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Investigations on a 3.3 K four-stage Stirling-type pulse tube cryocooler. Part A: Theoretical analyses and modeling
Journal Pre-proofs Investigations on a 3.3 K four-stage Stirling-type pulse tube cryocooler. Part A: Theoretical analyses and modeling Haizheng Dang, Rui Zha, Jun Tan, Tao Zhang, Jiaqi Li, Ning Li, Bangjian Zhao, Yongjiang Zhao, Han Tan, Renjun Xue PII: DOI: Reference:
S0011-2275(19)30080-3 https://doi.org/10.1016/j.cryogenics.2019.103014 JCRY 103014
To appear in:
Cryogenics
Received Date: Revised Date: Accepted Date:
4 March 2019 14 October 2019 2 December 2019
Please cite this article as: Dang, H., Zha, R., Tan, J., Zhang, T., Li, J., Li, N., Zhao, B., Zhao, Y., Tan, H., Xue, R., Investigations on a 3.3 K four-stage Stirling-type pulse tube cryocooler. Part A: Theoretical analyses and modeling, Cryogenics (2019), doi: https://doi.org/10.1016/j.cryogenics.2019.103014
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Investigations on a 3.3 K four-stage Stirling-type pulse tube cryocooler. Part A: Theoretical analyses and modeling Haizheng Danga,b,c,*, Rui Zhaa,b, Jun Tana,c, Tao Zhanga,b, Jiaqi Lia,b, Ning Lia, Bangjian Zhaoa,b, Yongjiang Zhaoa,b, Han Tana,b, Renjun Xuea,b a
State Key Laboratory of Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy
of Sciences, 500 Yutian Road, Shanghai 200083, China b University c Shanghai
*
of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China
Boreas Cryogenics Co., Ltd, 1388 Shuidian Road, Shanghai 200434, China
Corresponding author. State Key Laboratory of Infrared Physics, Shanghai Institute of Technical
Physics, Chinese Academy of Sciences, 500 Yutian Road, Shanghai 200083, China Tel.: +86-21-25051967 Fax: +86-21-65830734 Email: [email protected] (Haizheng Dang)
Abstract This paper conducts systematic theoretical analyses of a four-stage SPTC aimed at directly reaching a temperature of around 3 K and also simultaneously achieving the cooling capacities at four temperatures varying from 3.3 K to 80 K for multiple uses. The fourth stage cold finger is focused on and a 2-D axissymmetric CFD model is set up to study the irreversible losses and phase characteristics. The operating mechanisms in the fourth stage are modelled, in which the distributions of the losses along the whole regenerator are analysed quantitatively and the phase characteristics simulated as well. The interactions among the four stages are studied, in which the effect of parameters at each stage on the cooling performances of the other three stages is elaborated, respectively. The systematic optimization for the four-stage SPTC is then conducted. The detailed experimental verifications are presented in Part B.
Keywords: Four-stage Stirling-type pulse tube cryocooler; CFD model; Irreversible losses; Phase characteristics; Interactions among four stages.
Nomenclature
Afs
effective heat exchange area between fluid
T
temperature
and solid Ar
cross-sectional area of regenerator
Tf
temperature of fluid
f
frequency
Ts
temperature of solid
f1
frequency of compressor I
W1
input power of compressor I
f2
frequency of compressor II
W2
input power of compressor II
hfs
convective heat transfer coefficient
X
position in different components
Z
compressibility factor
between fluid and solid kf,x
thermal conductivity of fluid
ks,x
thermal conductivity of solid
L4
length of Reg IV
Greeks
L41
length of Er3Ni in Reg IV
ε
porosity of regenerator matrix
𝑚
mass flow rate
η
relative Carnot efficiency
p1
charge pressure of compressor I
p2
charge pressure of compressor IV
Subscripts
pd
dynamic pressure
1
first stage
pa
average pressure
2
second stage
Δp
pressure amplitude
3
third stage
R 𝑆hc
gas constant
4
fourth stage
entropy generation of heat conduction
c
cold end
𝑆ht
entropy generation of heat transfer
p
precooling power
𝑆pd
entropy generation of pressure drop
ps
phase-shifter
t
time
1. Introduction The pulse tube cryocooler (PTC) is a significant innovation in the regenerative refrigeration technology because it eliminates any moving component at the cold end, which endows it with evident advantages over the conventional regenerative cryocoolers such as GM and Stirling ones in terms of high reliability, long life and low vibration at the cold end [1, 2]. The Stirling-type PTC (SPTC) driven by the linear compressor employing the well-proven flexure springs and clearance technology further realizes the long operation time at the warm end, which thus makes it an attractive candidate for cooling cryogenic devices used in some special fields, especially in space [3, 4]. The past over three decades have seen a worldwide quest for the space-qualified SPTCs and also witnessed their successful in-orbit applications [5–11]. Up till now, single- and two-stage SPTCs which
cover the temperatures ranging from 20 K to 140 K have already been widely used in a variety of space missions, and the three-stage SPTCs developed for space applications with the cooling capacity at around 10 K are also becoming mature [5–18]. In the authors’ laboratory, a four-stage SPTC is being developed for the following three purposes. The first is for deep space exploration, in which the cryocooler is expected to provide the low-noise cooling at 4-6 K for the Si:As focal plane and/or Mid-Infrared Instrument under construction. The second purpose is to cool the superconducting nanowire single photon detector (SNSPD), which plays an important role in quantum key distribution, satellite laser ranging, and space quantum communication, etc., but the required cooling at around 2 K poses a serious challenge for the existing SPTCs. The third purpose is more ambitious because it is expected to simultaneously provide the cooling at four different temperatures varying from 3.3 K to 80 K for various applications, which will be much more compact and economical compared to using multiple cryocoolers. So far, only a few studies about the four-stage SPTCs have been reported. For example, in 2006, Olson et al. [19] reported a four-stage SPTC developed to provide cooling at 6 K and 18 K for MidInfrared Instrument in James Webb Space Telescope. In 2006 and 2007, Nast et al. [7, 20] introduced the continuous advances in the similar designs which reached a no-load temperature of 3.8 K. And In 2008, Nast et al. [21] continued to describe the improvement of a similar four-stage SPTC which reached a no-load temperature of 3.0 K with He-3 and provided simultaneous coolings at 4.5 K, 9.0 K, 30 K and 70 K for superconducting digital electronics communications programs. In 2017, Dang et al. [11] described the design of a four-stage SPTC for space applications and in 2018 Zha et al. [22] introduced its experimental results, in which a no-load temperature of 4.5 K with He-4 and the simultaneous cooling at 80 K, 60 K, 30 K and 5 K were achieved. In 2019, Dang et al. [23] reported the new progress of the
similar four-stage SPTC developed as an alternative of two choices expected to cool a SNSPD at 2 K, in which a no-load temperature of 4.38 K with He-4 was achieved. However, the previous studies mainly focused on briefly introducing structural design, describing practical fabrication and presenting experimental results, whereas the in-depth theoretical analyses of the inner process and operating mechanism have never been conducted. The four-stage SPTC cold finger (CF) has much more complicated physical structure and operating mechanism than those of single-, twoor three-stage ones. First, the staging configurations lead to difficulties in matching with compressors. Second, the irreversible losses in regenerators at much lower temperatures including pressure drop, axial heat conduction and ineffective heat transfer dramatically deteriorate the cooling performance. Third, the phase characteristics in regenerators become subtle, especially in the fourth stage. Fourth, the complex interactions among stages cause lots of difficulties in optimizing the cooling performance. In this part, the new advances in the four-stage SPTC will be reported together with the in-depth studies on the inner process, in which the fourth stage cold finger is focused on and a two-dimensional
(2-D) axis-symmetric CFD model is set up to study the irreversible losses and phase characteristics in its regenerator. The effects of precooling temperatures and operating parameters on the losses are discussed in detail. In view of the phase-shifting characteristics is critical to the performance improvement of the multi-stage cryocoolers [17, 24], the variations of phase difference with phaseshifter temperatures and operating frequencies are also analyzed. Then, the interactions among the four stages will be clarified and cooling performance optimized as well.
2. CFD simulations and analyses on the fourth stage CF
2.1 Model building
The establishment of the 2-D axis-symmetric CFD model of the fourth stage will draw on the experiences about the CFD models for the PTC in general [25–27], especially those developed in the same laboratory [28, 29]. As shown in figure 1, the fourth stage CF consists of an aftercooler, a segmented regenerator, three precooling heat exchangers (PHXs), a cold heat exchanger (CHX), a warm heat exchanger (WHX), a pulse tube (PT), a phase-shifter consists of two inertance tubes and a gas reservoir. The regenerator has four segments packed with #400 stainless steel (SS) meshes, #500 SS meshes, #635 SS meshes and rare-earth spherical materials, respectively. To investigate the effect of cryogenic phase-shifter, all of the WHX, inertance tubes and gas reservoir are regarded as the heat sink and they have isothermal boundary conditions in the model, of which the temperatures are variable in different cases. The compressor II is replaced by a sinusoidal pressure inlet setting at the left surface of the aftercooler, which is given as follows: 𝑝d = 𝑝a + ∆𝑝sin 2𝜋𝑓𝑡
(1)
where pd, pa, Δp and f are the dynamic pressure, average pressure (charge pressure), pressure amplitude and operating frequency, respectively. All the input powers mentioned in this paper are actually PV powers, which are calculated by integrating the dynamic pressures and volume flow rates at the outlet of the compressors. A first-order implicit unsteady pressure based segregated solver with PISO pressure-velocity is used in this CFD model. A second-order upwind scheme is applied to solve the continuity, momentum and energy equations. The convergence criterion is 10-6 for mass, velocity and energy. A time step of 1×10-5 s with 50 maximum iterations is set for cases.
The temperature profile among the fourth stage CF is shown in figure 2. A no-load temperature of 3.25 K is reached with a charge pressure of 1.2 MPa and an operating frequency of 32 Hz, while He-3 is used. The phase-shifter temperature is 30 K and the precooling temperatures are 70 K, 40 K and 10 K, respectively.
2.2 Regenerator losses
In the thermal non-equilibrium mode, three types of irreversible losses in the regenerator including axial heat conduction, pressure drop and ineffective heat transfer between fluid and solid are considered, which can be calculated by [30]: d〈𝑆hc〉 =
[(1 ― 𝜀)𝑘s,x + 𝜀𝑘f,x]𝐴r d𝑇
( ) d𝑥
𝑇2 d〈𝑆pd〉 = ―
𝑚𝑍𝑅 d𝑝 𝑝 𝑇f𝑇s
d𝑥
(2) (3)
(𝑇f ― 𝑇s)2
d〈𝑆ht〉 = ℎfs𝐴fs
2
d𝑥
(4)
where d〈𝑆hc〉, d〈𝑆pd〉 and d〈𝑆ht〉 are the entropy generations caused by axial heat conduction, pressure drop and ineffective heat transfer, respectively. 𝑚, Z and R are the mass flow rate, compressibility factor and gas constant of the fluid, respectively. Tf and Ts are the temperatures of fluid and solid. ε is the porosity of regenerator matrices. Ar is the cross-sectional area of the regenerator. kf,x and ks,x are the effective thermal conductivities of fluid and solid which vary with the position x in the regenerator. hfs,x and Afs are the heat transfer coefficient and the effective heat exchange area between fluid and solid, respectively. In addition, the non-ideal properties of the gas and solid matrices including the thermal conductivity and the heat capacity are considered. The real values of the properties are input into the model with the CFD’s User Defined Functions [28, 29]. Figure 3 presents the distributions of regenerator losses per unit length in the regenerator, in which
the simulated results of two conditions with He-4 and He-3 are compared. The operating frequencies and charge pressures for both He-4 and He-3 are the same, respectively. When He-4 is employed, the pressure drop loss and the heat conduction loss show a decreasing trend with the length of each segment of the regenerator, but the heat transfer loss increases during the first around 80% length of the regenerator and then decreases near the right end. The pressure drop loss accounts for the main part in Reg I and Reg II (more than 80% approximately) and is slightly reduced in Reg III. But in Reg IV it becomes larger than that in Reg III. This is because the porosity of HoCu2 is much smaller than that of SS meshes. The heat transfer loss is a negligible portion of the total loss in Reg I and Reg II, and it increases sharply near the right end of Reg III. In Reg IV, the heat transfer loss first increases and then decreases, and accounts for almost half of the gross losses. The values near the left end of Reg IV are smaller than those near the right end of Reg III, which means the efficiency of heat transfer between matrix and fluid is effectively enhanced by using HoCu2. The heat conduction loss accounts for a small part in Reg I and Reg II (around 15%, larger than the heat transfer loss) but turns to be negligible in Reg III and Reg IV (less than 5% approximately). The distributions of regenerator losses with He-3 are little different from those of He-4. As the dash lines shown in figure 3, the pressure drop loss is slightly reduced by using He-3, while the heat conduction loss increases a little in Reg I and Reg II, and decreases in Reg IV. The heat transfer loss is effectively reduced by using He-3 than by He-4 in Reg III and Reg IV. This is because the heat capacity of He-3 is apparently smaller than that of He-4 at below 10 K. According to the above results, Reg IV has the largest and most complicated loss among the four segments. Thus, it should be focused on and some methods are needed to reduce the loss. The heat conduction loss in Reg IV is quite small and accounts for only about 2% of the gross loss, thus it is ignored in the following analyses.
To reduce the heat transfer loss in Reg IV, mixed matrices of Er3Ni and HoCu2 can be applied. Er3Ni is packed near the left end (warm end) and HoCu2 near the right end (cold end). Then the optimal ratio between the two matrices should be found. Figures 4 (a) and 4 (b) show the variations of pressure drop loss and heat transfer loss in Reg IV with different L4-1/L4 and Tc3, respectively, in which L4 is the length of Reg IV and L4-1 stands for the length of Er3Ni. Obviously, the heat transfer loss is reduced by using mixed matrices than by only either Er3Ni or HoCu2, while the pressure drop loss almost keeps unchanged. When Tc3 decreases from 20 K to 10 K, the pressure drop loss decreases slightly by about 2.8% with either He-4 or He-3. Thus, the heat transfer loss is the major determinant of L4-1/L4. When Tc3 is 20 K, the optimal proportion of Er3Ni is 0.6 with He-4 and 0.5 with He-3. When Tc3 is 10 K, the optimal proportion is changed to 0.4 and 0.3, respectively. The results indicate that the proportion of Er3Ni should be smaller while Tc3 becomes lower or He-3 is employed. The temperature profiles of the four cases are shown in figure 5. The temperatures at the transition from Er3Ni to HoCu2 are marked in the figure, which are at around 8.2 K. This is probably due to the difference of thermal capacities between Er3Ni and HoCu2. The thermal capacity of Er3Ni is larger than that of HoCu2 when the temperature is above 8.2 K [17]. Below 8.2 K, the condition is contrary. Thus, Er3Ni is suitable for matrix at higher temperature where it causes less heat transfer loss than HoCu2 does. As shown in figure 5, when L4-1/L4 is 0.3 and Tc3 is 10 K, the cold end temperature of Reg IV is the lowest by using He-3. The effect of charge pressure on the regenerator losses are investigated, as shown in figure 6. The pressure ratio (pa+Δp)/pa is kept constant at 1.15. When the charge pressure increases from 0.6 MPa to 1.6 MPa, the pressure drop loss increases from 8.45 mW/K to 11.32 mW/K with He-4, and from 7.70
mW/K to 10.08 mW/K with He-3. The heat transfer loss reaches the minimum value at 1.0 MPa with He-4 and increases monotonously with He-3. The results indicate that to reduce the charge pressure can effectively reduce the losses with He-3, but with He-4, there exists an optimal charge pressure which makes the sum of the two losses minimum. The effect of operating frequency on the regenerator losses are also studied, as shown in figure 7. The pressure ratio (pa+Δp)/pa is still kept constant at 1.15. When the operating frequency varies from 26 Hz to 36 Hz, the pressure drop loss reduces from 12.70 mW/K to 8.64 mW/K with He-4 and from 10.91 mW/K to 7.68 mW/K with He-3. The heat transfer loss reaches the minimum value at 30 Hz with He-4 and at 32 Hz with He-3. The results indicate that increasing the operating frequency can effectively reduce the pressure drop loss with both He-4 and He-3, while the frequency has the optimal value to make the heat transfer loss minimum. In addition, the optimal frequency with He-4 is lower than that with He-3. Generally, the charge pressure and operating frequency not only affects the losses in regenerator but also does the cooling performance of the cold end. The detailed analyses will be introduced in section 3.2. In addition, the operating frequency is also related to the phase characteristics in the regenerator, which will be analyzed in the next section.
2.3 Analysis of phase characteristics in regenerator
The operating frequency is a critical parameter which affects the phase characteristics significantly. Figure 8 shows the variations of pressure amplitude and volume flow rate at the inlet and outlet of the regenerator with the operating frequency. When the frequency increases from 26 Hz to 36 Hz, the pressure amplitude at the inlet of regenerator increases from 1.28×105 Pa to 1.42×105 Pa and then
decreases to 1.20×105 Pa. The maximum value occurs at 30 Hz. At the outlet of regenerator, it increases monotonously from 0.51×105 Pa to 0.68×105 Pa. Accordingly, the volume flow rate decreases from 2.75×10-4 m3/s to 1.93×10-4 m3/s at the inlet of regenerator and from 0.68×10-4 m3/s to 0.38×10-4 m3/s at the outlet. The phase difference between the dynamic pressure and volume flow rate is shown in figure 9. When the frequency increases from 26 Hz to 36 Hz, the phase difference increases from 42° to 60° at the inlet of regenerator, and also increases from -38° to -56° at the outlet. The results indicate that a higher efficiency results in the larger variation amplitude of the phase difference between the two ends. The zero point of phase difference is closest to the middle position of the regenerator when the frequency is 30 Hz. As for a multi-stage SPTC aimed at reaching below 10 K, a cryogenic phase-shifter is always used to improve the phase shifting ability of the last stage inertance tubes [17]. In this section, the effects of phase-shifter temperature (Tps) on the phase characteristics are also investigated. Figure 10 shows the variations of phase difference with Tps. When Tps is 10 K, the zero point of phase difference is nearly at the middle position of the regenerator. When Tps increases from 10 K to 40 K, the phase difference decreases from 49° to 38° at the inlet of regenerator and increases from -40° to -55° at the outlet. It seems that the phase difference along the whole regenerator moves toward the warm end integrally with the increasing Tps. In addition, when Tps is 40 K, the curve has big difference from the other three ones. The results indicate that a lower Tps can result in a better phase condition in the regenerator but the change of Tps has much bigger influence on the phase condition at above 30 K than that at below 30 K. As shown in figure 11, when Tps increases from 10 K to 40 K, the minimum values of Tc4 decreases from 5.09 K to 4.23 K, while the optimal frequency increases from 30 Hz to 34 Hz. The reason might be that when the
phase-shifter is at a lower temperature, the density of gas becomes larger and then the inertial effect of the phase-shifter is enhanced, which needs a relatively lower frequency to keep the phase characteristics at an optimal condition.
3. Optimizations of the four stages After the operating mechanisms in the fourth stage CF are studied by CFD modeling, a theoretical model for the four-stage SPTC is built to clarify the complicated interactions among the four stages. A schematic of the four-stage SPTC is shown in figure 12. It has a thermally-coupled arrangement. The first two stages are driven by a compressor and the last two stages by the other one. The cooling powers of the four stages are Qc1, Qc2, Qc3 and Qc4, respectively. Qp12, Qp13 and Qp14 are the precooling powers from the first stage to the second, third and fourth stages, respectively. Qp23, Qp24 and Qp34 have the similar definitions. Qps is the precooling power into the fourth stage phase-shifter. p1, f1, W1 are the charge pressure, operating frequency and input PV power of compressor I, respectively, and p2, f2, W2 are the corresponding ones of compressor II. The first three stages all adopt the coaxial configurations while the fourth stage uses in-line one, for which the reasons are as follows. For the first three stages, the coaxial arrangement is employed mainly to achieve a very compact system and also to ease the integration between the cold heads and the cooled devices as well as the thermal straps. Another benefit for the configurations is that the warm ends of the pulse tubes (including the WHXs and the phase-shifters) can be arranged outside the vacuum chamber, which significantly simplifies the overall structure of the SPTC and also greatly facilitates the experiments. As for the fourth stage, the in-line configuration is chosen in order to maximize the cooling performance by avoiding the loss caused by the flow reversal such as in the coaxial CHX. Of course, the
in-line configuration will lead to a relatively loose system. However, in view of that the cooling power of the fourth stage is very small, the priority should be given to the cooling performance.
3.1 Cooling temperatures of the four stages
The cooling temperatures of the first three stages are crucial to the cooling performance of the fourth stage. Generally, to improve the performance of the last stage, the precooling temperatures should be as low as possible. However, in this study, the four-stage SPTC is aimed at simultaneously providing cooling powers at four typical temperatures of 70 K, 40 K, 15 K and 5 K, respectively. Thus, the interactions between one stage and the other three stages become crucial. In figure 13, the effect of the first stage cooling temperature (Tc1) on the other three cooling temperatures (Tc2, Tc3 and Tc4) is presented. The input PV powers of the two compressors are 200 W and 80 W, respectively. There are no extra heat loads to the second, third and fourth stage CHXs.
It is
observed that Tc2, Tc3 and Tc4 all increase with the increasing Tc1, in which Tc2 grows most rapidly. With He-3 in place of He-4 in the last two stages, Tc2, Tc3 and Tc4 all decrease while the reduction of Tc3 is the largest one. Figure 14 shows the variations of Qp12, Qp13, Qp14 and Qc1 with Tc1. When Tc1 increases from 60 K to 100 K, Qc1 increases from 0 W to 5.4 W. Qp12, Qp13, and Qp14 all decreases with the increasing Tc1, which means the precooling powers required from the first stage to the other three stages become smaller when the precooling temperature is higher. With He-3 in the last two stages, Qp12, Qp13, and Qp14 almost keep unchanged and Qc1 increases by a little. Figure 15 shows the effect of Tc1 on the cooling power of each stage. Tc2, Tc3 and Tc4 are fixed at 40 K, 15 K and 5 K, respectively. When Tc1 increases from 60 K to 87 K, Qc2 decreases from 1.87 W to
zero, which indicates that no cooling power can be acquired at the second stage when Tc1 is above 87 K. For the third stage, Qc3 decreases from 0.126 W to zero when Tc1 increases from 60 K to 80 K. And for the fourth stage, Qc4 decreases from 0.038 W to zero when Tc1 increases from 60 K to 77 K. With He-3 in the last two stages, Qc3 and Qc4 increase dramatically while both Qc1 and Qc2 change little. The above results indicate that to increase Tc1 can not only result in a larger cooling power obtained from the first stage but also deteriorate the cooling performances of the other three stages, in which the second stage is greatly affected. In figure 16, the effects of Tc2 on Tc1, Tc3 and Tc4 are presented. When Tc2 increases from 30 K to 50 K, Tc1 increases from 63.0 K to 65.2 K, Tc3 from 10.5 K to 16.5 K, and Tc4 from 4.12 K to 5.98 K, respectively. With He-3 in the last two stages, Tc3 and Tc4 decrease dramatically but Tc1 increases by a little. The reason for the increase of Tc1 might be that the lower Tc3 and Tc4 by using He-3 result in the larger Qp13 and Qp14, which will bring larger heat load to the first stage. Figure 17 shows the variations of Qp23, Qp24 and Qc2 with Tc2. When Tc2 increases from 30 K to 50 K, Qc2 increases from zero to 1.75 W. Both Qp23 and Qp24 decrease with the increasing Tc2. With He-3 in the last two stages, both Qp23, Qp24 increase by a little while Qc2 decreases slightly. Figure 18 shows the effect of Tc2 on the cooling power of each stage. Tc1, Tc3 and Tc4 are fixed at 70 K, 15 K and 5 K, respectively. When Tc2 increases from 30 K to 50 K, Qc1 decreases from 4.45 W to 4.26 W while Qc2 increases from zero to 1.75 W. For the third stage, when Tc2 increases from 30 K to 46.2 K, Qc3 decreases from 0.2 W to zero. For the fourth stage, when Tc2 increases from 30 K to 43.2 K, Qc4 decreases from 0.04 W to zero. With He-3 in the last two stages, Qc3 and Qc4 increase dramatically while both Qc1 and Qc2 change slightly. The above results indicate that the effect of Tc2 on the performance of either the third or the fourth stage is more significant than on that of the first stage.
In figure 19, the effect of Tc3 on Tc1, Tc2 and Tc4 is presented. When Tc3 increases from 12 K to 20 K, Tc1 and Tc2 almost keep unchanged while Tc4 increases from 4.02 K to 6.50 K. With He-3 in the last two stages, Tc4 decreases dramatically but Tc1 and Tc2 only decreases by a little. Figure 20 shows the variations of Qp34, Qps and Qc3 with Tc3. When Tc3 increases from 12 K to 20 K, Qc3 increases from zero to 0.35 W. Both Qps and Qp34 decrease with the increasing Tc3. With He-3 in the last two stages, both Qps and Qp34 slightly increase but Qc3 increases dramatically. Figure 21 shows the effect of Tc3 on the cooling power of each stage. Tc1, Tc2 and Tc4 are fixed at 70 K, 40 K and 5 K, respectively. When Tc3 increases from 12 K to 20 K, Qc1 and Qc2 almost keep unchanged, while Qc3 increases from zero to 0.30 W. When Tc3 is above 15.7 K, no cooling power can be acquired at the fourth stage. With He-3 in the last two stages, both Qc3 and Qc4 increase dramatically while Qc1 and Qc2 increase by a little. The above results indicate that Tc3 is crucial to the cooling performance of the fourth stage but has no apparent effect on the first two stages.
3.2 Operating parameters of compressor II
The operating conditions of the compressor include the operating frequency, charge pressure and input PV power. According to the previous studies, the optimal values of charge pressure and operating frequency of the first two stages have already been found, which are 3.3 MPa and 55 Hz, respectively. In this section, the three parameters of compressor II will be optimized, in which the cooling performances of the last two stages will be analyzed together. It is necessary to point out that the pressure ratio is kept constant at 1.15 as the charge pressure or operating frequency varies. Figure 22 (a) shows the effect of f2 on Tc3 and Tc4, respectively, while W2 is kept at 80 W. With He-4, the lowest temperatures of Tc3 and Tc4 are reached at 34 Hz and 30 Hz, respectively. With He-3,
the optimal frequencies increase to 35 Hz and 32 Hz, respectively. The results indicate that the optimal frequency of the third stage is higher than that of the fourth stage and using He-3 in place of He-4 also results in the increase of the optimal frequency. Figure 22 (b) shows the effect of f2 on Qc3 and Qc4, while Tc3 and Tc4 are at 15 K and 5 K, respectively. With He-4, the third stage has available cooling powers at between 27.9 Hz and 40.2 Hz, in which the maximum value of 0.11 W is reached at 34.0 Hz. The fourth stage has available cooling powers at between 26.5 Hz and 34.6 Hz and the maximum value is 0.032 W. With He-3, the improvement of the cooling performance is obvious. The maximum values of Qc3 and Qc4 are increased to 0.22 W and 0.041 W, respectively. The adjustable range of operating frequency is also expanded. Figure 23 (a) shows the effects of p2 and f2 on Qc3 with a constant pressure ratio. With He-4, Qc3 increases apparently with the increasing p2. When p2 increases from 0.8 MPa to 1.4 MPa, the maximum value of Qc3 increases from 0.08 W to 0.12 W in which the optimal frequency increases from 32.5 Hz to 34.2 Hz. Similar phenomena can also be observed with He-3. Figure 23 (b) shows the effects of p2 and f2 on Qc4 with a constant pressure ratio. Obviously, a larger value of p2 leads to a larger Qc4. When p2 increases from 0.8 MPa to 1.4 MPa, the maximum value of Qc4 increases from 0.028 W to 0.036 W in which the optimal frequency is near 30 Hz. With He-3, the optimal frequency with p2 is near 30 Hz. The above results indicate that increasing the charge pressure can enhance the cooling powers at the last two stages. However, with a fixed compressor, the input PV power of compressor II also increases with the increasing charge pressure, which might reduce the cooling efficiency of the last two stages and would also deteriorate the cooling performances of the first two stages.
Figure 24 (a) shows the effects of p2 on ηc3 and ηc4 with a constant pressure ratio. The results indicate that there exists an optimal value of p2 which results in the highest efficiency. With He-4, the maximum values of ηc3 and ηc4 are 5.1% and 3.2%, respectively, while the optimal values of p2 are 1.2 MPa and 1.1 MPa, respectively. With He-3, the maximum values of ηc3 and ηc4 increase to 6.4% and 5.2%, respectively, while the optimal values of p2 decrease to 1.15 MPa and 1.05 MPa, respectively. Figure 24 (b) shows the effects of p2 on Qc1 and Qc2 with a constant pressure ratio. Obviously, with the increasing p2, the cooling powers of both the first and second stages decrease dramatically. In consideration of the overall performance of the four stages, the operating frequency and charge pressure of compressor II are chosen as 1.2 MPa and 30 Hz, respectively, with He-4, and 1.0 MPa and 32 Hz, respectively, with He-3. Figure 25 shows the effect of W2 on the no-load temperatures of the four stages. W1 is kept constant at 200 W. When W2 increases from 10 W to 150 W, Tc1 increases from 57 K to 73 K and Tc2 from 17 K to 43 K. Tc3 first decreases with the increasing W2 and then increases very slowly when W2 is above 70 W. Similar phenomena can also be observed with Tc4. It first decreases with the increasing W2 and then shows an increasing trend when W2 is above 100 W. The possible reason is that when W2 is above 70 W, increasing the value of W2 leads to a limited increment of the third stage cooling power but does increase the precooling powers from the third stage to the fourth stage (Qp34 and Qp4), which deteriorate the cooling performance of the third stage and then that of the fourth stage as well. Figure 26 shows the variations of the cooling powers and the relative Carnot efficiencies of the last two stages with W2. W1 is kept constant at 200 W. When W2 increases from 20 W to 150 W, Qc3 enhances from zero to 0.13 W. The highest efficiency occurs when W2 is around 68 W. When W2 increases from 60 W to 150 W, Qc4 enhances from zero to 0.045 W. The highest efficiency occurs when W2 is around
93 W. The results indicate that W2 should not be as large as possible. There is an optimal W2 which makes the third or fourth stage work at the highest efficiency.
3.3 Regenerator geometries
Based on the above analyses, the regenerator geometry including the diameter and length of each segment will be optimized in this section. When the length and diameter of one segment regenerator are optimized, the dimensions of all the other segments are kept constant at the values shown in Table 1. Figure 27 shows the variations of ηc4 and Qp14 with the length and diameter of Reg I. According to the results, when the diameter of Reg I increases from 13 mm to 14 mm and then to 15 mm, the optimal length of Reg I decreases from 36.8 mm to 36.4 mm and then to 34.5 mm accordingly, and the maximum values of ηc4 are 5.56%, 5.79% and 5.82%, respectively. The reason is that, with the increasing length of Reg I, the heat transfer between the fluid and matrix becomes more sufficient, which significantly improves the cooling performance. However, a longer Reg I also results in a larger pressure drop loss, which accounts for the main part of losses in Reg I, as shown in figure 3, and thus deteriorates the cooling performance. In addition, a larger diameter of Reg I will reduce the pressure drop loss. Thus, with a given diameter of Reg I, there always exists an optimal length of Reg I which makes the cooling efficiency become the highest, and the cooling efficiency becomes higher with a larger diameter of Reg I. Qp14 decreases with the increasing length of Reg I, but increases with the increasing diameter of Reg I. The results indicate that increasing the diameter of Reg I can effectively enhance the cooling efficiency of the fourth stage but also increase the precooling power provided by the first stage. In view of that when the diameter increases from 14 mm to 15 mm, the increase of ηc4 is relatively small, so the diameter of Reg I is determined to be 14 mm, then the length of Reg I is 36.4 mm.
Figure 28 shows the variations of ηc4 and Qp24 with the length and diameter of Reg II, respectively. Based on the results, when the diameter of Reg II increases from 11 mm to 12 mm and then to 13 mm, the optimal length of Reg II decreases from 29.0 mm to 28.2 mm and then to 27.1 mm accordingly, and the maximum values of ηc4 are 5.69%, 5.74% and 5.79%, respectively. Qp24 decreases with the increasing length of Reg II but rises with the increasing diameter of Reg II. The results are similar to those in Reg I. In consideration with that when the diameter increases from 14 mm to 15 mm, Qp24 will increase sharply, so the diameter of Reg II is determined to be 12 mm, and then the length of Reg II should be 28.2 mm. Figure 29 shows the variations of ηc4 and Qp34 with the length and diameter of Reg III, respectively. Based on the results, when the diameter of Reg III increases from 11 mm to 12 mm and then to 13 mm, the optimal length of Reg III decreases from 33.7 mm to 32.5 mm and then to 31.2 mm accordingly, the maximum values of ηc4 are 5.74%, 5.85% and 5.77%, respectively. The results are quite different from those in terms of Reg I and Reg II. The possible reason is that when the diameter of Reg III is 13 mm, the increase of Qp34 is beyond the cooling ability at the third stage and then reduces the cooling performances at the third and fourth stages. Thus, the diameter has an optimal value which makes the cooling efficiency to become the highest. The diameter of Reg III is determined to be 12 mm, then the length of Reg III is 31.2 mm. Figure 30 shows the variations of ηc4 with the length and diameter of Reg IV, respectively. It is observed that, when the diameter of Reg IV increases from 11 mm to 12 mm and then to 13 mm, the optimal length of Reg IV decreases from 37.0 mm to 35.9 mm and then to 35.6 mm accordingly, and the maximum values of ηc4 are 5.68%, 5.74% and 5.40%, respectively. Thus, the diameter of Reg IV is determined to be 10 mm, and the length of Reg IV is 36.0 mm.
3.4 Phase-shifter dimensions and temperatures
The phase-shifter plays a vital role in adjusting the phase characteristics in the cold fingers which directly influence the cooling performance. In this section, the effects of dimensions and temperatures of the phase-shifters on the cooling performances will be simulated. Each phase-shifter consists of two segments of inertance tubes with different inner diameters and a gas reservoir. The thinner tube connects to the warm end of the pulse tube while the thicker one to the gas reservoir. The dimensions of both the first and the second stage phase shifters are determined according to the previous work in the same laboratory [31]. The dimensions of the third and the fourth stage phase-shifters are optimized, as shown in figures 31 and 32, respectively. When the length and diameter of one inertance tube are optimized, the dimensions of all the other components are kept constant at the values shown in Table 1. Figure 31 shows the variations of Tc3 with the dimensions of the third stage inertance tubes. According to the results, for a certain diameter, there is always an optimal length of inertance tube which results in the minimum value of Tc3. For the third stage inertance tube I, the optimal diameter and length are 3 mm and 800 mm, respectively. For the third stage inertance tube II, the optimal values are 4 mm and 1180 mm, respectively. Figure 32 shows the variations of Tc4 with the geometries of the fourth stage inertance tubes. The results are similar to the ones in figure 31. For the fourth stage inertance tube I, the optimal diameter and length are 3 mm and 320 mm, respectively. For the inertance tube II, the optimal ones are 4 mm and 1260 mm, respectively. Figure 33 shows the effects of the gas reservoir volumes on the cooling performances of the third and fourth stages. According to the results, when the volume of the third stage gas reservoir is 46 cm3,
Tc3 achieves the minimum value. When the volume of the fourth stage gas reservoir is 39 cm3, Tc4 achieves the minimum value. The dimensions of the two gas reservoirs are then determined, in which the inner diameter and height of the third stage reservoir are 30 mm and 65 mm, respectively, while the corresponding values of the fourth stage one is 30 mm and 55 mm, respectively. The actual dimensions of the phase-shifters in the experiments are the same as the simulated ones. The double-inlet would also be a useful phase-shifting approach for the four-stage SPTC. However, we avoid adopting it for the following two reasons. First, as well known, the double-inlet may introduce the DC flow and then lead to the instability of both the flow and the cooling performance, which is a thorny problem in the practical applications. Second, based on our simulations and experiments, even if the four-stage SPTC reaches a very low temperature such as 3.3 K with He-3, the inertance tube-type phase-shifter still works effectively. Therefore, in the present work, we only choose the inertance tubes with the companying reservoir as the phase-shifter. The effects of phase-shifter temperatures on the cooling performances are shown in figure 34. For the self-precooled four-stage SPTC to be studied, any additional outside heat sink is not permitted. In other words, no any additional GM cryocoolers or cryogens is available and it has to rely on it itself alone to achieve the needed extremely low temperature. Therefore, the first stage phase-shifter can only be put in the ambient environment. However, in the simulations, the second stage phase-shifter might be cooled by the first stage CHX indeed and the temperature range is herein set to be 70-300 K. Similarly, the temperature ranges of the third and fourth stage phase-shifters are set to be 40-300 K and 10-300 K, respectively. As one phase-shifter temperature varies, the parameters of other phase-shifters are kept constant.
For the second stage, when the phase-shifter temperature reduces from 300 K to 70 K, the simulated Tc2 decreases from 33.5 K to 30.6 K. For the third stage, when the phase-shifter temperature reduces from 300 K to 40 K, the simulated Tc3 reduces from 12.8 K to 10.9 K. And for the fourth stage, the simulated Tc4 decreases dramatically from 11.2 K to 4.0 K when the phase-shifter temperature decreases from 300 K to 10 K. The results indicate that a lower phase-shifter temperature can help improve the cooling performances of the second, third and fourth stages as well, and the improvement of the fourth stage is significantly larger than that of either of the second or the third stages. However, it should also be noted that, for the self-precooled SPTC, due to the unavailability of additional outside heat sink, to realize the cryogenic phase-shifters of either the second or the third stage also has to consume the cooling powers from its upper stage, which will greatly reduce the precooling powers into its regenerator and then subsequently deteriorates the cooling performance at the second or the third stage. Considering that one of the main purposes of developing the four-stage SPTC is to simultaneously provide the cooling at four different temperatures as discussed in the beginning of the paper, the cooling powers at the second and the third stages must also be guaranteed. Based on the above considerations, we work out the compromise that in the actual experiments the phase-shifters of both the second and the third stages are put in the ambient environment, though this handling will inevitably have an adverse effect on the no-load temperature and the cooling capacity as well of the fourth stage. As for the fourth stage, an ambient phase-shifter is obviously unacceptable because the coldhead temperature is expected to reach around 3 K. Therefore, the fourth stage phase-shifter is cooled by the third stage in both simulations and experiments.
4. Conclusions
This paper conducts systematic theoretical analyses of a four-stage SPTC aimed at directly
reaching a temperature of around 3 K and also simultaneously achieving the cooling capacities at four temperatures varying from 3.3 K to 80 K for multiple uses. The systematic optimization for the four-stage SPTC is then conducted. The fourth stage cold finger is focused on and a 2-D axis-symmetric CFD model is set up to study the irreversible losses and phase characteristics. The operating mechanisms in the fourth stage are modelled, in which the distributions of the losses along the whole regenerator are analyzed
quantitatively and the phase characteristics simulated as well. The distributions of regenerator losses are simulated and analysed in detail. The pressure drop loss accounts for the main part of gross losses in the first three segments of the regenerator. The heat transfer loss increases rapidly in Reg IV and almost accounts for 50%. Mixed matrices consist of Er3Ni and HoCu2 are proved to be effective for reducing heat transfer loss in Reg IV. The distributions of phase difference along the whole regenerator are presented, with different operating frequencies and phase-shifter temperatures.
The interactions among the four stages are studied, in which the effect of parameters at each stage on the cooling performances of the other three stages is elaborated, respectively. The cooling temperature of some a stage has much more influence on the cooling performance of its lower stage but is less effective on its upper stage. With He-3, the optimal frequency is relatively lower than that with He-4, while the optimal charge pressure becomes larger. Increasing the PV power of compressor II can improve the cooling performances of the last two stages within certain limits, while too large will deteriorate the whole cooling performance. In addition, the regenerator geometry is also optimized. The experimental verifications of this study will be presented in Part B [32].
5. Acknowledgements
This work is partly supported by the Chinese Academy of Sciences (No. 6141A01070102), Shanghai Municipality (Nos. 18511110100, 18511110101, 18511110102, 19511106800, 19511106801, 19511106802 and 19ZR1465300) and the Aeronautical Science Foundation of China (No. 20162490005).
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Figure Captions Figure 1.
Axis-symmetric geometry of the fourth stage CF.
Figure 2.
Temperature profile in the fourth stage CF.
Figure 3.
Distributions of regenerator losses per unit length in the regenerator.
Figure 4.
Variations of (a) heat transfer loss and (b) pressure drop loss in Reg IV with 𝐿4 ― 1/𝐿4.
Figure 5.
Temperature profiles in Reg IV.
Figure 6.
Effects of charge pressure on heat transfer loss and pressure drop loss with a constant pressure ratio.
Figure 7.
Effects of operating frequency on heat transfer loss and pressure drop loss with a constant pressure ratio.
Figure 8.
Effects of operating frequency on pressure amplitudes and volume flow rates at the inlet and outlet of the regenerator with a constant pressure ratio.
Figure 9.
Distributions of phase difference in the regenerator with operating frequency.
Figure 10.
Distributions of phase difference in the regenerator with phase-shifter temperature.
Figure 11.
Effects of operating frequency on Tc4 with different phase-shifter temperatures and a constant pressure ratio.
Figure 12.
Schematic of the four-stage SPTC.
Figure 13.
Variations of Tc2, Tc3 and Tc4 with Tc1.
Figure 14.
Variations of Qp12, Qp13, Qp14 and Qc1 with Tc1.
Figure 15.
Variations of the four cooling powers with Tc1.
Figure 16.
Variations of Tc1, Tc3 and Tc4 with Tc2.
Figure 17.
Variations of Qp23, Qp24 and Qc2 with Tc2.
Figure 18.
Variations of the four cooling powers with Tc2.
Figure 19.
Variations of Tc1, Tc2 and Tc4 with Tc3.
Figure 20.
Variations of Qp34, Qps and Qc3 with Tc3.
Figure 21.
Variations of the four cooling powers with Tc3.
Figure 22.
Effects of f2 on (a) Tc3 and Tc4, (b) Qc3 and Qc4.
Figure 23.
Effects of f2 and p2 on (a) Qc3 and (b) Qc4 with a constant pressure ratio.
Figure 24.
Effects of p2 on (a) ηc3 and ηc4, (b) Qc1 and Qc2 with a constant pressure ratio.
Figure 25.
Variations of the four cooling temperatures with W2.
Figure 26.
Variations of Qc3, Qc4, ηc3 and ηc4 with W2.
Figure 27.
Effects of geometry of Reg I on ηc4 and Qp14 when other segments are kept constant at the values shown in Table 1.
Figure 28.
Effects of geometry of Reg II on ηc4 and Qp24 when other segments are kept constant at the values shown in Table 1.
Figure 29.
Effects of geometry of Reg III on ηc4 and Qp34 when other segments are kept constant at the values shown in Table 1.
Figure 30.
Effects of geometry of Reg IV on ηc4 when other segments are kept constant at the values shown in Table 1.
Figure 31.
Variations of Tc3 with dimensions of the third stage (a) inertance tube I and (b) tube II.
Figure 32.
Variations of Tc4 with dimensions of the fourth stage (a) inertance tube I and (b) tube II.
Figure 33.
Effects of the gas reservoir volumes on the cooling performances of the third and fourth stages.
Figure 34.
Effects of phase-shifter temperature on the cooling performance of each stage.
Table list Table 1. Dimensions of each stage component (Inner diameter × Length).
Component
Regenerator
1st stage (mm)
24×64
2nd stage (mm) I: 20×40 II: 20×30
3rd stage (mm) I: 16×45 II: 16×40 III: 16×38
4th stage (mm) I: 14×36.4 II: 12×28.2 III: 12×31.2 IV: 10×36
Pulse tube
14×76
12×81
9×128
8×80
CHX
19×8
16×8
12.5×6
10×5
WHX
14×8
12×6
9×6
8×5
I: 3.5×2800
I: 3×3200
I: 3×800
I: 3×320
II: 4.5×1400
II: 4.5×1200
II: 4×1180
II: 4×1260
60×80
60×80
30×65
30×55
Inertance tube Gas reservoir
Highlights for CRYOGENICS_2019_64_R2_Part A
Theoretical investigations on a 3.3 K four-stage SPTC are conducted.
A 2-D axis-symmetric CFD model for the fourth stage cold finger is set up.
Irreversible losses and phase characteristics in the regenerator are studied.
Interactions among the four stages are elaborated.
Systematic optimization on the four-stage SPTC is presented.
Conflict of Interest Statement The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted
S0011-2275(19)30080-3 https://doi.org/10.1016/j.cryogenics.2019.103014 JCRY 103014
To appear in:
Cryogenics
Received Date: Revised Date: Accepted Date:
4 March 2019 14 October 2019 2 December 2019
Please cite this article as: Dang, H., Zha, R., Tan, J., Zhang, T., Li, J., Li, N., Zhao, B., Zhao, Y., Tan, H., Xue, R., Investigations on a 3.3 K four-stage Stirling-type pulse tube cryocooler. Part A: Theoretical analyses and modeling, Cryogenics (2019), doi: https://doi.org/10.1016/j.cryogenics.2019.103014
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Investigations on a 3.3 K four-stage Stirling-type pulse tube cryocooler. Part A: Theoretical analyses and modeling Haizheng Danga,b,c,*, Rui Zhaa,b, Jun Tana,c, Tao Zhanga,b, Jiaqi Lia,b, Ning Lia, Bangjian Zhaoa,b, Yongjiang Zhaoa,b, Han Tana,b, Renjun Xuea,b a
State Key Laboratory of Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy
of Sciences, 500 Yutian Road, Shanghai 200083, China b University c Shanghai
*
of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China
Boreas Cryogenics Co., Ltd, 1388 Shuidian Road, Shanghai 200434, China
Corresponding author. State Key Laboratory of Infrared Physics, Shanghai Institute of Technical
Physics, Chinese Academy of Sciences, 500 Yutian Road, Shanghai 200083, China Tel.: +86-21-25051967 Fax: +86-21-65830734 Email: [email protected] (Haizheng Dang)
Abstract This paper conducts systematic theoretical analyses of a four-stage SPTC aimed at directly reaching a temperature of around 3 K and also simultaneously achieving the cooling capacities at four temperatures varying from 3.3 K to 80 K for multiple uses. The fourth stage cold finger is focused on and a 2-D axissymmetric CFD model is set up to study the irreversible losses and phase characteristics. The operating mechanisms in the fourth stage are modelled, in which the distributions of the losses along the whole regenerator are analysed quantitatively and the phase characteristics simulated as well. The interactions among the four stages are studied, in which the effect of parameters at each stage on the cooling performances of the other three stages is elaborated, respectively. The systematic optimization for the four-stage SPTC is then conducted. The detailed experimental verifications are presented in Part B.
Keywords: Four-stage Stirling-type pulse tube cryocooler; CFD model; Irreversible losses; Phase characteristics; Interactions among four stages.
Nomenclature
Afs
effective heat exchange area between fluid
T
temperature
and solid Ar
cross-sectional area of regenerator
Tf
temperature of fluid
f
frequency
Ts
temperature of solid
f1
frequency of compressor I
W1
input power of compressor I
f2
frequency of compressor II
W2
input power of compressor II
hfs
convective heat transfer coefficient
X
position in different components
Z
compressibility factor
between fluid and solid kf,x
thermal conductivity of fluid
ks,x
thermal conductivity of solid
L4
length of Reg IV
Greeks
L41
length of Er3Ni in Reg IV
ε
porosity of regenerator matrix
𝑚
mass flow rate
η
relative Carnot efficiency
p1
charge pressure of compressor I
p2
charge pressure of compressor IV
Subscripts
pd
dynamic pressure
1
first stage
pa
average pressure
2
second stage
Δp
pressure amplitude
3
third stage
R 𝑆hc
gas constant
4
fourth stage
entropy generation of heat conduction
c
cold end
𝑆ht
entropy generation of heat transfer
p
precooling power
𝑆pd
entropy generation of pressure drop
ps
phase-shifter
t
time
1. Introduction The pulse tube cryocooler (PTC) is a significant innovation in the regenerative refrigeration technology because it eliminates any moving component at the cold end, which endows it with evident advantages over the conventional regenerative cryocoolers such as GM and Stirling ones in terms of high reliability, long life and low vibration at the cold end [1, 2]. The Stirling-type PTC (SPTC) driven by the linear compressor employing the well-proven flexure springs and clearance technology further realizes the long operation time at the warm end, which thus makes it an attractive candidate for cooling cryogenic devices used in some special fields, especially in space [3, 4]. The past over three decades have seen a worldwide quest for the space-qualified SPTCs and also witnessed their successful in-orbit applications [5–11]. Up till now, single- and two-stage SPTCs which
cover the temperatures ranging from 20 K to 140 K have already been widely used in a variety of space missions, and the three-stage SPTCs developed for space applications with the cooling capacity at around 10 K are also becoming mature [5–18]. In the authors’ laboratory, a four-stage SPTC is being developed for the following three purposes. The first is for deep space exploration, in which the cryocooler is expected to provide the low-noise cooling at 4-6 K for the Si:As focal plane and/or Mid-Infrared Instrument under construction. The second purpose is to cool the superconducting nanowire single photon detector (SNSPD), which plays an important role in quantum key distribution, satellite laser ranging, and space quantum communication, etc., but the required cooling at around 2 K poses a serious challenge for the existing SPTCs. The third purpose is more ambitious because it is expected to simultaneously provide the cooling at four different temperatures varying from 3.3 K to 80 K for various applications, which will be much more compact and economical compared to using multiple cryocoolers. So far, only a few studies about the four-stage SPTCs have been reported. For example, in 2006, Olson et al. [19] reported a four-stage SPTC developed to provide cooling at 6 K and 18 K for MidInfrared Instrument in James Webb Space Telescope. In 2006 and 2007, Nast et al. [7, 20] introduced the continuous advances in the similar designs which reached a no-load temperature of 3.8 K. And In 2008, Nast et al. [21] continued to describe the improvement of a similar four-stage SPTC which reached a no-load temperature of 3.0 K with He-3 and provided simultaneous coolings at 4.5 K, 9.0 K, 30 K and 70 K for superconducting digital electronics communications programs. In 2017, Dang et al. [11] described the design of a four-stage SPTC for space applications and in 2018 Zha et al. [22] introduced its experimental results, in which a no-load temperature of 4.5 K with He-4 and the simultaneous cooling at 80 K, 60 K, 30 K and 5 K were achieved. In 2019, Dang et al. [23] reported the new progress of the
similar four-stage SPTC developed as an alternative of two choices expected to cool a SNSPD at 2 K, in which a no-load temperature of 4.38 K with He-4 was achieved. However, the previous studies mainly focused on briefly introducing structural design, describing practical fabrication and presenting experimental results, whereas the in-depth theoretical analyses of the inner process and operating mechanism have never been conducted. The four-stage SPTC cold finger (CF) has much more complicated physical structure and operating mechanism than those of single-, twoor three-stage ones. First, the staging configurations lead to difficulties in matching with compressors. Second, the irreversible losses in regenerators at much lower temperatures including pressure drop, axial heat conduction and ineffective heat transfer dramatically deteriorate the cooling performance. Third, the phase characteristics in regenerators become subtle, especially in the fourth stage. Fourth, the complex interactions among stages cause lots of difficulties in optimizing the cooling performance. In this part, the new advances in the four-stage SPTC will be reported together with the in-depth studies on the inner process, in which the fourth stage cold finger is focused on and a two-dimensional
(2-D) axis-symmetric CFD model is set up to study the irreversible losses and phase characteristics in its regenerator. The effects of precooling temperatures and operating parameters on the losses are discussed in detail. In view of the phase-shifting characteristics is critical to the performance improvement of the multi-stage cryocoolers [17, 24], the variations of phase difference with phaseshifter temperatures and operating frequencies are also analyzed. Then, the interactions among the four stages will be clarified and cooling performance optimized as well.
2. CFD simulations and analyses on the fourth stage CF
2.1 Model building
The establishment of the 2-D axis-symmetric CFD model of the fourth stage will draw on the experiences about the CFD models for the PTC in general [25–27], especially those developed in the same laboratory [28, 29]. As shown in figure 1, the fourth stage CF consists of an aftercooler, a segmented regenerator, three precooling heat exchangers (PHXs), a cold heat exchanger (CHX), a warm heat exchanger (WHX), a pulse tube (PT), a phase-shifter consists of two inertance tubes and a gas reservoir. The regenerator has four segments packed with #400 stainless steel (SS) meshes, #500 SS meshes, #635 SS meshes and rare-earth spherical materials, respectively. To investigate the effect of cryogenic phase-shifter, all of the WHX, inertance tubes and gas reservoir are regarded as the heat sink and they have isothermal boundary conditions in the model, of which the temperatures are variable in different cases. The compressor II is replaced by a sinusoidal pressure inlet setting at the left surface of the aftercooler, which is given as follows: 𝑝d = 𝑝a + ∆𝑝sin 2𝜋𝑓𝑡
(1)
where pd, pa, Δp and f are the dynamic pressure, average pressure (charge pressure), pressure amplitude and operating frequency, respectively. All the input powers mentioned in this paper are actually PV powers, which are calculated by integrating the dynamic pressures and volume flow rates at the outlet of the compressors. A first-order implicit unsteady pressure based segregated solver with PISO pressure-velocity is used in this CFD model. A second-order upwind scheme is applied to solve the continuity, momentum and energy equations. The convergence criterion is 10-6 for mass, velocity and energy. A time step of 1×10-5 s with 50 maximum iterations is set for cases.
The temperature profile among the fourth stage CF is shown in figure 2. A no-load temperature of 3.25 K is reached with a charge pressure of 1.2 MPa and an operating frequency of 32 Hz, while He-3 is used. The phase-shifter temperature is 30 K and the precooling temperatures are 70 K, 40 K and 10 K, respectively.
2.2 Regenerator losses
In the thermal non-equilibrium mode, three types of irreversible losses in the regenerator including axial heat conduction, pressure drop and ineffective heat transfer between fluid and solid are considered, which can be calculated by [30]: d〈𝑆hc〉 =
[(1 ― 𝜀)𝑘s,x + 𝜀𝑘f,x]𝐴r d𝑇
( ) d𝑥
𝑇2 d〈𝑆pd〉 = ―
𝑚𝑍𝑅 d𝑝 𝑝 𝑇f𝑇s
d𝑥
(2) (3)
(𝑇f ― 𝑇s)2
d〈𝑆ht〉 = ℎfs𝐴fs
2
d𝑥
(4)
where d〈𝑆hc〉, d〈𝑆pd〉 and d〈𝑆ht〉 are the entropy generations caused by axial heat conduction, pressure drop and ineffective heat transfer, respectively. 𝑚, Z and R are the mass flow rate, compressibility factor and gas constant of the fluid, respectively. Tf and Ts are the temperatures of fluid and solid. ε is the porosity of regenerator matrices. Ar is the cross-sectional area of the regenerator. kf,x and ks,x are the effective thermal conductivities of fluid and solid which vary with the position x in the regenerator. hfs,x and Afs are the heat transfer coefficient and the effective heat exchange area between fluid and solid, respectively. In addition, the non-ideal properties of the gas and solid matrices including the thermal conductivity and the heat capacity are considered. The real values of the properties are input into the model with the CFD’s User Defined Functions [28, 29]. Figure 3 presents the distributions of regenerator losses per unit length in the regenerator, in which
the simulated results of two conditions with He-4 and He-3 are compared. The operating frequencies and charge pressures for both He-4 and He-3 are the same, respectively. When He-4 is employed, the pressure drop loss and the heat conduction loss show a decreasing trend with the length of each segment of the regenerator, but the heat transfer loss increases during the first around 80% length of the regenerator and then decreases near the right end. The pressure drop loss accounts for the main part in Reg I and Reg II (more than 80% approximately) and is slightly reduced in Reg III. But in Reg IV it becomes larger than that in Reg III. This is because the porosity of HoCu2 is much smaller than that of SS meshes. The heat transfer loss is a negligible portion of the total loss in Reg I and Reg II, and it increases sharply near the right end of Reg III. In Reg IV, the heat transfer loss first increases and then decreases, and accounts for almost half of the gross losses. The values near the left end of Reg IV are smaller than those near the right end of Reg III, which means the efficiency of heat transfer between matrix and fluid is effectively enhanced by using HoCu2. The heat conduction loss accounts for a small part in Reg I and Reg II (around 15%, larger than the heat transfer loss) but turns to be negligible in Reg III and Reg IV (less than 5% approximately). The distributions of regenerator losses with He-3 are little different from those of He-4. As the dash lines shown in figure 3, the pressure drop loss is slightly reduced by using He-3, while the heat conduction loss increases a little in Reg I and Reg II, and decreases in Reg IV. The heat transfer loss is effectively reduced by using He-3 than by He-4 in Reg III and Reg IV. This is because the heat capacity of He-3 is apparently smaller than that of He-4 at below 10 K. According to the above results, Reg IV has the largest and most complicated loss among the four segments. Thus, it should be focused on and some methods are needed to reduce the loss. The heat conduction loss in Reg IV is quite small and accounts for only about 2% of the gross loss, thus it is ignored in the following analyses.
To reduce the heat transfer loss in Reg IV, mixed matrices of Er3Ni and HoCu2 can be applied. Er3Ni is packed near the left end (warm end) and HoCu2 near the right end (cold end). Then the optimal ratio between the two matrices should be found. Figures 4 (a) and 4 (b) show the variations of pressure drop loss and heat transfer loss in Reg IV with different L4-1/L4 and Tc3, respectively, in which L4 is the length of Reg IV and L4-1 stands for the length of Er3Ni. Obviously, the heat transfer loss is reduced by using mixed matrices than by only either Er3Ni or HoCu2, while the pressure drop loss almost keeps unchanged. When Tc3 decreases from 20 K to 10 K, the pressure drop loss decreases slightly by about 2.8% with either He-4 or He-3. Thus, the heat transfer loss is the major determinant of L4-1/L4. When Tc3 is 20 K, the optimal proportion of Er3Ni is 0.6 with He-4 and 0.5 with He-3. When Tc3 is 10 K, the optimal proportion is changed to 0.4 and 0.3, respectively. The results indicate that the proportion of Er3Ni should be smaller while Tc3 becomes lower or He-3 is employed. The temperature profiles of the four cases are shown in figure 5. The temperatures at the transition from Er3Ni to HoCu2 are marked in the figure, which are at around 8.2 K. This is probably due to the difference of thermal capacities between Er3Ni and HoCu2. The thermal capacity of Er3Ni is larger than that of HoCu2 when the temperature is above 8.2 K [17]. Below 8.2 K, the condition is contrary. Thus, Er3Ni is suitable for matrix at higher temperature where it causes less heat transfer loss than HoCu2 does. As shown in figure 5, when L4-1/L4 is 0.3 and Tc3 is 10 K, the cold end temperature of Reg IV is the lowest by using He-3. The effect of charge pressure on the regenerator losses are investigated, as shown in figure 6. The pressure ratio (pa+Δp)/pa is kept constant at 1.15. When the charge pressure increases from 0.6 MPa to 1.6 MPa, the pressure drop loss increases from 8.45 mW/K to 11.32 mW/K with He-4, and from 7.70
mW/K to 10.08 mW/K with He-3. The heat transfer loss reaches the minimum value at 1.0 MPa with He-4 and increases monotonously with He-3. The results indicate that to reduce the charge pressure can effectively reduce the losses with He-3, but with He-4, there exists an optimal charge pressure which makes the sum of the two losses minimum. The effect of operating frequency on the regenerator losses are also studied, as shown in figure 7. The pressure ratio (pa+Δp)/pa is still kept constant at 1.15. When the operating frequency varies from 26 Hz to 36 Hz, the pressure drop loss reduces from 12.70 mW/K to 8.64 mW/K with He-4 and from 10.91 mW/K to 7.68 mW/K with He-3. The heat transfer loss reaches the minimum value at 30 Hz with He-4 and at 32 Hz with He-3. The results indicate that increasing the operating frequency can effectively reduce the pressure drop loss with both He-4 and He-3, while the frequency has the optimal value to make the heat transfer loss minimum. In addition, the optimal frequency with He-4 is lower than that with He-3. Generally, the charge pressure and operating frequency not only affects the losses in regenerator but also does the cooling performance of the cold end. The detailed analyses will be introduced in section 3.2. In addition, the operating frequency is also related to the phase characteristics in the regenerator, which will be analyzed in the next section.
2.3 Analysis of phase characteristics in regenerator
The operating frequency is a critical parameter which affects the phase characteristics significantly. Figure 8 shows the variations of pressure amplitude and volume flow rate at the inlet and outlet of the regenerator with the operating frequency. When the frequency increases from 26 Hz to 36 Hz, the pressure amplitude at the inlet of regenerator increases from 1.28×105 Pa to 1.42×105 Pa and then
decreases to 1.20×105 Pa. The maximum value occurs at 30 Hz. At the outlet of regenerator, it increases monotonously from 0.51×105 Pa to 0.68×105 Pa. Accordingly, the volume flow rate decreases from 2.75×10-4 m3/s to 1.93×10-4 m3/s at the inlet of regenerator and from 0.68×10-4 m3/s to 0.38×10-4 m3/s at the outlet. The phase difference between the dynamic pressure and volume flow rate is shown in figure 9. When the frequency increases from 26 Hz to 36 Hz, the phase difference increases from 42° to 60° at the inlet of regenerator, and also increases from -38° to -56° at the outlet. The results indicate that a higher efficiency results in the larger variation amplitude of the phase difference between the two ends. The zero point of phase difference is closest to the middle position of the regenerator when the frequency is 30 Hz. As for a multi-stage SPTC aimed at reaching below 10 K, a cryogenic phase-shifter is always used to improve the phase shifting ability of the last stage inertance tubes [17]. In this section, the effects of phase-shifter temperature (Tps) on the phase characteristics are also investigated. Figure 10 shows the variations of phase difference with Tps. When Tps is 10 K, the zero point of phase difference is nearly at the middle position of the regenerator. When Tps increases from 10 K to 40 K, the phase difference decreases from 49° to 38° at the inlet of regenerator and increases from -40° to -55° at the outlet. It seems that the phase difference along the whole regenerator moves toward the warm end integrally with the increasing Tps. In addition, when Tps is 40 K, the curve has big difference from the other three ones. The results indicate that a lower Tps can result in a better phase condition in the regenerator but the change of Tps has much bigger influence on the phase condition at above 30 K than that at below 30 K. As shown in figure 11, when Tps increases from 10 K to 40 K, the minimum values of Tc4 decreases from 5.09 K to 4.23 K, while the optimal frequency increases from 30 Hz to 34 Hz. The reason might be that when the
phase-shifter is at a lower temperature, the density of gas becomes larger and then the inertial effect of the phase-shifter is enhanced, which needs a relatively lower frequency to keep the phase characteristics at an optimal condition.
3. Optimizations of the four stages After the operating mechanisms in the fourth stage CF are studied by CFD modeling, a theoretical model for the four-stage SPTC is built to clarify the complicated interactions among the four stages. A schematic of the four-stage SPTC is shown in figure 12. It has a thermally-coupled arrangement. The first two stages are driven by a compressor and the last two stages by the other one. The cooling powers of the four stages are Qc1, Qc2, Qc3 and Qc4, respectively. Qp12, Qp13 and Qp14 are the precooling powers from the first stage to the second, third and fourth stages, respectively. Qp23, Qp24 and Qp34 have the similar definitions. Qps is the precooling power into the fourth stage phase-shifter. p1, f1, W1 are the charge pressure, operating frequency and input PV power of compressor I, respectively, and p2, f2, W2 are the corresponding ones of compressor II. The first three stages all adopt the coaxial configurations while the fourth stage uses in-line one, for which the reasons are as follows. For the first three stages, the coaxial arrangement is employed mainly to achieve a very compact system and also to ease the integration between the cold heads and the cooled devices as well as the thermal straps. Another benefit for the configurations is that the warm ends of the pulse tubes (including the WHXs and the phase-shifters) can be arranged outside the vacuum chamber, which significantly simplifies the overall structure of the SPTC and also greatly facilitates the experiments. As for the fourth stage, the in-line configuration is chosen in order to maximize the cooling performance by avoiding the loss caused by the flow reversal such as in the coaxial CHX. Of course, the
in-line configuration will lead to a relatively loose system. However, in view of that the cooling power of the fourth stage is very small, the priority should be given to the cooling performance.
3.1 Cooling temperatures of the four stages
The cooling temperatures of the first three stages are crucial to the cooling performance of the fourth stage. Generally, to improve the performance of the last stage, the precooling temperatures should be as low as possible. However, in this study, the four-stage SPTC is aimed at simultaneously providing cooling powers at four typical temperatures of 70 K, 40 K, 15 K and 5 K, respectively. Thus, the interactions between one stage and the other three stages become crucial. In figure 13, the effect of the first stage cooling temperature (Tc1) on the other three cooling temperatures (Tc2, Tc3 and Tc4) is presented. The input PV powers of the two compressors are 200 W and 80 W, respectively. There are no extra heat loads to the second, third and fourth stage CHXs.
It is
observed that Tc2, Tc3 and Tc4 all increase with the increasing Tc1, in which Tc2 grows most rapidly. With He-3 in place of He-4 in the last two stages, Tc2, Tc3 and Tc4 all decrease while the reduction of Tc3 is the largest one. Figure 14 shows the variations of Qp12, Qp13, Qp14 and Qc1 with Tc1. When Tc1 increases from 60 K to 100 K, Qc1 increases from 0 W to 5.4 W. Qp12, Qp13, and Qp14 all decreases with the increasing Tc1, which means the precooling powers required from the first stage to the other three stages become smaller when the precooling temperature is higher. With He-3 in the last two stages, Qp12, Qp13, and Qp14 almost keep unchanged and Qc1 increases by a little. Figure 15 shows the effect of Tc1 on the cooling power of each stage. Tc2, Tc3 and Tc4 are fixed at 40 K, 15 K and 5 K, respectively. When Tc1 increases from 60 K to 87 K, Qc2 decreases from 1.87 W to
zero, which indicates that no cooling power can be acquired at the second stage when Tc1 is above 87 K. For the third stage, Qc3 decreases from 0.126 W to zero when Tc1 increases from 60 K to 80 K. And for the fourth stage, Qc4 decreases from 0.038 W to zero when Tc1 increases from 60 K to 77 K. With He-3 in the last two stages, Qc3 and Qc4 increase dramatically while both Qc1 and Qc2 change little. The above results indicate that to increase Tc1 can not only result in a larger cooling power obtained from the first stage but also deteriorate the cooling performances of the other three stages, in which the second stage is greatly affected. In figure 16, the effects of Tc2 on Tc1, Tc3 and Tc4 are presented. When Tc2 increases from 30 K to 50 K, Tc1 increases from 63.0 K to 65.2 K, Tc3 from 10.5 K to 16.5 K, and Tc4 from 4.12 K to 5.98 K, respectively. With He-3 in the last two stages, Tc3 and Tc4 decrease dramatically but Tc1 increases by a little. The reason for the increase of Tc1 might be that the lower Tc3 and Tc4 by using He-3 result in the larger Qp13 and Qp14, which will bring larger heat load to the first stage. Figure 17 shows the variations of Qp23, Qp24 and Qc2 with Tc2. When Tc2 increases from 30 K to 50 K, Qc2 increases from zero to 1.75 W. Both Qp23 and Qp24 decrease with the increasing Tc2. With He-3 in the last two stages, both Qp23, Qp24 increase by a little while Qc2 decreases slightly. Figure 18 shows the effect of Tc2 on the cooling power of each stage. Tc1, Tc3 and Tc4 are fixed at 70 K, 15 K and 5 K, respectively. When Tc2 increases from 30 K to 50 K, Qc1 decreases from 4.45 W to 4.26 W while Qc2 increases from zero to 1.75 W. For the third stage, when Tc2 increases from 30 K to 46.2 K, Qc3 decreases from 0.2 W to zero. For the fourth stage, when Tc2 increases from 30 K to 43.2 K, Qc4 decreases from 0.04 W to zero. With He-3 in the last two stages, Qc3 and Qc4 increase dramatically while both Qc1 and Qc2 change slightly. The above results indicate that the effect of Tc2 on the performance of either the third or the fourth stage is more significant than on that of the first stage.
In figure 19, the effect of Tc3 on Tc1, Tc2 and Tc4 is presented. When Tc3 increases from 12 K to 20 K, Tc1 and Tc2 almost keep unchanged while Tc4 increases from 4.02 K to 6.50 K. With He-3 in the last two stages, Tc4 decreases dramatically but Tc1 and Tc2 only decreases by a little. Figure 20 shows the variations of Qp34, Qps and Qc3 with Tc3. When Tc3 increases from 12 K to 20 K, Qc3 increases from zero to 0.35 W. Both Qps and Qp34 decrease with the increasing Tc3. With He-3 in the last two stages, both Qps and Qp34 slightly increase but Qc3 increases dramatically. Figure 21 shows the effect of Tc3 on the cooling power of each stage. Tc1, Tc2 and Tc4 are fixed at 70 K, 40 K and 5 K, respectively. When Tc3 increases from 12 K to 20 K, Qc1 and Qc2 almost keep unchanged, while Qc3 increases from zero to 0.30 W. When Tc3 is above 15.7 K, no cooling power can be acquired at the fourth stage. With He-3 in the last two stages, both Qc3 and Qc4 increase dramatically while Qc1 and Qc2 increase by a little. The above results indicate that Tc3 is crucial to the cooling performance of the fourth stage but has no apparent effect on the first two stages.
3.2 Operating parameters of compressor II
The operating conditions of the compressor include the operating frequency, charge pressure and input PV power. According to the previous studies, the optimal values of charge pressure and operating frequency of the first two stages have already been found, which are 3.3 MPa and 55 Hz, respectively. In this section, the three parameters of compressor II will be optimized, in which the cooling performances of the last two stages will be analyzed together. It is necessary to point out that the pressure ratio is kept constant at 1.15 as the charge pressure or operating frequency varies. Figure 22 (a) shows the effect of f2 on Tc3 and Tc4, respectively, while W2 is kept at 80 W. With He-4, the lowest temperatures of Tc3 and Tc4 are reached at 34 Hz and 30 Hz, respectively. With He-3,
the optimal frequencies increase to 35 Hz and 32 Hz, respectively. The results indicate that the optimal frequency of the third stage is higher than that of the fourth stage and using He-3 in place of He-4 also results in the increase of the optimal frequency. Figure 22 (b) shows the effect of f2 on Qc3 and Qc4, while Tc3 and Tc4 are at 15 K and 5 K, respectively. With He-4, the third stage has available cooling powers at between 27.9 Hz and 40.2 Hz, in which the maximum value of 0.11 W is reached at 34.0 Hz. The fourth stage has available cooling powers at between 26.5 Hz and 34.6 Hz and the maximum value is 0.032 W. With He-3, the improvement of the cooling performance is obvious. The maximum values of Qc3 and Qc4 are increased to 0.22 W and 0.041 W, respectively. The adjustable range of operating frequency is also expanded. Figure 23 (a) shows the effects of p2 and f2 on Qc3 with a constant pressure ratio. With He-4, Qc3 increases apparently with the increasing p2. When p2 increases from 0.8 MPa to 1.4 MPa, the maximum value of Qc3 increases from 0.08 W to 0.12 W in which the optimal frequency increases from 32.5 Hz to 34.2 Hz. Similar phenomena can also be observed with He-3. Figure 23 (b) shows the effects of p2 and f2 on Qc4 with a constant pressure ratio. Obviously, a larger value of p2 leads to a larger Qc4. When p2 increases from 0.8 MPa to 1.4 MPa, the maximum value of Qc4 increases from 0.028 W to 0.036 W in which the optimal frequency is near 30 Hz. With He-3, the optimal frequency with p2 is near 30 Hz. The above results indicate that increasing the charge pressure can enhance the cooling powers at the last two stages. However, with a fixed compressor, the input PV power of compressor II also increases with the increasing charge pressure, which might reduce the cooling efficiency of the last two stages and would also deteriorate the cooling performances of the first two stages.
Figure 24 (a) shows the effects of p2 on ηc3 and ηc4 with a constant pressure ratio. The results indicate that there exists an optimal value of p2 which results in the highest efficiency. With He-4, the maximum values of ηc3 and ηc4 are 5.1% and 3.2%, respectively, while the optimal values of p2 are 1.2 MPa and 1.1 MPa, respectively. With He-3, the maximum values of ηc3 and ηc4 increase to 6.4% and 5.2%, respectively, while the optimal values of p2 decrease to 1.15 MPa and 1.05 MPa, respectively. Figure 24 (b) shows the effects of p2 on Qc1 and Qc2 with a constant pressure ratio. Obviously, with the increasing p2, the cooling powers of both the first and second stages decrease dramatically. In consideration of the overall performance of the four stages, the operating frequency and charge pressure of compressor II are chosen as 1.2 MPa and 30 Hz, respectively, with He-4, and 1.0 MPa and 32 Hz, respectively, with He-3. Figure 25 shows the effect of W2 on the no-load temperatures of the four stages. W1 is kept constant at 200 W. When W2 increases from 10 W to 150 W, Tc1 increases from 57 K to 73 K and Tc2 from 17 K to 43 K. Tc3 first decreases with the increasing W2 and then increases very slowly when W2 is above 70 W. Similar phenomena can also be observed with Tc4. It first decreases with the increasing W2 and then shows an increasing trend when W2 is above 100 W. The possible reason is that when W2 is above 70 W, increasing the value of W2 leads to a limited increment of the third stage cooling power but does increase the precooling powers from the third stage to the fourth stage (Qp34 and Qp4), which deteriorate the cooling performance of the third stage and then that of the fourth stage as well. Figure 26 shows the variations of the cooling powers and the relative Carnot efficiencies of the last two stages with W2. W1 is kept constant at 200 W. When W2 increases from 20 W to 150 W, Qc3 enhances from zero to 0.13 W. The highest efficiency occurs when W2 is around 68 W. When W2 increases from 60 W to 150 W, Qc4 enhances from zero to 0.045 W. The highest efficiency occurs when W2 is around
93 W. The results indicate that W2 should not be as large as possible. There is an optimal W2 which makes the third or fourth stage work at the highest efficiency.
3.3 Regenerator geometries
Based on the above analyses, the regenerator geometry including the diameter and length of each segment will be optimized in this section. When the length and diameter of one segment regenerator are optimized, the dimensions of all the other segments are kept constant at the values shown in Table 1. Figure 27 shows the variations of ηc4 and Qp14 with the length and diameter of Reg I. According to the results, when the diameter of Reg I increases from 13 mm to 14 mm and then to 15 mm, the optimal length of Reg I decreases from 36.8 mm to 36.4 mm and then to 34.5 mm accordingly, and the maximum values of ηc4 are 5.56%, 5.79% and 5.82%, respectively. The reason is that, with the increasing length of Reg I, the heat transfer between the fluid and matrix becomes more sufficient, which significantly improves the cooling performance. However, a longer Reg I also results in a larger pressure drop loss, which accounts for the main part of losses in Reg I, as shown in figure 3, and thus deteriorates the cooling performance. In addition, a larger diameter of Reg I will reduce the pressure drop loss. Thus, with a given diameter of Reg I, there always exists an optimal length of Reg I which makes the cooling efficiency become the highest, and the cooling efficiency becomes higher with a larger diameter of Reg I. Qp14 decreases with the increasing length of Reg I, but increases with the increasing diameter of Reg I. The results indicate that increasing the diameter of Reg I can effectively enhance the cooling efficiency of the fourth stage but also increase the precooling power provided by the first stage. In view of that when the diameter increases from 14 mm to 15 mm, the increase of ηc4 is relatively small, so the diameter of Reg I is determined to be 14 mm, then the length of Reg I is 36.4 mm.
Figure 28 shows the variations of ηc4 and Qp24 with the length and diameter of Reg II, respectively. Based on the results, when the diameter of Reg II increases from 11 mm to 12 mm and then to 13 mm, the optimal length of Reg II decreases from 29.0 mm to 28.2 mm and then to 27.1 mm accordingly, and the maximum values of ηc4 are 5.69%, 5.74% and 5.79%, respectively. Qp24 decreases with the increasing length of Reg II but rises with the increasing diameter of Reg II. The results are similar to those in Reg I. In consideration with that when the diameter increases from 14 mm to 15 mm, Qp24 will increase sharply, so the diameter of Reg II is determined to be 12 mm, and then the length of Reg II should be 28.2 mm. Figure 29 shows the variations of ηc4 and Qp34 with the length and diameter of Reg III, respectively. Based on the results, when the diameter of Reg III increases from 11 mm to 12 mm and then to 13 mm, the optimal length of Reg III decreases from 33.7 mm to 32.5 mm and then to 31.2 mm accordingly, the maximum values of ηc4 are 5.74%, 5.85% and 5.77%, respectively. The results are quite different from those in terms of Reg I and Reg II. The possible reason is that when the diameter of Reg III is 13 mm, the increase of Qp34 is beyond the cooling ability at the third stage and then reduces the cooling performances at the third and fourth stages. Thus, the diameter has an optimal value which makes the cooling efficiency to become the highest. The diameter of Reg III is determined to be 12 mm, then the length of Reg III is 31.2 mm. Figure 30 shows the variations of ηc4 with the length and diameter of Reg IV, respectively. It is observed that, when the diameter of Reg IV increases from 11 mm to 12 mm and then to 13 mm, the optimal length of Reg IV decreases from 37.0 mm to 35.9 mm and then to 35.6 mm accordingly, and the maximum values of ηc4 are 5.68%, 5.74% and 5.40%, respectively. Thus, the diameter of Reg IV is determined to be 10 mm, and the length of Reg IV is 36.0 mm.
3.4 Phase-shifter dimensions and temperatures
The phase-shifter plays a vital role in adjusting the phase characteristics in the cold fingers which directly influence the cooling performance. In this section, the effects of dimensions and temperatures of the phase-shifters on the cooling performances will be simulated. Each phase-shifter consists of two segments of inertance tubes with different inner diameters and a gas reservoir. The thinner tube connects to the warm end of the pulse tube while the thicker one to the gas reservoir. The dimensions of both the first and the second stage phase shifters are determined according to the previous work in the same laboratory [31]. The dimensions of the third and the fourth stage phase-shifters are optimized, as shown in figures 31 and 32, respectively. When the length and diameter of one inertance tube are optimized, the dimensions of all the other components are kept constant at the values shown in Table 1. Figure 31 shows the variations of Tc3 with the dimensions of the third stage inertance tubes. According to the results, for a certain diameter, there is always an optimal length of inertance tube which results in the minimum value of Tc3. For the third stage inertance tube I, the optimal diameter and length are 3 mm and 800 mm, respectively. For the third stage inertance tube II, the optimal values are 4 mm and 1180 mm, respectively. Figure 32 shows the variations of Tc4 with the geometries of the fourth stage inertance tubes. The results are similar to the ones in figure 31. For the fourth stage inertance tube I, the optimal diameter and length are 3 mm and 320 mm, respectively. For the inertance tube II, the optimal ones are 4 mm and 1260 mm, respectively. Figure 33 shows the effects of the gas reservoir volumes on the cooling performances of the third and fourth stages. According to the results, when the volume of the third stage gas reservoir is 46 cm3,
Tc3 achieves the minimum value. When the volume of the fourth stage gas reservoir is 39 cm3, Tc4 achieves the minimum value. The dimensions of the two gas reservoirs are then determined, in which the inner diameter and height of the third stage reservoir are 30 mm and 65 mm, respectively, while the corresponding values of the fourth stage one is 30 mm and 55 mm, respectively. The actual dimensions of the phase-shifters in the experiments are the same as the simulated ones. The double-inlet would also be a useful phase-shifting approach for the four-stage SPTC. However, we avoid adopting it for the following two reasons. First, as well known, the double-inlet may introduce the DC flow and then lead to the instability of both the flow and the cooling performance, which is a thorny problem in the practical applications. Second, based on our simulations and experiments, even if the four-stage SPTC reaches a very low temperature such as 3.3 K with He-3, the inertance tube-type phase-shifter still works effectively. Therefore, in the present work, we only choose the inertance tubes with the companying reservoir as the phase-shifter. The effects of phase-shifter temperatures on the cooling performances are shown in figure 34. For the self-precooled four-stage SPTC to be studied, any additional outside heat sink is not permitted. In other words, no any additional GM cryocoolers or cryogens is available and it has to rely on it itself alone to achieve the needed extremely low temperature. Therefore, the first stage phase-shifter can only be put in the ambient environment. However, in the simulations, the second stage phase-shifter might be cooled by the first stage CHX indeed and the temperature range is herein set to be 70-300 K. Similarly, the temperature ranges of the third and fourth stage phase-shifters are set to be 40-300 K and 10-300 K, respectively. As one phase-shifter temperature varies, the parameters of other phase-shifters are kept constant.
For the second stage, when the phase-shifter temperature reduces from 300 K to 70 K, the simulated Tc2 decreases from 33.5 K to 30.6 K. For the third stage, when the phase-shifter temperature reduces from 300 K to 40 K, the simulated Tc3 reduces from 12.8 K to 10.9 K. And for the fourth stage, the simulated Tc4 decreases dramatically from 11.2 K to 4.0 K when the phase-shifter temperature decreases from 300 K to 10 K. The results indicate that a lower phase-shifter temperature can help improve the cooling performances of the second, third and fourth stages as well, and the improvement of the fourth stage is significantly larger than that of either of the second or the third stages. However, it should also be noted that, for the self-precooled SPTC, due to the unavailability of additional outside heat sink, to realize the cryogenic phase-shifters of either the second or the third stage also has to consume the cooling powers from its upper stage, which will greatly reduce the precooling powers into its regenerator and then subsequently deteriorates the cooling performance at the second or the third stage. Considering that one of the main purposes of developing the four-stage SPTC is to simultaneously provide the cooling at four different temperatures as discussed in the beginning of the paper, the cooling powers at the second and the third stages must also be guaranteed. Based on the above considerations, we work out the compromise that in the actual experiments the phase-shifters of both the second and the third stages are put in the ambient environment, though this handling will inevitably have an adverse effect on the no-load temperature and the cooling capacity as well of the fourth stage. As for the fourth stage, an ambient phase-shifter is obviously unacceptable because the coldhead temperature is expected to reach around 3 K. Therefore, the fourth stage phase-shifter is cooled by the third stage in both simulations and experiments.
4. Conclusions
This paper conducts systematic theoretical analyses of a four-stage SPTC aimed at directly
reaching a temperature of around 3 K and also simultaneously achieving the cooling capacities at four temperatures varying from 3.3 K to 80 K for multiple uses. The systematic optimization for the four-stage SPTC is then conducted. The fourth stage cold finger is focused on and a 2-D axis-symmetric CFD model is set up to study the irreversible losses and phase characteristics. The operating mechanisms in the fourth stage are modelled, in which the distributions of the losses along the whole regenerator are analyzed
quantitatively and the phase characteristics simulated as well. The distributions of regenerator losses are simulated and analysed in detail. The pressure drop loss accounts for the main part of gross losses in the first three segments of the regenerator. The heat transfer loss increases rapidly in Reg IV and almost accounts for 50%. Mixed matrices consist of Er3Ni and HoCu2 are proved to be effective for reducing heat transfer loss in Reg IV. The distributions of phase difference along the whole regenerator are presented, with different operating frequencies and phase-shifter temperatures.
The interactions among the four stages are studied, in which the effect of parameters at each stage on the cooling performances of the other three stages is elaborated, respectively. The cooling temperature of some a stage has much more influence on the cooling performance of its lower stage but is less effective on its upper stage. With He-3, the optimal frequency is relatively lower than that with He-4, while the optimal charge pressure becomes larger. Increasing the PV power of compressor II can improve the cooling performances of the last two stages within certain limits, while too large will deteriorate the whole cooling performance. In addition, the regenerator geometry is also optimized. The experimental verifications of this study will be presented in Part B [32].
5. Acknowledgements
This work is partly supported by the Chinese Academy of Sciences (No. 6141A01070102), Shanghai Municipality (Nos. 18511110100, 18511110101, 18511110102, 19511106800, 19511106801, 19511106802 and 19ZR1465300) and the Aeronautical Science Foundation of China (No. 20162490005).
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Figure Captions Figure 1.
Axis-symmetric geometry of the fourth stage CF.
Figure 2.
Temperature profile in the fourth stage CF.
Figure 3.
Distributions of regenerator losses per unit length in the regenerator.
Figure 4.
Variations of (a) heat transfer loss and (b) pressure drop loss in Reg IV with 𝐿4 ― 1/𝐿4.
Figure 5.
Temperature profiles in Reg IV.
Figure 6.
Effects of charge pressure on heat transfer loss and pressure drop loss with a constant pressure ratio.
Figure 7.
Effects of operating frequency on heat transfer loss and pressure drop loss with a constant pressure ratio.
Figure 8.
Effects of operating frequency on pressure amplitudes and volume flow rates at the inlet and outlet of the regenerator with a constant pressure ratio.
Figure 9.
Distributions of phase difference in the regenerator with operating frequency.
Figure 10.
Distributions of phase difference in the regenerator with phase-shifter temperature.
Figure 11.
Effects of operating frequency on Tc4 with different phase-shifter temperatures and a constant pressure ratio.
Figure 12.
Schematic of the four-stage SPTC.
Figure 13.
Variations of Tc2, Tc3 and Tc4 with Tc1.
Figure 14.
Variations of Qp12, Qp13, Qp14 and Qc1 with Tc1.
Figure 15.
Variations of the four cooling powers with Tc1.
Figure 16.
Variations of Tc1, Tc3 and Tc4 with Tc2.
Figure 17.
Variations of Qp23, Qp24 and Qc2 with Tc2.
Figure 18.
Variations of the four cooling powers with Tc2.
Figure 19.
Variations of Tc1, Tc2 and Tc4 with Tc3.
Figure 20.
Variations of Qp34, Qps and Qc3 with Tc3.
Figure 21.
Variations of the four cooling powers with Tc3.
Figure 22.
Effects of f2 on (a) Tc3 and Tc4, (b) Qc3 and Qc4.
Figure 23.
Effects of f2 and p2 on (a) Qc3 and (b) Qc4 with a constant pressure ratio.
Figure 24.
Effects of p2 on (a) ηc3 and ηc4, (b) Qc1 and Qc2 with a constant pressure ratio.
Figure 25.
Variations of the four cooling temperatures with W2.
Figure 26.
Variations of Qc3, Qc4, ηc3 and ηc4 with W2.
Figure 27.
Effects of geometry of Reg I on ηc4 and Qp14 when other segments are kept constant at the values shown in Table 1.
Figure 28.
Effects of geometry of Reg II on ηc4 and Qp24 when other segments are kept constant at the values shown in Table 1.
Figure 29.
Effects of geometry of Reg III on ηc4 and Qp34 when other segments are kept constant at the values shown in Table 1.
Figure 30.
Effects of geometry of Reg IV on ηc4 when other segments are kept constant at the values shown in Table 1.
Figure 31.
Variations of Tc3 with dimensions of the third stage (a) inertance tube I and (b) tube II.
Figure 32.
Variations of Tc4 with dimensions of the fourth stage (a) inertance tube I and (b) tube II.
Figure 33.
Effects of the gas reservoir volumes on the cooling performances of the third and fourth stages.
Figure 34.
Effects of phase-shifter temperature on the cooling performance of each stage.
Table list Table 1. Dimensions of each stage component (Inner diameter × Length).
Component
Regenerator
1st stage (mm)
24×64
2nd stage (mm) I: 20×40 II: 20×30
3rd stage (mm) I: 16×45 II: 16×40 III: 16×38
4th stage (mm) I: 14×36.4 II: 12×28.2 III: 12×31.2 IV: 10×36
Pulse tube
14×76
12×81
9×128
8×80
CHX
19×8
16×8
12.5×6
10×5
WHX
14×8
12×6
9×6
8×5
I: 3.5×2800
I: 3×3200
I: 3×800
I: 3×320
II: 4.5×1400
II: 4.5×1200
II: 4×1180
II: 4×1260
60×80
60×80
30×65
30×55
Inertance tube Gas reservoir
Highlights for CRYOGENICS_2019_64_R2_Part A
Theoretical investigations on a 3.3 K four-stage SPTC are conducted.
A 2-D axis-symmetric CFD model for the fourth stage cold finger is set up.
Irreversible losses and phase characteristics in the regenerator are studied.
Interactions among the four stages are elaborated.
Systematic optimization on the four-stage SPTC is presented.
Conflict of Interest Statement The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted