- Email: [email protected]
Numerical study on the working performance of a G-M cryocooler with a mechanically driven displacer
International Journal of Heat and Mass Transfer 115 (2017) 611–618
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical study on the working performance of a G-M cryocooler with a mechanically driven displacer Xiaoqin Zhi a, John M. Pfotenhauer b,⇑, Franklin Miller b, Vladimir Gershtein c a
Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, China Department of Mechanical Engineering, University of Wisconsin, Madison 53706, USA c Sumitomo (SHI) Cryogenics of America, Inc. 1833 Vultee St., Allentown, PA, USA b
a r t i c l e
i n f o
Article history: Received 24 March 2017 Received in revised form 25 May 2017 Accepted 12 July 2017
Keywords: G-M cryocooler CFD simulation Temperature distribution Heat transfer Flow channel
a b s t r a c t Single stage G-M cryocoolers (GMCs) with high cooling capacity at 20–50 K provide significant benefit for the field of high temperature superconductors, however, their working efficiency can still be improved for commercial applications. Numerical simulations can serve as a valuable guide for precise optimization of the GMC because they allow one to study its internal operating characteristics. In this research, a twodimensional, transient model of a single stage GMC with a mechanically driven displacer is built and studied using computational fluid dynamics (CFD) simulation. The modelling method has been tested and experimentally verified. It enables a view of the instantaneous non-uniform flow and temperature distributions inside the GMC. Additionally, the effects of cold end channels on the heat transfer efficiency of the cold heat exchanger while operating at a large cooling power are analyzed. The results show that during certain periods of a cycle, most of the regenerator near the cold end remains at almost the same low temperature with no temperature gradient due to the small heat capacity of the materials compared to that of the helium gas; while most of the pressure drop occurs in the region near the hot end due to the high viscosity of helium-4. For a good cooling performance at low temperatures, the cold end channels should be distributed uniformly to guarantee a uniform flow and temperature distribution in the regenerator; while for a high output cooling power at high temperatures, they should be designed to enhance the heat exchange between the gas and the cold heat exchanger as much as possible. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The regenerative cryocoolers including the G-M, Stirling and pulse tube cryocoolers have been rapidly developed due to their advantages of compactness, flexibility and reliability for cooling superconductors, electron devices, infrared detectors, physical experiment rigs and so on. Among them, the G-M cryocooler (GMC) is the earliest commercial cryocooler and also the most widely used because of its high cooling performance at low temperatures and the low manufacturing costs of commercial helium compressors. At the present state of their development, a single stage GMC can operate down to 11 K while a two-stage version can provide about 2 W cooling power at 4.2 K [1–3]. Recently there has been a push to develop single stage GMCs that can provide cooling capacities as large as 40–200 W at 20–50 K for an increasing number of applications in the high temperature superconductor field. Great efforts are being made by the cryogenic ⇑ Corresponding author. E-mail address: [email protected] (J.M. Pfotenhauer). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.07.058 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
community to develop large cooling power GMCs with high efficiency through theoretical and experimental studies [1,4–8]. Yamada analyzed the influence of the inclination angle on a GMC by experiment, and demonstrated that the reduction in cooling power caused by the improper orientation of the cryocooler can be as large as 24% [8]. Wang et al. built a one-dimensional simulation model to study the geometry of a GMC, and predicted a remarkable increase of the cooling performance by optimizing the structural parameters [1]. Xu et al. studied the internal dynamic parameters of the regenerator by numerical simulation, and found that the performance of the GMC can be improved by shifting the temperature profile of the regenerator to a higher level [9]. Hao et al. studied the effect of the expansion volume rate and operating conditions on the cooling performance of a 4 K GMC. They demonstrated a cooling power of 1.75 W at 4.2 K, higher than other commercial pneumatic GMCs, could be obtained by optimizing the displacer stroke, operation speed and input power [10]. Liu et al. proposed a new regeneration-labyrinth sealing structure and developed a numerical program to study its internal thermal processes, showing that the heat shuttle loss is the primary thermal
612
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
loss caused by the new type of sealing structure [11]. Hao et al. numerically investigated the effects of the clearance size on different heat losses, and by using a 0.01 mm sealing clearance the cooling performance of a two-stage GMC below 10 K was observably improved in practice [12]. As stated above, numerical simulations have played an essential role in guiding the performance optimization of the GMC. However, the existing simulations for the GMC are based on assumptions of one-dimensional, idealized pressure variation and displacer motion, or steady state conditions. Such simplifications do not allow the models to reveal the instantaneous and nonuniform field distributions inside the GMC during a cycle, especially the heat transfer characteristics of the complicated channels near the cold heat exchanger under large cooling powers. The moving displacer, which functions as a regenerator with a large transient heat exchange further complicates the picture. In order to make further precise optimizations for GMCs, a detailed investigation of the real dynamic operating processes of the GMC is necessary. However, it is difficult to obtain these details by traditional thermodynamic programs. Nevertheless, CFD simulations represent an increasingly utilized method for transient, multipledimensional, dynamic mesh modelling of GMCs. Although it has been widely used to study pulse tube cryocoolers, it is seldom used to model GMCs [13–15]. In this research, a two-dimensional GMC model with a mechanically driven displacer is constructed and studied using CFD simulation. The internal instantaneous characteristics such as the temperature and pressure drop variation in the regenerator were revealed and analyzed. Additionally, the effects of various cold end flow channels on the heat exchange between the working fluid and the cold heat exchanger (CHX) were studied in order to improve the heat transfer efficiency of the CHX under conditions of large cooling power.
2. Numerical simulation model of the single stage GMC 2.1. Physical model and boundary conditions A single stage GMC developed by Sumitomo Heavy Industries, Ltd (SHI). is studied in this research, and the simplified 2-D, axissymmetric model is shown in Fig. 1. The structural details are given in Table 1. The model includes the hot end inlet channel a, three sections of the regenerator b, c, d with different materials, the displacer wall e, the cold end flow channels f, g, i, k, the cold end solid regions h, j, the cold end expanding chamber l and the CHX (cold heat exchanger) m. The displacer is composed of parts b, c, d, e, f, h, i, j, k. The clearance between the displacer and the cylindrical outer wall of the cryocooler is neglected in this model. For all materials, the specific heat and thermal conductivity (from the
Table 1 Details for different parts of the GMC model. Model part Inlet channel a (fluid area) Regenerator (fluid area, porous media)
Porosity
Material/size
Phenolic resin for the wall Phosphor bronze mesh screen c 0.49 Phosphor bronze mesh screen d 0.39 Lead sphere Solid wall e 0 Phenolic resin Cold end channels f, g, i, k (fluid 1 0.7 mm gap for g, f, i; areas) 1.6 mm for k Cold end blocks h, j (solid area) 0 Phenolic resin Cold end chamber l (fluid area) 1 Length range: 1–18 mm Cold heat exchanger (CHX) m 0 Copper, 1 mm for thickness Note: Displacer is composed of b, c, d, e, f, h, i, j, k b
0 0.66
NIST and EES database) are treated as piecewise linear functions of temperature between 10 and 300 K. Since a large thermal contact resistance exists between the screens and lead spheres in the regenerator, 10% of the bulk thermal conductivity for phosphor bronze is used along the axial direction, while 30% of the bulk thermal conductivity for lead is used isotropically [15]. In order to artificially decrease the initial cool-down time (thereby saving computing time) at the CHX, the specific heat of copper at 50 K is used for all temperatures greater than 50 K, but the real values of specific heat are used for temperatures below 50 K. Also the thickness of component m is artificially reduced to decrease the cool-down time. Both the ideal gas and real gas versions of helium-4 are used and compared here. For the real gas model, the Aungier-Redlich-Kwong equation of state is used instead of calling the NIST database for the density of helium-4. This approach also significantly reduces computing time, and introduces a negligible density error of only 4.47% compared to that from the NIST database in the range of 9–21 bar and 8–30 K. The viscosity and thermal conductivity of helium-4 are also treated as piecewise linear functions of temperature. The operating frequency is 2 Hz. The general operation of this GMC can be typically described as four processes: (1) the pressure increasing process, in which the high pressure valve opens, gas flows into the GMC and releases heat to the regenerator; (2) the intake process, during which the high pressure valve remains open at the early portion of this process, and the displacer begins to move away from the cold end. As a result, gas flows into the cold end chamber with a temperature close to that at the cold end of the regenerator; (3) the expanding process, in which the low pressure valve opens and gas flows out of the GMC. The gas remaining in the cold end chamber expands and produces the cooling effect; (4) the exhaust process, during which the low pressure valve remains open at the early portion of this
Fig. 1. Structure of the GMC model: inlet channel a; regenerator sections b, c, and d; displacer wall e; cold end flow channels f, g, i, and k; cold end solid regions h, j, cold end expanding chamber l and cold heat exchanger m.
613
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
process and the displacer begins to move back to the cold end. As a result, the cold gas is pushed out of the cold end chamber, and one cycle is finished. An inlet pressure Pin at the warm-end channel a and a constant displacement speed +U or U of the displacer are used as the boundary conditions for this model, as shown in Fig. 2. The time dependent value of Pin used in the model is simplified compared to the real pressure measured by SHI at room temperature. During the pressure increasing and decreasing periods, Pin changes periodically between 9.5 and 20.3 bar; during the constant pressure periods, Pin keeps constant 20.3 bar or 9.5 bar. The time ratio of the pressure increasing period to the pressure decreasing period is 0.745. Throughout the cycle, the inlet temperature of the gas flow remains constant at 280 K. The motion of the displacer is controlled to move with a constant speed U as reported by SHI. In Fig. 2, U represents the displacer moving away from the cold end, causing an increase of the cold chamber volume l; +U stands for the displacer moving back towards the cold end, causing a decrease of the cold chamber volume l. The largest displacement of the displacer to one side is 17 mm. All outside walls of the cryocooler are assumed adiabatic, and the initial temperature of the model is 280 K. The cool-down process of Tc (the average temperature of the CHX m) from 280 K to a final low temperature is a primary objective of the simulation. 2.2. Mathematical model and solution settings For all the flow areas with constant control volume, the mass, momentum and energy equations can be described as follows:
@ ðuqf Þ þ rðuqf ~ uÞ ¼ 0 @t
ð1Þ
@ uÞ þ rðuqf ~ u~ uÞ ¼ urp þ r ðus~ uÞ þ SXðYÞ ðuqf ~ @t
ð2Þ
@ ðuqf Ef þ ð1 uÞqs Es Þ þ r ð~ uðqf Ef pÞÞ @t uÞ ¼ r ½ðukf þ ð1 uÞks ÞrT þ ðus ~ Ef ¼ H f þ
SXðYÞ ¼
u2 p ; Es ¼ C v ;s T 2 qf
1 f q j~ uj~ u 2dh r f
ð3Þ ð4Þ
ð5Þ
The explanations of each item in the above equations are given in references [16,17]. For areas a, f, g, i, k and l, the porosity u ¼ 1, and the momentum source term SxðyÞ¼0 ; for the porous areas b, c
and d, their porosity is given in Table 1, and the momentum source term is calculated by Eq. (5), in which fr is the friction resistance coefficient of the porous media. The empirical formulas of fr for mesh screens and lead spheres are from NASA’s regenerator testing report, it is treated isotropically. For areas b and c, it can be calculated by [18]:
f r ¼ 129=Re þ 2:91Re0:103
ð6Þ
Area d is packed with spheres, fr can be calculated by [19]:
f r ¼ ð157=Re þ 5:15Re0:137 Þðu=0:39Þ3:48
ð7Þ
For the dynamic meshes, the conservation equation considering the volume change can be described generally as [16]:
@ @t
Z V
qf vdV þ
Z
¼
@V
Z @V
qf vð~ u U m Þ dA
Cvð~ u U m Þ dA þ
Z V
Sv dV
ð8Þ
where v is a certain general scalar, V is an arbitrary control volume, Um is the mesh velocity of the moving mesh, Cis the diffusion coefficient, and Sv is the source term of v. The layering method is used for the dynamic mesh updating, in which the structured meshes near the moving boundary will disappear and generate with its motion. In early versions of the model only the displacer was treated as a dynamic mesh area, conforming to the actual situation. However, the calculation easily diverged due to the interlaced nodes generating at the interface between the narrow channel g and the displacer (areas f, h) during the motion. Apparently such interlaced nodes increased the residuals of the dynamic mesh calculation. Therefore, channel g is also configured as a dynamic mesh and moves together with the displacer to avoid interlaced nodes. Using this dynamic mesh scheme, the model converges well. A double precision model is necessary in the GMC simulation in order to achieve numerical stability and reasonable calculation results. For the turbulence calculation, the SST k-e model is used. The PISO scheme is used to solve the pressure-velocity coupling. The second order discretization is used for the pressure, energy and density, while the first order discretization is used for k and e. The residual criterion for convergence is 106 for the energy equation, and 103 for the continuity, x-velocity, y-velocity, turbulent kinetic energy k and specific dissipation rate x. The largest time step size to guarantee a complete convergence in every time step is found to be 0.00025 s, i.e. calculating each cycle requires 2000 steps, much more than the time steps needed in the simulation of a pulse tube cryocooler with no moving parts and narrow channels [13]. During the calculation, the model can converge well within 30 iterations per time step 3. Simulation results and discussions 3.1. Model tests and experiment verification
Fig. 2. Inlet pressure and the displacer motion speed for the GMC model (1, 2, 3 and 4 represent the moments at 0.1 s, 0.23 s, 0.37 s and 0.5 s of a cycle respectively).
One of the first steps in building a reliable CFD model is to verify mesh independence. Fig. 3 shows the cool-down curve using different mesh models along with ideal gas for helium-4. The 3 mesh, 5 mesh, and 7 mesh models represent models with 3, 5, 7 meshes respectively in the small channels f, g, and i. The total grid numbers of these models are 15000, 35000 and 52000 respectively, and the associated computing time required for each model to generate 20 min of simulated time is about 22 days, 38 days and 75 days respectively on a high-performance computing center. As shown in Fig. 3, all the models almost have the same cool-down speed and their final no-load temperature Tc is between 11.2 K and 11.4 K. Therefore, the 3 mesh model is qualified for the simulation.
614
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
Fig. 3. Cool-down curves of different mesh models.
Fig. 5. Cool-down curves from the simulation and experiment result.
Fig. 4. Cooling performance of the GMC with real gas and ideal gas model.
Fig. 6. Cooling performance from the simulation and experiment result.
Fig. 4 shows the cooling performance of the GMC with real gas and ideal gas versions. The heat load of the CHX m, i.e. the cooling power Qc, is determined by adding an average heat flux at the outside wall of the CHX. It can be seen, the no-load temperature of the real gas model is 12.8 K, 1.6 K higher than that of the ideal gas model. Furthermore, the cooling power in the real gas version is lower than that of the ideal gas model, especially below 16 K. This might be because the ideal gas model ignores the enthalpy flow caused by the pressure change under low temperatures, which contributes a kind of heat loss in the regenerator. As can be seen from Fig. 4, for this GMC operating below about 16 K, the ideal gas model will predict a higher cooling performance than that of the real gas model. Fig. 5 shows the cool-down curves of the experiment and simulation results. The no-load temperature Tc of 12.8 K obtained from this simulation is quite close to the 12.0 K result from the experiment. Because a smaller CHX with a smaller specific heat of copper at high temperatures is used in the simulation as compared to the real GMC (and as stated in Section 2.1), the cool-down speed of Tc here is faster than that in the experiment. Figs. 6 and 7 further show the comparisons of the cooling power and the cold end pressure variation between experiment and simulation, respectively. From Figs. 5–7, it can be seen that both the cooling performance and the pressure variation obtained from the model agree well with that from the experiment.
Fig. 7. Cold end pressures from the simulation and experiment result.
3.2. Temperature and pressure distributions in the GMC The internal gas and solid temperature distributions of the GMC at four typical moments (moment 1, 2, 3 and 4 represent the 0.1 s, 0.23 s, 0.37 s and 0.5 s of a cycle respectively, as shown in Fig. 2) during a cycle are shown in Fig. 8. Fig. 9 further shows the crosssectional average temperature distribution along the regenerator
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
615
Fig. 8. Temperature distributions of the GMC at typical moments (Tc = 12.8 K, Moments 1, 2, 3 and 4 represent the moments at 0.1 s, 0.23 s, 0.37 s and 0.5 s of a cycle respectively).
Fig. 9. Temperature distributions on the middle cross-section of the regenerator sections b, c and d (Moments 1, 2, 3 and 4 represent the moments at 0.1 s, 0.23 s, 0.37 s and 0.5 s of a cycle respectively).
at the four moments. At moment 1, during the compression process, Pin increases and the displacer remains at the cold end; at moment 2, during the cold chamber intake process, Pin = Ph and the displacer moves away from the cold end; at moment 3, during the expansion process, Pin decreases and the displacer remains at the hot end; at moment 4, during the cold chamber exhausting process, Pin = Pl and the displacer moves back to the cold end. From Figs. 8 and 9 it can be seen that the temperature in the regenerator at moment 2 increases significantly compared to moment 1 due to the compression and the intake process; its high temperature region becomes larger and the low temperature region becomes smaller. Compared to moment 2, the temperature in the regenerator at moment 3 decreases due to the expansion process; its high temperature region becomes smaller and its low temperature region becomes larger. Compared to moment 3, the temperature in the regenerator at moment 4 further decreases substantially due to the cold chamber exhausting process in which the cold gas is pushed out from the cold chamber to the regenerator. The temperature profile at moment 4 displays a large temperature inhomogeneity at the middle of the regenerator. This is caused by the jet flows of the cold gas with large density that generate from channels f, i, and k during the cold chamber exhausting process. As shown in Figs. 8 and 9, at certain times during a cycle, most of the regenerator near the cold end remains at very low temperatures with small temperature gradient. Indeed, at moment 4,
more than half of the regenerator remains at almost the same, low temperature. Furthermore, most of the temperature drop along the regenerator occurs in the middle area c which is a little shorter than both areas b and d. From Fig. 9 it can be seen that there is a large temperature fluctuation, which occurs over most of the regenerator during one cycle. Especially in area c, the temperature change at the middle cross section (X/L = 0.5) can be larger than 120 K. An ideal regenerator should have as small temperature change as possible for any position during a cycle and its temperature gradient should be consistent along the regenerator. The above results reveal that the heat capacity of the regenerator is insufficient, and a large enthalpy flow is likely caused by the large temperature fluctuation in the regenerator, generating a type of heat loss in the GMC. Fig. 10 shows the pressure drop variations of area b, c, and d in the regenerator during a cycle, i.e. pressure differences between surfaces S1 and S2, S2 and S3, and S3 and S4 (as shown in Fig. 1), respectively. As can be seen, the pressure drop in area c and d is much smaller than that in area b. Indeed, in area d there is almost no pressure drop during the entire cycle. It should be noted that the length of area b, c and d are almost the same. As noted in Table 1, area b has the largest porosity, secondly area c, and thirdly area d. Therefore, the length and porosity are not the main reason for the pressure drop distribution in the different areas. The primary cause is the different viscosity of helium-4 in the various temperature ranges. As shown in Figs. 8 and 9, area b remains at
Fig. 10. Pressure drop variations at different areas of the regenerator.
616
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
temperatures between about 200 and 280 K; area c remains at temperatures between 50 and 200 K; area d remains at temperatures between 13 and 50 K. Therefore, area b has the highest pressure drop due to the large viscosity of helium-4 at high temperatures, while area c has almost no pressure drop due to the correspondingly low viscosity below 50 K. From the above temperature and pressure drop distributions, it can be inferred that using regenerator materials with high porosity at high temperature regions and low porosity, high heat capacity at low temperature regions can be an effective way to improve the cooling performance of this single stage GMC. 3.3. Heat transfer efficiency of the CHX under different situations Besides reducing the total regenerator losses of the system, increasing the cooling power transfer efficiency at the CHX is also a significant concern for cryocoolers, especially for the ones working with large cooling power. As shown in Fig. 1, wall-1 is the wall between channel g and the CHX m, wall-2 is the wall between channel l and CHX m, and the heat load Qc is transferred from the CHX to the inside cold gas through wall-1 and wall-2. Apparently, the heat transfer at the CHX is primarily determined by the outflow distribution of the cold gases through the various channels (g, i and k in Fig. 1) during the expansion and displacer exhausting period. Therefore, the effects of different cold end channel configurations on the heat transfer efficiency of the CHX are further investigated: case 1 is the original model with three channels g, i, k; case 2 is the model with only channel g at the cold end, while channels i and k are set as solid area. Figs. 11 and 12 show the temperature difference between the CHX and its nearby gas in case 1 and case 2 during a cycle. DT1 is the difference between the average temperature of wall-1 and that of the gas in channel g; DT2 is the difference between the average temperature of wall-2 and that of the gas in channel l; DTa is the cycle-averaged value of DT1 or alternatively DT2. From Fig. 11 it can be seen that the values ofDT1 for case 2 under large cooling powers are noticeably smaller than those of case 1. The cycle-averaged value DTa decreases from 4.2 K to 2.9 K and from 8.4 K to 5.4 K at respective cooling powers of 80 W and 120 W. As shown in Fig. 12, the values ofDT2 for case 2 are also noticeably reduced compared to those of case 1; its cycle-averaged value decreases from 5.6 K to 3.4 K and from 10.9 K to 7.3 K at respective cooling powers of 80 W and 120 W. In Fig. 12, there are frequent fluctuations of DT2 occurring at around 0.12–0.23 s and 0.34–
Fig. 11. Temperature difference DT1 between the cold heat exchanger (wall-1) and gas in channel g (Case 1 and Case 2 represent three channel configuration and single channel configuration respectively).
Fig. 12. Temperature difference DT2 between the cold heat exchanger (wall-2) and gas in channel l (Case 1 and Case 2 represent three channel configuration and single channel configuration respectively).
0.5 s when the displacer is moving. This may be due to the complicated flow inside the cold end chamber l caused by the motion of the displacer. During the intake and exhaust process, some turbulence or non-uniform flows can exist at the junctions between the cold end channels (g, i, k) and the cold chamber l. These unsteady flows may lead to a fluctuation of DT2 due to the mixing of gas with different temperatures and also an instantaneous heat transfer variation between the gas and wall-2. While for channel g, it is narrow and long, so its internal flow can be more regular and smooth, as reflected by the values of DT1. Fig. 13 further shows the average heat transfer efficiency of wall-1 and wall-2 in case 1 and case 2 under different cooling powers. As can be seen, the heat transfer efficiency of CHX in case 2 is obviously increased compared to that in case 1 during a cycle. These observations result because in case 1 the cold gas can flow from the cold chamber l to the regenerator through three paths: channels g and f, channel i, and channel k, among which the path of channel g and f has the largest flow resistance. Therefore, most of the cold gas would like to flow through channels i and k instead of g and f, which significantly reduces
Fig. 13. Average heat transfer coefficient h between cold chamber gas and heat exchanger wall (Case 1 and Case 2 represent three channel configuration and single channel configuration respectively).
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
617
non-uniform flow and large density of the gas at low temperatures. This will lead to larger heat losses in the regenerator. Therefore, at low temperatures, the performance of case 1 is better than that of case 2 due to its smaller irreversible losses. On the other hand, at the high temperatures resulting from a large cooling power, such as 80 W, since the density of helium-4 is smaller, the flow and temperature inhomogeneity caused by the side channel g is not as pronounced, see Fig. 15. Furthermore, in this situation, the heat transfer efficiency of the CHX is more significant. Therefore, at large cooling powers the performance of case 2 is better than case 1. As stated above, it can be concluded that when the cryocooler is designed to work at low temperatures with a small cooling power, the flow and the temperature distribution at the cold end should be considered more than the heat transfer at the CHX; however, when it operates at large cooling powers, the heat transfer at the CHX is more important for the cooling performance and should be a more dominant factor for designing the cold end channels. Fig. 14. Cooling performance comparison of case 1 and case 2 (Case 1 and Case 2 represent three channel configuration and single channel configuration respectively).
the total heat transfer between the CHX and gas, since the heat transfer area of wall-2 is much smaller than that of wall-1. In contrast, for case 2 all the cold gas will flow back to the regenerator through channel g with sufficient time and area to exchange heat with the CHX. Therefore, optimizing the cold channels as in case 2 will increase the available cooling power of the GMC. Fig. 14 shows the calculated cooling powers at different temperatures for case 1 and case 2. It can be seen that above about 20 K the cooling power of case 2 is larger than that of case 1, which is consistent with the above results. Especially at 30 K, the calculated cooling power of case 2 is larger by 20% compared to that of case 1. However, it can be seen that the cooling power of case 2 is smaller than that of case 1 at low temperatures, below about 20 K. This is because the flow channels not only affect the heat transfer at the CHX but they also influence the flow distribution and temperature distribution in the regenerator. Fig. 15 shows the temperature distributions of case 2 at moment 4 for two different heat loads (Qc), 0 W and 80 W. Compared to case 1 at moment 4 (in Fig. 8), the temperature distribution for case 2, where there is only side channel g at the cold end, is more inhomogeneous due to the
4. Conclusions The single stage G-M cryocooler (GMC) with a mechanically driven displacer has been simulated and studied for the first time using a CFD method with a dynamic mesh scheme. The instantaneous temperature and pressure distributions inside the GMC are displayed, and the effects of various cold end channels on the heat transfer efficiency of the cold heat exchanger (CHX) are analyzed. The model results agree well with the experimental measurements. The results demonstrate that for this single stage GMC working below 13 K, half of its regenerator near the cold end will remain at very low temperatures with a small temperature gradient during a cycle. At the same time, there exist large temperature fluctuations, larger than about 100 K, at most middle parts of the regenerator during a cycle. Both of these features indicate an inefficient use of the heat capacity of the materials working at low temperatures. Additionally, more than 80% of the total pressure drop occurs in the regenerator’s upper region which operates at high temperatures. The results imply that using regenerative materials with high porosity at high temperature region and low porosity and high heat capacity at low temperature region can improve the cooling performance of the GMC. The design of the cold end channels will vary for the GMC, depending on the operating goals. When it operates at low
Fig. 15. Temperature distributions of case 2 (Case 2: three channel configuration, Tc = 14.6 K) with different cooling powers at Moment 4 (at 0.5 s, the end of a cycle).
618
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
temperatures with a small cooling power, the arrangement of the cold end channels should be driven more by the uniform flow and temperature distribution at the cold end than the heat transfer efficiency of the CHX. However, when it operates at high temperatures with a large cooling power, the heat transfer at the CHX is more important for the cooling performance and the cold end channels should be designed to allow as much cold gas flowing past the CHX as possible. Acknowledgement The authors appreciate the helpful discussions with Ralph Longsworth. This work is supported by SHI Cryogenics of America, Inc. Conflict of interest None declared. References [1] C. Wang, P.E. Gifford, High efficiency, single-stage GM cryorefrigerators optimized for 20 to 40K, Cryocoolers 11 (2001) 387–392. [2] I. Takashi, N. Masashi, N. Kouki, Y. Hideto, Development of a 2 W class 4 K Gifford- McMahon cycle cryocooler, Cryocoolers 9 (1997) 617–626. [3] X.H. Hao, S.H. Yao, T. Schilling, Design and experimental investigation of the high efficiency 1.5 W/4.2 K pneumatic-drive G-M cryocooler, Cryogenics 70 (2015) 28–33. [4] W. Huang, P.Y. Wu, S.L. Hu, L. Zhang, et al., Dynamic simulation of one-stage GM refrigerator and comparison with experiment, Cryogenics 36 (1996) 643– 647.
[5] A. Fiedler, J. Gerban, H.U. Haefner, Efficient single stage Gifford-McMahon refrigerator operating at 20 K, Adv. Cryogenic Eng. 43 (1998) 1823–1830. [6] G. Green, W. Superczynski, Ribbon regenerator performance in a single-stage GM cryocooler, Cryocoolers 14 (2007) 373–379. [7] C. Wang, P.E. Gifford, Performance improvement of a single stage GM cryocooler at 25 K, Adv. Cryogenic Eng. 53 (2008) 26–33. [8] K. Yamada, Development of a large cooling capacity single stage GM cryocooler, Cryogenics 63 (2014) 110–113. [9] M.Y. Xu, T. Morie, Numerical simulation of the second stage regenerator in a 4 K GM cryocooler, Adv. Cryogenic Eng. 1573 (1) (2014) 1143–1148. [10] X.H. Hao, S.H. Yao, T. Schilling, Design and experimental investigation of the high efficiency 1.5 W/4.2 K pneumatic-drive G-M cryocooler, Cryogenics 70 (2014) 28–33. [11] L.Q. Liu, L. Zhang, Study on a new type of sealing – regeneration-labyrinth sealing for displacer in cryocoolers: Part I - Theoretical study, Cryogenics 40 (2000) 75–83. [12] X.H. Hao, Y.L. Ju, Y.J. Lu, Experimental study on the sealing clearance between the labyrinth sealing displacer and cylinder in the 10 K G-M refrigerator, Cryogenics 51 (2011) 203–208. [13] X.Q. Zhi, L.M. Qiu, J.M. Pfotenhauer, et al., Refrigeration mechanism of the gas parcels in pulse tube cryocoolers under different phase angles, Int. J. Heat Mass Transfer 103 (2016) 382–389. [14] Y. Zhao, H. Dang, CFD simulation of a miniature coaxial Stirling-type pulse tube cryocooler operating at 128 Hz, Cryogenics 73 (2015) 53–59. [15] B. Flake, A. Razani, Modeling pulse tube cryocoolers with CFD, Adv. Cryogenic Eng. 710 (2004) 1493–1499. [16] ANSYS INC, ANSYS Fluent 14.5 User’s Guide, 2012. [17] L.M. Qiu, X.Q. Zhi, Z.H. Gan, et al., Function of gas parcels in the pulse tube, Int. J. Refrigeration 38 (2014) 358–366. [18] D. Gedeon, G. Wood, Oscillating-flow regenerator test rig: hardware and theory with correlations for screens and felts, NASA-Lewis Contractor Report 198422, 1996. [19] David Gedeon, Stirling-cycle model class reference guide (Sage v5 Edition), 2007, pp. 114.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical study on the working performance of a G-M cryocooler with a mechanically driven displacer Xiaoqin Zhi a, John M. Pfotenhauer b,⇑, Franklin Miller b, Vladimir Gershtein c a
Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, China Department of Mechanical Engineering, University of Wisconsin, Madison 53706, USA c Sumitomo (SHI) Cryogenics of America, Inc. 1833 Vultee St., Allentown, PA, USA b
a r t i c l e
i n f o
Article history: Received 24 March 2017 Received in revised form 25 May 2017 Accepted 12 July 2017
Keywords: G-M cryocooler CFD simulation Temperature distribution Heat transfer Flow channel
a b s t r a c t Single stage G-M cryocoolers (GMCs) with high cooling capacity at 20–50 K provide significant benefit for the field of high temperature superconductors, however, their working efficiency can still be improved for commercial applications. Numerical simulations can serve as a valuable guide for precise optimization of the GMC because they allow one to study its internal operating characteristics. In this research, a twodimensional, transient model of a single stage GMC with a mechanically driven displacer is built and studied using computational fluid dynamics (CFD) simulation. The modelling method has been tested and experimentally verified. It enables a view of the instantaneous non-uniform flow and temperature distributions inside the GMC. Additionally, the effects of cold end channels on the heat transfer efficiency of the cold heat exchanger while operating at a large cooling power are analyzed. The results show that during certain periods of a cycle, most of the regenerator near the cold end remains at almost the same low temperature with no temperature gradient due to the small heat capacity of the materials compared to that of the helium gas; while most of the pressure drop occurs in the region near the hot end due to the high viscosity of helium-4. For a good cooling performance at low temperatures, the cold end channels should be distributed uniformly to guarantee a uniform flow and temperature distribution in the regenerator; while for a high output cooling power at high temperatures, they should be designed to enhance the heat exchange between the gas and the cold heat exchanger as much as possible. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The regenerative cryocoolers including the G-M, Stirling and pulse tube cryocoolers have been rapidly developed due to their advantages of compactness, flexibility and reliability for cooling superconductors, electron devices, infrared detectors, physical experiment rigs and so on. Among them, the G-M cryocooler (GMC) is the earliest commercial cryocooler and also the most widely used because of its high cooling performance at low temperatures and the low manufacturing costs of commercial helium compressors. At the present state of their development, a single stage GMC can operate down to 11 K while a two-stage version can provide about 2 W cooling power at 4.2 K [1–3]. Recently there has been a push to develop single stage GMCs that can provide cooling capacities as large as 40–200 W at 20–50 K for an increasing number of applications in the high temperature superconductor field. Great efforts are being made by the cryogenic ⇑ Corresponding author. E-mail address: [email protected] (J.M. Pfotenhauer). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.07.058 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
community to develop large cooling power GMCs with high efficiency through theoretical and experimental studies [1,4–8]. Yamada analyzed the influence of the inclination angle on a GMC by experiment, and demonstrated that the reduction in cooling power caused by the improper orientation of the cryocooler can be as large as 24% [8]. Wang et al. built a one-dimensional simulation model to study the geometry of a GMC, and predicted a remarkable increase of the cooling performance by optimizing the structural parameters [1]. Xu et al. studied the internal dynamic parameters of the regenerator by numerical simulation, and found that the performance of the GMC can be improved by shifting the temperature profile of the regenerator to a higher level [9]. Hao et al. studied the effect of the expansion volume rate and operating conditions on the cooling performance of a 4 K GMC. They demonstrated a cooling power of 1.75 W at 4.2 K, higher than other commercial pneumatic GMCs, could be obtained by optimizing the displacer stroke, operation speed and input power [10]. Liu et al. proposed a new regeneration-labyrinth sealing structure and developed a numerical program to study its internal thermal processes, showing that the heat shuttle loss is the primary thermal
612
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
loss caused by the new type of sealing structure [11]. Hao et al. numerically investigated the effects of the clearance size on different heat losses, and by using a 0.01 mm sealing clearance the cooling performance of a two-stage GMC below 10 K was observably improved in practice [12]. As stated above, numerical simulations have played an essential role in guiding the performance optimization of the GMC. However, the existing simulations for the GMC are based on assumptions of one-dimensional, idealized pressure variation and displacer motion, or steady state conditions. Such simplifications do not allow the models to reveal the instantaneous and nonuniform field distributions inside the GMC during a cycle, especially the heat transfer characteristics of the complicated channels near the cold heat exchanger under large cooling powers. The moving displacer, which functions as a regenerator with a large transient heat exchange further complicates the picture. In order to make further precise optimizations for GMCs, a detailed investigation of the real dynamic operating processes of the GMC is necessary. However, it is difficult to obtain these details by traditional thermodynamic programs. Nevertheless, CFD simulations represent an increasingly utilized method for transient, multipledimensional, dynamic mesh modelling of GMCs. Although it has been widely used to study pulse tube cryocoolers, it is seldom used to model GMCs [13–15]. In this research, a two-dimensional GMC model with a mechanically driven displacer is constructed and studied using CFD simulation. The internal instantaneous characteristics such as the temperature and pressure drop variation in the regenerator were revealed and analyzed. Additionally, the effects of various cold end flow channels on the heat exchange between the working fluid and the cold heat exchanger (CHX) were studied in order to improve the heat transfer efficiency of the CHX under conditions of large cooling power.
2. Numerical simulation model of the single stage GMC 2.1. Physical model and boundary conditions A single stage GMC developed by Sumitomo Heavy Industries, Ltd (SHI). is studied in this research, and the simplified 2-D, axissymmetric model is shown in Fig. 1. The structural details are given in Table 1. The model includes the hot end inlet channel a, three sections of the regenerator b, c, d with different materials, the displacer wall e, the cold end flow channels f, g, i, k, the cold end solid regions h, j, the cold end expanding chamber l and the CHX (cold heat exchanger) m. The displacer is composed of parts b, c, d, e, f, h, i, j, k. The clearance between the displacer and the cylindrical outer wall of the cryocooler is neglected in this model. For all materials, the specific heat and thermal conductivity (from the
Table 1 Details for different parts of the GMC model. Model part Inlet channel a (fluid area) Regenerator (fluid area, porous media)
Porosity
Material/size
Phenolic resin for the wall Phosphor bronze mesh screen c 0.49 Phosphor bronze mesh screen d 0.39 Lead sphere Solid wall e 0 Phenolic resin Cold end channels f, g, i, k (fluid 1 0.7 mm gap for g, f, i; areas) 1.6 mm for k Cold end blocks h, j (solid area) 0 Phenolic resin Cold end chamber l (fluid area) 1 Length range: 1–18 mm Cold heat exchanger (CHX) m 0 Copper, 1 mm for thickness Note: Displacer is composed of b, c, d, e, f, h, i, j, k b
0 0.66
NIST and EES database) are treated as piecewise linear functions of temperature between 10 and 300 K. Since a large thermal contact resistance exists between the screens and lead spheres in the regenerator, 10% of the bulk thermal conductivity for phosphor bronze is used along the axial direction, while 30% of the bulk thermal conductivity for lead is used isotropically [15]. In order to artificially decrease the initial cool-down time (thereby saving computing time) at the CHX, the specific heat of copper at 50 K is used for all temperatures greater than 50 K, but the real values of specific heat are used for temperatures below 50 K. Also the thickness of component m is artificially reduced to decrease the cool-down time. Both the ideal gas and real gas versions of helium-4 are used and compared here. For the real gas model, the Aungier-Redlich-Kwong equation of state is used instead of calling the NIST database for the density of helium-4. This approach also significantly reduces computing time, and introduces a negligible density error of only 4.47% compared to that from the NIST database in the range of 9–21 bar and 8–30 K. The viscosity and thermal conductivity of helium-4 are also treated as piecewise linear functions of temperature. The operating frequency is 2 Hz. The general operation of this GMC can be typically described as four processes: (1) the pressure increasing process, in which the high pressure valve opens, gas flows into the GMC and releases heat to the regenerator; (2) the intake process, during which the high pressure valve remains open at the early portion of this process, and the displacer begins to move away from the cold end. As a result, gas flows into the cold end chamber with a temperature close to that at the cold end of the regenerator; (3) the expanding process, in which the low pressure valve opens and gas flows out of the GMC. The gas remaining in the cold end chamber expands and produces the cooling effect; (4) the exhaust process, during which the low pressure valve remains open at the early portion of this
Fig. 1. Structure of the GMC model: inlet channel a; regenerator sections b, c, and d; displacer wall e; cold end flow channels f, g, i, and k; cold end solid regions h, j, cold end expanding chamber l and cold heat exchanger m.
613
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
process and the displacer begins to move back to the cold end. As a result, the cold gas is pushed out of the cold end chamber, and one cycle is finished. An inlet pressure Pin at the warm-end channel a and a constant displacement speed +U or U of the displacer are used as the boundary conditions for this model, as shown in Fig. 2. The time dependent value of Pin used in the model is simplified compared to the real pressure measured by SHI at room temperature. During the pressure increasing and decreasing periods, Pin changes periodically between 9.5 and 20.3 bar; during the constant pressure periods, Pin keeps constant 20.3 bar or 9.5 bar. The time ratio of the pressure increasing period to the pressure decreasing period is 0.745. Throughout the cycle, the inlet temperature of the gas flow remains constant at 280 K. The motion of the displacer is controlled to move with a constant speed U as reported by SHI. In Fig. 2, U represents the displacer moving away from the cold end, causing an increase of the cold chamber volume l; +U stands for the displacer moving back towards the cold end, causing a decrease of the cold chamber volume l. The largest displacement of the displacer to one side is 17 mm. All outside walls of the cryocooler are assumed adiabatic, and the initial temperature of the model is 280 K. The cool-down process of Tc (the average temperature of the CHX m) from 280 K to a final low temperature is a primary objective of the simulation. 2.2. Mathematical model and solution settings For all the flow areas with constant control volume, the mass, momentum and energy equations can be described as follows:
@ ðuqf Þ þ rðuqf ~ uÞ ¼ 0 @t
ð1Þ
@ uÞ þ rðuqf ~ u~ uÞ ¼ urp þ r ðus~ uÞ þ SXðYÞ ðuqf ~ @t
ð2Þ
@ ðuqf Ef þ ð1 uÞqs Es Þ þ r ð~ uðqf Ef pÞÞ @t uÞ ¼ r ½ðukf þ ð1 uÞks ÞrT þ ðus ~ Ef ¼ H f þ
SXðYÞ ¼
u2 p ; Es ¼ C v ;s T 2 qf
1 f q j~ uj~ u 2dh r f
ð3Þ ð4Þ
ð5Þ
The explanations of each item in the above equations are given in references [16,17]. For areas a, f, g, i, k and l, the porosity u ¼ 1, and the momentum source term SxðyÞ¼0 ; for the porous areas b, c
and d, their porosity is given in Table 1, and the momentum source term is calculated by Eq. (5), in which fr is the friction resistance coefficient of the porous media. The empirical formulas of fr for mesh screens and lead spheres are from NASA’s regenerator testing report, it is treated isotropically. For areas b and c, it can be calculated by [18]:
f r ¼ 129=Re þ 2:91Re0:103
ð6Þ
Area d is packed with spheres, fr can be calculated by [19]:
f r ¼ ð157=Re þ 5:15Re0:137 Þðu=0:39Þ3:48
ð7Þ
For the dynamic meshes, the conservation equation considering the volume change can be described generally as [16]:
@ @t
Z V
qf vdV þ
Z
¼
@V
Z @V
qf vð~ u U m Þ dA
Cvð~ u U m Þ dA þ
Z V
Sv dV
ð8Þ
where v is a certain general scalar, V is an arbitrary control volume, Um is the mesh velocity of the moving mesh, Cis the diffusion coefficient, and Sv is the source term of v. The layering method is used for the dynamic mesh updating, in which the structured meshes near the moving boundary will disappear and generate with its motion. In early versions of the model only the displacer was treated as a dynamic mesh area, conforming to the actual situation. However, the calculation easily diverged due to the interlaced nodes generating at the interface between the narrow channel g and the displacer (areas f, h) during the motion. Apparently such interlaced nodes increased the residuals of the dynamic mesh calculation. Therefore, channel g is also configured as a dynamic mesh and moves together with the displacer to avoid interlaced nodes. Using this dynamic mesh scheme, the model converges well. A double precision model is necessary in the GMC simulation in order to achieve numerical stability and reasonable calculation results. For the turbulence calculation, the SST k-e model is used. The PISO scheme is used to solve the pressure-velocity coupling. The second order discretization is used for the pressure, energy and density, while the first order discretization is used for k and e. The residual criterion for convergence is 106 for the energy equation, and 103 for the continuity, x-velocity, y-velocity, turbulent kinetic energy k and specific dissipation rate x. The largest time step size to guarantee a complete convergence in every time step is found to be 0.00025 s, i.e. calculating each cycle requires 2000 steps, much more than the time steps needed in the simulation of a pulse tube cryocooler with no moving parts and narrow channels [13]. During the calculation, the model can converge well within 30 iterations per time step 3. Simulation results and discussions 3.1. Model tests and experiment verification
Fig. 2. Inlet pressure and the displacer motion speed for the GMC model (1, 2, 3 and 4 represent the moments at 0.1 s, 0.23 s, 0.37 s and 0.5 s of a cycle respectively).
One of the first steps in building a reliable CFD model is to verify mesh independence. Fig. 3 shows the cool-down curve using different mesh models along with ideal gas for helium-4. The 3 mesh, 5 mesh, and 7 mesh models represent models with 3, 5, 7 meshes respectively in the small channels f, g, and i. The total grid numbers of these models are 15000, 35000 and 52000 respectively, and the associated computing time required for each model to generate 20 min of simulated time is about 22 days, 38 days and 75 days respectively on a high-performance computing center. As shown in Fig. 3, all the models almost have the same cool-down speed and their final no-load temperature Tc is between 11.2 K and 11.4 K. Therefore, the 3 mesh model is qualified for the simulation.
614
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
Fig. 3. Cool-down curves of different mesh models.
Fig. 5. Cool-down curves from the simulation and experiment result.
Fig. 4. Cooling performance of the GMC with real gas and ideal gas model.
Fig. 6. Cooling performance from the simulation and experiment result.
Fig. 4 shows the cooling performance of the GMC with real gas and ideal gas versions. The heat load of the CHX m, i.e. the cooling power Qc, is determined by adding an average heat flux at the outside wall of the CHX. It can be seen, the no-load temperature of the real gas model is 12.8 K, 1.6 K higher than that of the ideal gas model. Furthermore, the cooling power in the real gas version is lower than that of the ideal gas model, especially below 16 K. This might be because the ideal gas model ignores the enthalpy flow caused by the pressure change under low temperatures, which contributes a kind of heat loss in the regenerator. As can be seen from Fig. 4, for this GMC operating below about 16 K, the ideal gas model will predict a higher cooling performance than that of the real gas model. Fig. 5 shows the cool-down curves of the experiment and simulation results. The no-load temperature Tc of 12.8 K obtained from this simulation is quite close to the 12.0 K result from the experiment. Because a smaller CHX with a smaller specific heat of copper at high temperatures is used in the simulation as compared to the real GMC (and as stated in Section 2.1), the cool-down speed of Tc here is faster than that in the experiment. Figs. 6 and 7 further show the comparisons of the cooling power and the cold end pressure variation between experiment and simulation, respectively. From Figs. 5–7, it can be seen that both the cooling performance and the pressure variation obtained from the model agree well with that from the experiment.
Fig. 7. Cold end pressures from the simulation and experiment result.
3.2. Temperature and pressure distributions in the GMC The internal gas and solid temperature distributions of the GMC at four typical moments (moment 1, 2, 3 and 4 represent the 0.1 s, 0.23 s, 0.37 s and 0.5 s of a cycle respectively, as shown in Fig. 2) during a cycle are shown in Fig. 8. Fig. 9 further shows the crosssectional average temperature distribution along the regenerator
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
615
Fig. 8. Temperature distributions of the GMC at typical moments (Tc = 12.8 K, Moments 1, 2, 3 and 4 represent the moments at 0.1 s, 0.23 s, 0.37 s and 0.5 s of a cycle respectively).
Fig. 9. Temperature distributions on the middle cross-section of the regenerator sections b, c and d (Moments 1, 2, 3 and 4 represent the moments at 0.1 s, 0.23 s, 0.37 s and 0.5 s of a cycle respectively).
at the four moments. At moment 1, during the compression process, Pin increases and the displacer remains at the cold end; at moment 2, during the cold chamber intake process, Pin = Ph and the displacer moves away from the cold end; at moment 3, during the expansion process, Pin decreases and the displacer remains at the hot end; at moment 4, during the cold chamber exhausting process, Pin = Pl and the displacer moves back to the cold end. From Figs. 8 and 9 it can be seen that the temperature in the regenerator at moment 2 increases significantly compared to moment 1 due to the compression and the intake process; its high temperature region becomes larger and the low temperature region becomes smaller. Compared to moment 2, the temperature in the regenerator at moment 3 decreases due to the expansion process; its high temperature region becomes smaller and its low temperature region becomes larger. Compared to moment 3, the temperature in the regenerator at moment 4 further decreases substantially due to the cold chamber exhausting process in which the cold gas is pushed out from the cold chamber to the regenerator. The temperature profile at moment 4 displays a large temperature inhomogeneity at the middle of the regenerator. This is caused by the jet flows of the cold gas with large density that generate from channels f, i, and k during the cold chamber exhausting process. As shown in Figs. 8 and 9, at certain times during a cycle, most of the regenerator near the cold end remains at very low temperatures with small temperature gradient. Indeed, at moment 4,
more than half of the regenerator remains at almost the same, low temperature. Furthermore, most of the temperature drop along the regenerator occurs in the middle area c which is a little shorter than both areas b and d. From Fig. 9 it can be seen that there is a large temperature fluctuation, which occurs over most of the regenerator during one cycle. Especially in area c, the temperature change at the middle cross section (X/L = 0.5) can be larger than 120 K. An ideal regenerator should have as small temperature change as possible for any position during a cycle and its temperature gradient should be consistent along the regenerator. The above results reveal that the heat capacity of the regenerator is insufficient, and a large enthalpy flow is likely caused by the large temperature fluctuation in the regenerator, generating a type of heat loss in the GMC. Fig. 10 shows the pressure drop variations of area b, c, and d in the regenerator during a cycle, i.e. pressure differences between surfaces S1 and S2, S2 and S3, and S3 and S4 (as shown in Fig. 1), respectively. As can be seen, the pressure drop in area c and d is much smaller than that in area b. Indeed, in area d there is almost no pressure drop during the entire cycle. It should be noted that the length of area b, c and d are almost the same. As noted in Table 1, area b has the largest porosity, secondly area c, and thirdly area d. Therefore, the length and porosity are not the main reason for the pressure drop distribution in the different areas. The primary cause is the different viscosity of helium-4 in the various temperature ranges. As shown in Figs. 8 and 9, area b remains at
Fig. 10. Pressure drop variations at different areas of the regenerator.
616
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
temperatures between about 200 and 280 K; area c remains at temperatures between 50 and 200 K; area d remains at temperatures between 13 and 50 K. Therefore, area b has the highest pressure drop due to the large viscosity of helium-4 at high temperatures, while area c has almost no pressure drop due to the correspondingly low viscosity below 50 K. From the above temperature and pressure drop distributions, it can be inferred that using regenerator materials with high porosity at high temperature regions and low porosity, high heat capacity at low temperature regions can be an effective way to improve the cooling performance of this single stage GMC. 3.3. Heat transfer efficiency of the CHX under different situations Besides reducing the total regenerator losses of the system, increasing the cooling power transfer efficiency at the CHX is also a significant concern for cryocoolers, especially for the ones working with large cooling power. As shown in Fig. 1, wall-1 is the wall between channel g and the CHX m, wall-2 is the wall between channel l and CHX m, and the heat load Qc is transferred from the CHX to the inside cold gas through wall-1 and wall-2. Apparently, the heat transfer at the CHX is primarily determined by the outflow distribution of the cold gases through the various channels (g, i and k in Fig. 1) during the expansion and displacer exhausting period. Therefore, the effects of different cold end channel configurations on the heat transfer efficiency of the CHX are further investigated: case 1 is the original model with three channels g, i, k; case 2 is the model with only channel g at the cold end, while channels i and k are set as solid area. Figs. 11 and 12 show the temperature difference between the CHX and its nearby gas in case 1 and case 2 during a cycle. DT1 is the difference between the average temperature of wall-1 and that of the gas in channel g; DT2 is the difference between the average temperature of wall-2 and that of the gas in channel l; DTa is the cycle-averaged value of DT1 or alternatively DT2. From Fig. 11 it can be seen that the values ofDT1 for case 2 under large cooling powers are noticeably smaller than those of case 1. The cycle-averaged value DTa decreases from 4.2 K to 2.9 K and from 8.4 K to 5.4 K at respective cooling powers of 80 W and 120 W. As shown in Fig. 12, the values ofDT2 for case 2 are also noticeably reduced compared to those of case 1; its cycle-averaged value decreases from 5.6 K to 3.4 K and from 10.9 K to 7.3 K at respective cooling powers of 80 W and 120 W. In Fig. 12, there are frequent fluctuations of DT2 occurring at around 0.12–0.23 s and 0.34–
Fig. 11. Temperature difference DT1 between the cold heat exchanger (wall-1) and gas in channel g (Case 1 and Case 2 represent three channel configuration and single channel configuration respectively).
Fig. 12. Temperature difference DT2 between the cold heat exchanger (wall-2) and gas in channel l (Case 1 and Case 2 represent three channel configuration and single channel configuration respectively).
0.5 s when the displacer is moving. This may be due to the complicated flow inside the cold end chamber l caused by the motion of the displacer. During the intake and exhaust process, some turbulence or non-uniform flows can exist at the junctions between the cold end channels (g, i, k) and the cold chamber l. These unsteady flows may lead to a fluctuation of DT2 due to the mixing of gas with different temperatures and also an instantaneous heat transfer variation between the gas and wall-2. While for channel g, it is narrow and long, so its internal flow can be more regular and smooth, as reflected by the values of DT1. Fig. 13 further shows the average heat transfer efficiency of wall-1 and wall-2 in case 1 and case 2 under different cooling powers. As can be seen, the heat transfer efficiency of CHX in case 2 is obviously increased compared to that in case 1 during a cycle. These observations result because in case 1 the cold gas can flow from the cold chamber l to the regenerator through three paths: channels g and f, channel i, and channel k, among which the path of channel g and f has the largest flow resistance. Therefore, most of the cold gas would like to flow through channels i and k instead of g and f, which significantly reduces
Fig. 13. Average heat transfer coefficient h between cold chamber gas and heat exchanger wall (Case 1 and Case 2 represent three channel configuration and single channel configuration respectively).
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
617
non-uniform flow and large density of the gas at low temperatures. This will lead to larger heat losses in the regenerator. Therefore, at low temperatures, the performance of case 1 is better than that of case 2 due to its smaller irreversible losses. On the other hand, at the high temperatures resulting from a large cooling power, such as 80 W, since the density of helium-4 is smaller, the flow and temperature inhomogeneity caused by the side channel g is not as pronounced, see Fig. 15. Furthermore, in this situation, the heat transfer efficiency of the CHX is more significant. Therefore, at large cooling powers the performance of case 2 is better than case 1. As stated above, it can be concluded that when the cryocooler is designed to work at low temperatures with a small cooling power, the flow and the temperature distribution at the cold end should be considered more than the heat transfer at the CHX; however, when it operates at large cooling powers, the heat transfer at the CHX is more important for the cooling performance and should be a more dominant factor for designing the cold end channels. Fig. 14. Cooling performance comparison of case 1 and case 2 (Case 1 and Case 2 represent three channel configuration and single channel configuration respectively).
the total heat transfer between the CHX and gas, since the heat transfer area of wall-2 is much smaller than that of wall-1. In contrast, for case 2 all the cold gas will flow back to the regenerator through channel g with sufficient time and area to exchange heat with the CHX. Therefore, optimizing the cold channels as in case 2 will increase the available cooling power of the GMC. Fig. 14 shows the calculated cooling powers at different temperatures for case 1 and case 2. It can be seen that above about 20 K the cooling power of case 2 is larger than that of case 1, which is consistent with the above results. Especially at 30 K, the calculated cooling power of case 2 is larger by 20% compared to that of case 1. However, it can be seen that the cooling power of case 2 is smaller than that of case 1 at low temperatures, below about 20 K. This is because the flow channels not only affect the heat transfer at the CHX but they also influence the flow distribution and temperature distribution in the regenerator. Fig. 15 shows the temperature distributions of case 2 at moment 4 for two different heat loads (Qc), 0 W and 80 W. Compared to case 1 at moment 4 (in Fig. 8), the temperature distribution for case 2, where there is only side channel g at the cold end, is more inhomogeneous due to the
4. Conclusions The single stage G-M cryocooler (GMC) with a mechanically driven displacer has been simulated and studied for the first time using a CFD method with a dynamic mesh scheme. The instantaneous temperature and pressure distributions inside the GMC are displayed, and the effects of various cold end channels on the heat transfer efficiency of the cold heat exchanger (CHX) are analyzed. The model results agree well with the experimental measurements. The results demonstrate that for this single stage GMC working below 13 K, half of its regenerator near the cold end will remain at very low temperatures with a small temperature gradient during a cycle. At the same time, there exist large temperature fluctuations, larger than about 100 K, at most middle parts of the regenerator during a cycle. Both of these features indicate an inefficient use of the heat capacity of the materials working at low temperatures. Additionally, more than 80% of the total pressure drop occurs in the regenerator’s upper region which operates at high temperatures. The results imply that using regenerative materials with high porosity at high temperature region and low porosity and high heat capacity at low temperature region can improve the cooling performance of the GMC. The design of the cold end channels will vary for the GMC, depending on the operating goals. When it operates at low
Fig. 15. Temperature distributions of case 2 (Case 2: three channel configuration, Tc = 14.6 K) with different cooling powers at Moment 4 (at 0.5 s, the end of a cycle).
618
X. Zhi et al. / International Journal of Heat and Mass Transfer 115 (2017) 611–618
temperatures with a small cooling power, the arrangement of the cold end channels should be driven more by the uniform flow and temperature distribution at the cold end than the heat transfer efficiency of the CHX. However, when it operates at high temperatures with a large cooling power, the heat transfer at the CHX is more important for the cooling performance and the cold end channels should be designed to allow as much cold gas flowing past the CHX as possible. Acknowledgement The authors appreciate the helpful discussions with Ralph Longsworth. This work is supported by SHI Cryogenics of America, Inc. Conflict of interest None declared. References [1] C. Wang, P.E. Gifford, High efficiency, single-stage GM cryorefrigerators optimized for 20 to 40K, Cryocoolers 11 (2001) 387–392. [2] I. Takashi, N. Masashi, N. Kouki, Y. Hideto, Development of a 2 W class 4 K Gifford- McMahon cycle cryocooler, Cryocoolers 9 (1997) 617–626. [3] X.H. Hao, S.H. Yao, T. Schilling, Design and experimental investigation of the high efficiency 1.5 W/4.2 K pneumatic-drive G-M cryocooler, Cryogenics 70 (2015) 28–33. [4] W. Huang, P.Y. Wu, S.L. Hu, L. Zhang, et al., Dynamic simulation of one-stage GM refrigerator and comparison with experiment, Cryogenics 36 (1996) 643– 647.
[5] A. Fiedler, J. Gerban, H.U. Haefner, Efficient single stage Gifford-McMahon refrigerator operating at 20 K, Adv. Cryogenic Eng. 43 (1998) 1823–1830. [6] G. Green, W. Superczynski, Ribbon regenerator performance in a single-stage GM cryocooler, Cryocoolers 14 (2007) 373–379. [7] C. Wang, P.E. Gifford, Performance improvement of a single stage GM cryocooler at 25 K, Adv. Cryogenic Eng. 53 (2008) 26–33. [8] K. Yamada, Development of a large cooling capacity single stage GM cryocooler, Cryogenics 63 (2014) 110–113. [9] M.Y. Xu, T. Morie, Numerical simulation of the second stage regenerator in a 4 K GM cryocooler, Adv. Cryogenic Eng. 1573 (1) (2014) 1143–1148. [10] X.H. Hao, S.H. Yao, T. Schilling, Design and experimental investigation of the high efficiency 1.5 W/4.2 K pneumatic-drive G-M cryocooler, Cryogenics 70 (2014) 28–33. [11] L.Q. Liu, L. Zhang, Study on a new type of sealing – regeneration-labyrinth sealing for displacer in cryocoolers: Part I - Theoretical study, Cryogenics 40 (2000) 75–83. [12] X.H. Hao, Y.L. Ju, Y.J. Lu, Experimental study on the sealing clearance between the labyrinth sealing displacer and cylinder in the 10 K G-M refrigerator, Cryogenics 51 (2011) 203–208. [13] X.Q. Zhi, L.M. Qiu, J.M. Pfotenhauer, et al., Refrigeration mechanism of the gas parcels in pulse tube cryocoolers under different phase angles, Int. J. Heat Mass Transfer 103 (2016) 382–389. [14] Y. Zhao, H. Dang, CFD simulation of a miniature coaxial Stirling-type pulse tube cryocooler operating at 128 Hz, Cryogenics 73 (2015) 53–59. [15] B. Flake, A. Razani, Modeling pulse tube cryocoolers with CFD, Adv. Cryogenic Eng. 710 (2004) 1493–1499. [16] ANSYS INC, ANSYS Fluent 14.5 User’s Guide, 2012. [17] L.M. Qiu, X.Q. Zhi, Z.H. Gan, et al., Function of gas parcels in the pulse tube, Int. J. Refrigeration 38 (2014) 358–366. [18] D. Gedeon, G. Wood, Oscillating-flow regenerator test rig: hardware and theory with correlations for screens and felts, NASA-Lewis Contractor Report 198422, 1996. [19] David Gedeon, Stirling-cycle model class reference guide (Sage v5 Edition), 2007, pp. 114.