ={\langle \stackrel{·}{H}\rangle }_{\mathrm{pt}}-{\langle \stackrel{·}{H}\rangle }_{\mathrm{reg}}-{\stackrel{·}{Q}}_{\mathrm{cond,reg}}-{\stackrel{·}{Q}}_{\mathrm{cond,pt}}-{\stackrel{·}{Q}}_{\mathrm{rad,ch}}\\ \mathrm{=}{\displaystyle \frac{Z_{\mathrm{c}}{T}_{\mathrm{c}}}{Z_{\mathrm{h}}{T}_{\mathrm{h}}}}\cdot {\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{reg,h}}-{\left(\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle \mathrm{+}{\stackrel{·}{Q}}_{\mathrm{rad}}\right)}_{{}_{\mathrm{ch}}}-{\left(\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle \mathrm{+}{\stackrel{·}{Q}}_{\mathrm{cond,w}}\mathrm{+}{\stackrel{·}{Q}}_{\mathrm{cond,g}}\right)}_{\mathrm{pt}}\\ \mathrm{}-{\left({\displaystyle \frac{Z_{\mathrm{c}}{T}_{\mathrm{c}}}{Z_{\mathrm{h}}{T}_{\mathrm{h}}}}\cdot {\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{lost}}\mathrm{+}{\stackrel{·}{Q}}_{\mathrm{cond,w}}\mathrm{+}{\stackrel{·}{Q}}_{\mathrm{cond,r}}\mathrm{+}{\stackrel{·}{Q}}_{\mathrm{cond,g}}\mathrm{+}\langle \stackrel{·}{H}\rangle \right)}_{{}_{\mathrm{reg}}},\end{array}

(9){\langle \stackrel{·}{W_{\mathrm{a}}}\rangle }_{\mathrm{reg,h}}=\langle \stackrel{·}{W_{\mathrm{e}}}\rangle -{\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle }_{\mathrm{co}}-{\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle }_{\mathrm{du}}-{\langle \stackrel{·}{W_{\mathrm{lost}}}\rangle }_{\mathrm{ac}}.

3 Numerical simulation and analysis

This paper uses SAGE10 developed by D. Gedeon to build a simulation model [28]. SAGE connects each component through mass flow, pressure wave, and energy flow, which can realize the calculation of the whole cryocooler. SAGE also uses abundant empirical factors to consider various actual losses of each component. By discretizing the continuity equation, momentum equation, and energy equation, the calculation results of each node can be given. The output parameters include mass flow, pressure, temperature, and others, which can be processed to calculate all required losses.

3.1 Radiation model of the gas-coupled multi-bypass HPTC

The gas-coupled multi-bypass cooling process is illustrated in Fig. 1. The arrangement between all regenerators and pulse tubes is coaxial, and the aftercooler also functions as the hot-end heat exchanger to achieve a more compact structure. One or more radiation shields, cooled by the transition section between the hot end and the cold head of the gas-coupled model, are employed to reduce radiation heat leakage between the colder stage and the ambient. In the present model, the diameter and length of the regenerator 1, 2, and 3 are 26 and 31 mm, 18 and 43 mm, 12 and 46 mm, respectively. The diameter and length of the radiation shield 1 and 2 are 83 and 140 mm, 45 and 64 mm, respectively.

It can be known from Eq. (9) that gross losses from the compressor, the duct, and the aftercooler only affect the acoustic power at the cold part entrance. Compared with the first-stage cold head and the multi-bypass, achieving a high performance for the second-stage cold head is more significant. Therefore, the acoustic power into regenerator 1 is kept constant, and only the losses in the lower-temperature section (in regenerator 3, cold-end exchanger 2, and pulse tube 3) are specifically analyzed.

3.2 Effect of cooling position on various losses

The cooling of the radiation shield cannot only reduce the radiation heat leakage from cold-end exchanger 2, which can directly increase its net refrigeration power, but also reduce the radiation heat leakage from regenerator 3, which is beneficial to reducing the internal losses in the lower-temperature section. However, it also increases the partial thermal load of the higher-temperature section, which adversely affects the lower-temperature section. Therefore, an appropriate cooling position should be optimized to achieve minimum loss. In this section, the cooling of radiation shield 1 at different positions of regenerator 2 is simulated to study its impact on different internal losses in the lower-temperature section.

Figure 2(a) shows the change of radiation heat leakage from cold-end heat exchanger 2, regenerator 3, and regenerator 2 with different cooling positions of radiation shield 1. The ordinate represents the ratio of radiation heat leakage after cooling the radiation shield to the original value without any insulation. The abscissa represents the relative position of radiation shield 1 on regenerator 2. It can be seen that as the radiation shield approaches the cold end of the regenerator, the radiation heat leakage from cold-end heat exchanger 2 and regenerator 3 decreases rapidly due to the lower ambient temperature, which also implies that more cold energy is needed to cool the radiation shield. Conversely, the radiation heat leakage from regenerator 2 increases dramatically, which is also not conducive to the performance improvement of the lower-temperature section.

Figure 2(b) displays the changes of different losses in cold-end heat exchanger 2. As the radiation shield is cooled successively from the hot end to the cold end of regenerator 2, all losses increase first and then decrease gradually, of which the deterioration of heat transfer loss is the most significant. The reasons for these phenomena are specifically explained in the following analysis. The variations of different losses in regenerator 3 and pulse tube 3 are exhibited in Figs. 2(c) and 2(d). It can be seen that except for pressure reduction loss, other losses are reduced variously after cooling the radiation shield. Compared with the loss in pulse tube 3, the cooling of the radiation shield has a greater impact on that in regenerator 3. Moreover, as radiation shield 1 approaches the cold end of regenerator 2, axial heat conduction losses deteriorate significantly, especially for the gas in the pulse tube.

To explore the main constraints on the performance of the second-stage cold head, the composition of different losses in the lower-temperature section is demonstrated in Fig. 3. As can be observed in Fig. 3, the loss in the regenerator accounts for a large proportion of the total loss, especially the average enthalpy flow. It can be seen from Fig. 2(c) that the average enthalpy flow in regenerator 3 is always lower than the original value when radiation shield 1 is cooled at different positions of regenerator 2. Moreover, the gas-coupled model has a large temperature span, which means the cold energy of the system can be multi-output to cool more radiation shields simultaneously, leading a higher performance enhancement.

3.3 Effect of different insulation solutions on refrigeration performance

The operation of the gas-coupled cryocooler with 5 different thermal insulation solutions is simulated separately in this section. In Case 1, without any thermal insulation, the cold part of the cryocooler directly exchanges radiant heat with the environment. In Case 2, the surface emissivity of the cold part is reduced by half, simulating the traditional passive method by using multi-layer insulation. In Case 3, radiation shield 2 is cooled by the multi-bypass. In Case 4, radiation shield 1 is cooled by the first-stage cold head. In Case 5, radiation shields 1 and 2 are cooled simultaneously at the two positions mentioned above. Additionally, the surface emissivities in Case 3, 4, and 5 are identical to that of the cold part in Case 1.

In the present cases, the working conditions are 22 Hz and 2.7 MPa. The ambient temperature is 290 K. The second-stage cold head operates at 15 K, with 175 W acoustic power (the corresponding input electric power is 300 W) at the entrance of regenerator 1.

3.3.1 Radiation heat leakage and various losses

Figure 4 depicts the radiation heat leakage from the colder stage with different insulation solutions. It can be seen that without any thermal insulation, the radiation heat leakage of cold-end heat exchanger 2, regenerator 3, and regenerator 2 are 12, 170, and 230 mW, respectively. Compared with the passive insulation by reducing the surface emissivity, the active insulation by lowering the surrounding temperature can more effectively reduce the heat leakage. Specifically, in Case 5, the radiation heat leakage from the colder stage achieves the minimum simultaneously.

Figure 5(a) presents the total loss in cold-end heat exchanger 2 with different insulation solutions. To show the difference of various losses clearly, the pressure reduction loss and heat conduction loss are increased by a factor of 100. Compared with the heat transfer loss, the pressure reduction and heat conduction loss can be ignored, as shown in Fig. 5(a). Compared with the original value, the heat transfer loss in the passive or active insulation solutions increases, which is consistent with the phenomenon in Fig. 2(b). In the second-stage cold head, the heat transfer loss produced by the temperature difference between the working gas and the wall is primarily correlated with the mass and heat transfer capacity of the gas, which will be given later.

The change of the total loss in regenerator 3 is plotted in Fig. 5(b). The main constraints on the refrigeration performance are the average enthalpy flow and pressure reduction loss, as shown in Fig. 3. It can be seen from Fig. 5(b) that with the reduction of radiation heat leakage from regenerator 3, the average enthalpy flow decreases and pressure reduction loss increases, except in Case 3. The pressure reduction loss is affected by the mass of the working gas. Aside from radiation heat leakage, the average enthalpy flow is also determined by the enthalpy in the hotter-stage regenerator. Besides, compared to the cooling of radiation shield 2, the cooling of radiation shield 1 can reduce the heat leakage from regenerators 2 and 3, thereby reducing their internal enthalpy simultaneously.

Figure 5(c) shows the total loss in pulse tube 3. Compared with the change of the total loss in regenerator 3, the change in pulse tube 3 is negligible in different insulation solutions. Generally, with the decrease of radiation heat leakage, the total loss is reduced gradually, except in Case 3 where the radiation heat leakage from regenerator 3 is less than that in Case 4. However, the cooling of radiation shield 2 increases the thermal load of the hot end in pulse tube 3, thereby producing a larger gas and wall heat conduction loss, as shown in Fig. 5(c).

3.3.2 Mass flow and acoustic power

Different from the thermal-coupled structure, the gas-coupled cryocooler adjusts the temperature distribution and working ability of each stage by directly adjusting the gas distribution of the system. Besides, it can be known from Section 2.2 that the net refrigeration power is also determined by the acoustic power entering this part. Figure 6 shows the variations of mass flow and acoustic power at the entrance of regenerator 3. As shown in Fig. 6, the mass flow and acoustic power in the passive or active insulation solutions increase variously, which results in a distinct increase in net refrigeration power. The variations explain why the pressure reduction loss in regenerator 3 in Case 3 is smaller than that in Case 2, and the heat transfer loss in cold-end heat exchanger 2 with thermal insulation is larger than the original value without any thermal insulation.

3.3.3 Total loss and net refrigeration power

Figure 7 shows the total loss and actual cooling power in different insulation solutions. It can be seen that without any thermal insulation, the second-stage cold head can initially provide a cooling power of 69.1 mW at 15 K (a relative Carnot efficiency of 0.438%), with a total loss of 870 mW in the lower-temperature section. Compared with the passive insulation, the active insulation of cooling one or two radiation shields can more effectively improve the performance. Specifically, when radiation shields 1 and 2 are cooled simultaneously, the net refrigeration power at 15 K is increased to 251.3 mW (a relative Carnot efficiency of 1.592%), and the total loss is reduced to 730 mW.

Generally, for the gas-coupled multi-bypass model, the radiation heat leakage accounts for a large proportion of the original cooling power. Compared with the passive insulation by reducing the surface emissivity, the active insulation by lowering the ambient temperature can more effectively improve the refrigeration performance of the gas-coupled HPTC. Moreover, the cascade utilization of the cold energy at different positions can lead to a higher enhancement of the refrigeration performance.

4 Experimental verifications

4.1 Experimental set-up

The photo of the overall experimental setup with the HPTC experimental assemble (without a vacuum enclosure) is given in Fig. 8. The actual structural parameters of each component are consistent with those in the simulation model. All temperatures are measured by rhodium-iron thermometers calibrated from 1.3 to 295 K with an accuracy of 0.1 K. The refrigeration power is evaluated by measuring the Joule heat input to the coil heater, which is tightly attached to the second-stage cold head. The LIHAN CP2221 compressor is used as the only driving source, which is controlled by a KIKUSUI PCR-4000L power supply. The pressure of the compression chamber is measured by the JYB-KO-HVG sensor. The temperature and pressure are acquired by National Instruments (NI), whose models are NI 9203 and NI 9217, respectively. The data-recording is implemented by a Labview’s programming.

4.2 Comparisons between simulated and experimental results in passive insulation

The influence of the conventional passive method by using multi-layer insulation on the cooling performance of the experimental prototype was studied first. Figure 9 shows the performance of the developed prototype under different operating conditions with different insulation layers. The model simulates the traditional passive insulation by reducing the surface emissivity of the cold part. Figure 10 compares the simulated and experimental results. It can be seen from Fig. 9 that when 5 layers of thermal insulation were employed, the prototype obtained a no-load temperature of 27.4 K with an input power of 300 W and a frequency of 22 Hz. With 19 layers of thermal insulation, the corresponding no-load temperature was reduced to 20.9 K. With the further increase of insulation layers, the no-load temperature was not reduced significantly. Specifically, when the 450 W electric power was input, the no-load temperature with 33 layers was only reduced to 15.1 K from 15.8 K with 19 layers of thermal insulation, as shown in Fig. 10. However, as the surface emissivity of the cold part decreases, the no-load temperature of the model decreases monotonically. The reason for the deviation of the calculated results from the experimental results is that the apparent thermal conductivity of the multilayer insulation is not linearly related to the surface emissivity, and it is also affected by parameters such as contact thermal resistance, vacuum, and surface area [31]. However, since SAGE cannot simulate multilayer insulation, only the effect of the surface emissivity on the cooling performance was calculated. Overall, the calculation results and experimental results in Fig. 10 illustrate that the radiation leakage heat has a great influence on the cooling performance of the HPTC with a small cooling capacity. Therefore, it is necessary to adopt new measures to reduce the radiation leakage heat.

4.3 Comparison of simulated and experimental results in active insulation

The influence of different active insulation methods by cooling one or two radiation shields at different positions on the performance of the experimental prototype is studied in this section. Figure 11 shows the performance of the experimental prototype with different active insulation methods. The simulated and experimental results are also compared in Fig. 12. It can be seen that compared with the cooling of radiation shield 2, the cooling of radiation shield 1 further improved the cooling performance. When radiation shields 1 and 2 were cooled simultaneously, the cooling performance of the experimental prototype was promoted to a higher enhancement. Specifically, when the 220 W electric power was input, the experimental cryocooler provided a cooling power of around 50 mW at 16.3 K with radiation shield 2, and the same cooling power at 15.1 K with radiation shield 1. In addition, the corresponding temperature was further reduced to 13.6 K when both radiation shields 1 and 2 were employed. It can be seen from Fig. 12 that there is a good agreement between the simulated and experimental results of the gas-coupled HPTC with different active insulation methods. The cooling performance from the simulation is better than that from the experiment, which may be due to the fact that SAGE is a one-dimensional calculation software, which makes a lot of assumptions and simplifications.

4.4 Comparison of experimental results in passive and active insulation

Figure 13 shows the optimal refrigeration performance of the developed cryocooler with passive and active thermal insulation, respectively. It can be seen that when 33 layers of thermal insulation materials were wrapped on the cold part of the prototype, at an input power of 450 W, the cryocooler obtained the lowest temperature of 15.1 K and provided a cooling power of 7 mW at 15.5 K (a relative Carnot efficiency of 0.029%). When radiations 1 and 2 were cooled simultaneously by the first-stage cold head and the multi-bypass, at an input power of 350 W, the cryocooler obtained the lowest temperature of 6.4 K and provided a cooling power of 190 mW at 15.5 K (a relative Carnot efficiency of 0.996%), which indicates that compared with the traditional passive method by using multi-layer thermal insulation materials, the active thermal insulation method can more effectively improve the refrigeration performance through cascade utilization of the cold energy of the system at different positions to simultaneously cool more radiation shields (which is also consistent with the simulation results in Section 3.3).

5 Conclusions

An active thermal insulation method was proposed for the gas-coupled HPTC through the cascade utilization of the cold energy of the system, and a gas-coupled HPTC calculation model was established to study the effects of radiation heat leakage on the apparent cooling performance and the internal parameters of the cryocooler. The numerical calculation results were verified by experiments, and some conclusions are obtained from the quantitative analyses and comparisons between the simulation and experiment.

Numerical simulations indicate that the external radiation heat leakage has a prominent effect on the performance of the gas-coupled HPTC, whose reduction cannot only directly increase the net cooling power, but also decrease the internal gross losses and increase the mass and acoustic power in the lower-temperature section. When two radiation shields are cooled simultaneously at different positions, the calculated relative Carnot efficiency increases from 0.438% to 1.592% at 15 K.

The numerical calculation results were further verified by experiments, and the test results showed that the no-load temperature of the developed prototype decreased from 15.1 K to 6.4 K, and its relative Carnot efficiency at 15.5 K increased from 0.029% to 0.996% when substituting the proposed active method for the traditional passive method with multi-layer thermal insulation materials.

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