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Refrigeration mechanism of the gas parcels in pulse tube cryocoolers under different phase angles
International Journal of Heat and Mass Transfer 103 (2016) 382–389
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Refrigeration mechanism of the gas parcels in pulse tube cryocoolers under different phase angles Xiaoqin Zhi a, Limin Qiu a,⇑, John M. Pfotenhauer b, Zhihua Gan a, Yan Yan c a
Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, China College of Mechanic Engineering, University of Wisconsin, Madison 53706, USA c School of Mechanical and Engineering, Southeast University, Nanjing 21000, China b
a r t i c l e
i n f o
Article history: Received 8 April 2016 Received in revised form 21 June 2016 Accepted 12 July 2016
Keywords: Pulse tube cryocooler Heat transfer Refrigeration mechanism Thermodynamic cycles Phase angle
a b s t r a c t The thermodynamic behavior of gas parcels in the key parts of the pulse tube cryocooler (PTC) is studied. The refrigeration mechanism of PTCs under different cold end phase angles is revealed by analyzing and comparing the heat transfer characteristics of the gas parcels at both sides of the cold end heat exchanger. Results show that the cold end phase angle, an important parameter of the PTC, determines the cooling performance of the PTC by affecting the heat transfer characteristics of the gas parcels in a cycle. In PTCs with a cold end phase angle normally between 30° and 60°, the gases pump heat from the cold end heat exchanger to the aftercooler all the way through the regenerator. In contrast, for the basic PTC with a cold end phase angle close to 90°, the heat transfer direction is reversed, the gases pump heat from the cold end heat exchanger to the hot end heat exchanger all the way through the pulse tube. This study unifies the various perspectives on the PTC’s refrigeration mechanism by revealing the inherent effect of the cold end phase angle, indicating that enhancing the heat pumping function of the gas in the regenerator is an essential way to improve the cooling performance. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The pulse tube cryocooler (PTC) with no moving parts at the cold chamber has attracted increasing attention due to its promising applications in the military, space and superconductor fields, each having rigorous requirements of compactness and high reliability for the cooling system. The recently published literature shows that the PTC is receiving more interest as a cryocooler than the traditional J-T, Stirling and G-M cryocoolers. It is likely to replace other cryocoolers in more applications because of its simplified structure and reliable performance [1]. However, the simplified structure of the PTC leads to complex working mechanisms due to the phase shifting effect of the gas inside. As a member of the regenerative cryocooler family, the PTC has a ‘‘gas piston” inside the pulse tube that causes gas expansion and decreasing temperatures at the cold end like the piston expander in the Stirling and G-M cryocoolers. Therefore, the early studies of the PTC mainly focused on the pulse tube component, for example, the surface heat pumping theory [2], the phase shifting theory [3], the adiabatic expansion principle, and the thermodynamic non-symmetry analysis [4,5]. These studies consider that it is the ⇑ Corresponding author. Fax: +86 571 87952793. E-mail address: [email protected] (L. Qiu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.07.032 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
gas in the pulse tube that produces cooling by expanding and absorbing heat at the cold end heat exchanger. However, with the development of the thermoacoustic theory, people realized that the regenerator is not only a heat exchanger, but also a work-heat converter, working either as an engine or a refrigerator under different thermoacoustic fields. Since then, theoretical investigations have been carried out on the regenerator based on classic thermodynamic principles [6,7] and thermoacoustic theory [8,9], these demonstrating that the gas in the regenerator can produce cooling by pumping heat away from the cold end heat exchanger. The contradictory nature of the above studies, as to whether the pulse tube or the regenerator produce cooling at the cold end is mainly caused by the over-simplification associated with the one dimensional, ideal regenerator with no dead volume, and the ideal isothermal or adiabatic analyses. These inappropriate methods of analysis may lead to the limitations and uncertain authenticity of the results obtained. Also some results analyses are shown in Euler way of focusing on the time-averaged parameters at fixed cross section, which is not so visual to reveal the periodical behavior of the oscillating gases. In a PTC, the various gas parcels each experience differing thermodynamic processes through periodic oscillation, and the inhomogeneous characteristics of energy, mass transfer and temperature distribution, especially inside the pulse tube, cannot
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be ignored [10]. It is helpful to reveal the refrigeration mechanism of the PTC by studying the gas parcels’ thermodynamic behavior through the use of a multi-dimensional, quantitative approach such as is possible at a micro scale in a Lagrangian way. Such research has been realized in the regenerator and the pulse tube by CFD simulation methods [11,12]. The cooling power is actually the result of a cooperative interaction between the different parts in a PTC. However, most of the previous investigations studying the refrigeration mechanism are carried out on a single component (either the regenerator or the pulse tube) under different working conditions, and these are unable to explain how the net cooling effect generated in the cold end heat exchanger is a result of the combined work of the gas in the regenerator and pulse tube. Additionally, although it is an essential parameter of the oscillating flow, no research has yet investigated how the cold end phase angle affects the thermodynamic cycles of the gas in the regenerator at a micro scale, connecting that behavior to the macro-scale cooling performance. In this study, the refrigeration mechanism of the PTC is investigated by analyzing the thermodynamic behavior of the gas parcels in its key locations. The heat transfer characteristics of the gas parcels in the pulse tube, regenerator and cold end heat exchanger are compared in order to explain the net cooling effect that is produced. Furthermore, the effects of different cold end phase angles on the gas parcels’ periodic heat transfer characteristics are studied at a micro-scale to reveal its influence on the cooling performance.
@ ! !! ! /qf u þ r /qf u u ¼ /rp þ r /s u þ SxðyÞ @t @ ! ð/qf Ef þ ð1 /Þqs Es Þ þ r u ðqf Ef þ pÞ @t h i ! ¼ r ð/kf þ ð1 /Þks ÞrT þ ð/s u Þ Ef ¼ Hf þ SxðyÞ ¼
u2 p ; 2 qf
Es ¼ C v s T
2.1. Mathematic model and boundary conditions The structure of the two-dimentional, axis-symmetric single stage PTC is shown in Fig. 1. The details are given in Table 1. The PTC model contains two kinds of flow areas, the porous media area as the aftercooler a, regenerator b, cold end heat exchanger (CHX) c and hot end heat exchanger (HHX) e; and the non-porous media area as the pulse tube d. For all the flow areas, the mass, momentum and energy equations can be described as follows [13]:
@ ! ð/qf Þ þ r /qf u ¼ 0 @t
ð1Þ
ð3Þ ð4Þ
1 !! f q juju 2Dh r f
ð5Þ
The explanations of each item in the above equations are given in reference [12,13]. For the pulse tube d, the porosityu = 1, and the momentum source term Sx(y) = 0; for the areas a, b, c, e, their porosity is given in Table 1, the momentum source term is calculated by Eq. (5), in which fr is the flow resistance coefficient of the porous media. In the regenerator and the hot end heat exchanger, which are packed with stacked screens, the term fr is treated as isotropic and it can be calculated by the following empirical formula [14]:
f r ¼ 129=Re þ 2:91Re0:103
ð6Þ
For the areas a, c which are slit type heat exchangers, the term fr can be calculated as [15]:
f r ¼ 0:11ð/=dh þ 68=ReÞ
2. Numerical model of the single stage PTCs
ð2Þ
0:25
ð7Þ
In order to study the influence of the cold end phase angle hC, its value is determined for both the phase shifting type and the basic type of PTCs. The cold end phase angle is defined as the phase angle between the mass flow oscillation and the pressure oscillation at the cold end. It is positive when the mass flow oscillation leads the pressure oscillation. For the phase shifting PTCs, the mass flow _ in and the pressure wave pout are respectively used as the rate m inlet and outlet boundary conditions:
m_in ¼ ma sinð2pft þ hh p=180Þ
ð8Þ
pout ¼ pa sinð2pftÞ
ð9Þ
Fig. 1. Sketch of the pulse tube cryocooler and the typical gas parcels at different positions.
Table 1 Details for different parts of the pulse tube cryocooler. Items
Porosity
Material
Boundary conditions
Aftercooler a Regenerator b Cold end heat exchanger (CHX) c Pulse tube d Hot end heat exchanger (HHX) e
0.2 0.72 0.24 1 0.78
Slit type, copper SS304 mesh screen Slit type, copper Stainless steel Copper mesh screen
Isothermal, 280 K Adiabatic 80 K (160 K for basic PTC) Adiabatic Isothermal, 280 K
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where ma (4 g/s) and pa (1.6 bar) are the amplitudes of the oscillating mass flow rate and the pressure wave (with an average pressure of 2 MPa), f (40 Hz) is the operating frequency, and hH is the phase angle between m_in and pout at the hot end of the regenerator. For the basic PTC where the hot end heat exchanger e is closed, only the mass flow rate is used as the inlet boundary condition. Various values for the cold end phase angle hC can be obtained by using different values of hH. Here the input values of hH used for phase shifting type PTCs are 30°, 55°, 72° and 81°, the corresponding values of hC obtained based on hH are 30°, 0°, 30° and 60°. For the basic PTC the value of hC directly calculated from the model is 83°. In all PTC models, the temperature of the aftercooler and the HHX are fixed at 280 K, while the temperature for the CHX is fixed at 80 K for the phase shifting PTCs and 160 K for the basic PTC. The heat transfer between the wall and the gas inside the pulse tube is considered and the wall thickness is included in the model. The walls of the heat exchangers are set as isothermal while that of other components is set as adiabatic. For the heat exchangers, the heat capacity and thermal conductivity are set to the respective constant values corresponding to the local temperatures; for the regenerator they are treated as piecewise linear functions of temperature. The ideal gas version of helium-4 is used as the working fluid. Because the pressure change is small, the viscosity, heat capacity and thermal conductivity of the helium-4 are also treated as piecewise linear functions of temperature. In all the models, the charge pressure is 2.0 MPa, the operating frequency is 40 Hz, and each cycle is divided to 100 time steps in the computation. Table 2 shows a comparison of this work to the previous study of the same PTC (with the same operating condition and geometric parameters) [16]. The internal fluid properties of the PTC obtained here agree with those obtained by the common cryocooler design software SAGE. Also, the cooling power, in this case is close to that obtained in a previous experiment (8.8 W at 80 K) [17]. 2.2. Data processing method for visualizing the thermodynamic cycle of the gas parcel CFD simulations based on the Euler algorithm cannot provide the thermodynamic cycles of the gases directly. However, due to the instantaneous results available with CFD, the thermodynamic cycles of individual gas parcels can be extracted according to the Lagrange perspective by tracking their motion at each time step. For example, at moment n, the gas parcel’s location is Xn, and its velocity is Un. After one time step Dt, the gas parcel’s location Xn+1 at the next moment n + 1 is calculated as follows:
X nþ1 ¼ X n þ
ðU nþ1 þ U n Þ Dt 2
period, for example from moment i to j, can be calculated as follows:
Z
j
W ij ¼ i
Z Q ij ¼
pdv
n¼j
1X ðT n þ T nþ1 Þðsnþ1 sn Þ 2 n¼i
ð11Þ
n¼j
j
Tds i
1X ðp þ pnþ1 Þðv nþ1 v n Þ 2 n¼i n
ð12Þ
The typical gas parcel in each of the key sections of PTC are analyzed, and their equilibrium positions are shown in Fig. 1. In order to avoid calculating results at the junction between the exchangers and its nearby parts, the equilibrium position of gas parcels 1, 5, 7, 8, 9, 10, 12, 13 and 15 is about 1 mm away from the heat exchangers, which is similar to the treatment in a previous study [12]. Their movement in a cycle can be divided into two processes: the BCD process, in which the gas parcel moves at the right side of the equilibrium position; and the DAB process, in which it moves at the left side. In the pulse tube, the BCD process belongs to the hot side and the DAB process belongs to the cold side. The opposite situation exists in the regenerator due to the reverse temperature gradient compared to the pulse tube. In each cycle, the gas parcels move left and right around the equilibrium position and exchange energy with the surroundings, resulting in a heat and work transfer between the cold side and hot side.
3. Thermodynamic cycle of the gas parcels under hC = 0° In a PTC, where the displacement of the gas lags the mass flow rate by 90°, we can infer the associated pressure values for the typical positions A, B, C, D of a gas parcel oscillation for different values of hC, as shown in Fig. 2. A phase angle of hC = 0° (pressure and mass flow rate are in phase) is a common value in a PTC. When hC = 0°, as shown in Fig. 2, the section BCD represents a complete expansion process because the pressure drops throughout this section, while the DAB section is associated completely with a compression process because the pressure increases throughout this section.
ð10Þ
where an average velocity from two adjacent moments is used in order to track the gas parcels accurately. From Eq. (10), the gas parcel’s locations at each time step can be obtained, so its corresponding pressure p, specific volume v, temperature T and specific entropy s at each time step can be known to get the p-v and T-s diagrams in a cycle. In the p-v and T-s diagrams, the amount of heat and power exchange between the gas parcels during any time
Table 2 Results comparison between previous SAGE simulation and this work. Items
Previous simulation
This work
Cold end mass flow amplitude (g/s) Cold end pressure ratio Cold end phase angle (°) Cold end PV power (W) Cold end enthalpy flow (W) Cooling power (W)
3.53 1.190 29.1 26.7 18.29 9.56
3.32 1.181 30 23.9 16.2 9.12
Fig. 2. Relationship between pressure wave and mass flow rate under different hC.
X. Zhi et al. / International Journal of Heat and Mass Transfer 103 (2016) 382–389
385
Fig. 3. The p-v (a) and T-s (b) diagram of gas parcel 5.
Fig. 4. The p-v (a) and T-s (b) diagram of gas parcel 6.
Table 3 Work and heat exchange in the four processes of gas parcel 5,6 and 7 (hC = 0°, unit: kJ/kg).
heat isothermally by expanding. In process BC, GP 5 absorbs a net amount of heat and produces a net amount work (as shown in Table 3). In process CD, GP 5 continues to expand and absorbs heat in the CHX. After it leaves the CHX, it moves toward the hot side in the regenerator where the local temperature is increasing, so gas parcel 5 absorbs a large amount of heat from the regenerator and its temperature increases, even though it is expanding. Therefore, in process BCD, GP 5 expands and absorbs a net amount of heat at the cold side. In process DA, as GP 5 moves towards the hot side of the regenerator, the compression period starts. Because its temperature at the early part of the compression period is lower than the temperature of the regenerator, it continues to absorb heat, and its specific entropy increases. Later, it releases heat because the compression process elevates the gas temperature above the local matrix temperature in the regenerator. In process DA, GP 5 consumes a net amount of work and absorbs a net amount of heat. In process AB, GP 5 moves back to the equilibrium position, and continues to be compressed. Since the local temperature of the regenerator is decreasing, gas parcel 5 releases a large amount of heat to the surroundings. The heat it releases is larger than the work it consumes (as shown in Table 3), so its temperature decreases. In process DAB, GP 5 consumes a net amount of work and releases a net amount of heat at the hot side. Therefore, as stated above, GP 5 expands and absorbs heat through the cold portion of its cycle in the CHX, and is compressed, releasing heat during the hot portion of its cycle. For the complete cycle, a net amount of work is consumed as the cost of transferring heat from the cold side to hot side.
Process
BC CD DA AB BCD DAB BCDAB
GP 5
GP 6
GP 7
W
Q
W
Q
W
Q
9.31 23.35 8.29 27.82 32.66 36.11 3.45
0.62 31.47 29.52 65.07 32.09 35.55 3.46
15.87 11.84 16.00 11.71 27.71 27.71 0.00
15.87 11.84 16.00 11.71 27.71 27.71 0.00
11.05 7.15 7.44 11.35 18.20 18.79 0.58
3.23 1.48 6.04 11.35 4.72 5.30 0.58
3.1. The gas parcels at the CHX Figs. 3–5 show the T-s and p-v diagrams of gas parcel 5, 6, 7, and the corresponding amounts of heat and work exchanged in processes BC, CD, DA and AB in a cycle are shown in Table 3. Gas parcel 5 (GP 5) moves between the regenerator and the CHX. In process BC, as shown in Fig. 3, GP 5 moves from the equilibrium position to the CHX, and its pressure drops. The initial rate of temperature drop during the early expanding period is lower than the decrease of temperature in the regenerator matrix in the direction of the CHX (i.e. the regenerator temperature gradient), therefore, before the gas parcel enters the CHX, it releases some heat, and its specific entropy and volume decrease. Upon entering the CHX, its temperature remains at 80 K and it absorbs
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Fig. 6. T-s diagram of gas parcels 1 to 5 in the regenerator.
Fig. 5. The p-v (a) and T-s (b) diagram of gas parcel 7.
Gas parcel 6 (GP 6) moves completely within the CHX, experiences isothermal compression and expansion, and produces no net energy transfer within a cycle. In process BCD it expands and absorbs heat, and in process DAB it is compressed and releases the same amount heat there. Therefore, gas parcels in the CHX transfer heat from the pulse tube side to the regenerator side equally, as shown in Fig. 4 and Table 3. Gas parcel 7 (GP 7) moves between the CHX and pulse tube. In process BCD, GP 7 moves within the pulse tube, and as shown in Fig. 5, it expands throughout this period and absorbs heat. In process DAB, GP 7 moves into the CHX, and experiences a compression process. At the beginning, its temperature is lower than the CHX because it is expanding, so when it enters the CHX, it first absorbs heat from the CHX. But later, since its temperature remains 80 K as it is compressed, it releases heat back to the CHX. In process DAB,
the gas parcel releases a net amount of heat to the CHX and consumes a net amount of work, as shown in Table 3. Thus, GP 7 does not produce cooling at the CHX even though its temperature decreases by expanding in the pulse tube. Rather, it transfers heat loss from the pulse tube to the CHX, which is in keeping with the previous study [12]. The total heat released to the CHX (QDAB) is the sum of the heat absorbed in process BCD (QBCD) and the net work consumed in the cycle (WBCDAB). In an ideal pulse tube where the gas expands adiabatically, there is no heat absorbed in process BCD. However, there is always a net amount of work consumed due to the transition between the adiabatic and isothermal boundary. Therefore, there is always some heat loss caused by the gas as it moves between the pulse tube and the CHX, which isn’t revealed in the existing studies. It should be noted here that the GP 7 only moves between the CHX and the pulse tube, so it cannot produce cooling at the CHX as it is compressed in the CHX. In some cases where the CHX is short enough and the mass flow rate is large, there are gas parcels moving between the pulse tube and the regenerator by crossing the CHX. These gases move across three components in a cycle and do not individually strictly belong to any single component. They act as a bridge between the gases at the regenerator side and the pulse tube side. During a cycle, these gas parcels mainly expand in the pulse tube, and then absorb a large amount of heat in the CHX after expanding; later they mainly compress in the regenerator and reject a large amount of heat there. As a result, these gas parcels still produce net cooling at the CHX. Therefore, it is not true that all the gases expanding in the pulse tube will produce a heat loss rather than a cooling effect at the CHX. However, it is true of the gas parcels discussed above, that is those gas parcels on the pulse tube side that are moving between the CHX and the pulse tube alone, and those moving completely within the pulse tube.
Table 4 Heat exchange in the four processes of gas parcels at cold end (hC = 0°). Q (kJ/kg)
BC CD DA AB BCD DAB BCDAB
Regenerator side
Pulse tube side
GP 5
GP 8
GP 9
GP 10
GP 7
GP 11
GP 12
GP 13
0.62 31.47 29.52 65.07 32.09 35.55 3.46
0.68 31.12 29.79 65.13 31.80 35.35 3.55
0.70 31.07 29.77 65.24 31.77 35.47 3.70
0.74 31.35 29.39 65.21 32.09 35.82 3.73
3.62 1.10 6.05 11.35 4.72 5.30 0.58
3.33 3.23 4.10 11.20 6.61 7.10 0.53
3.12 4.21 3.33 11.20 7.33 7.87 0.54
6.07 10.23 7.20 9.70 16.30 16.90 0.60
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X. Zhi et al. / International Journal of Heat and Mass Transfer 103 (2016) 382–389 Table 5 Heat exchange amount of gas parcels along the regenerator and pulse tube (hC = 0°). Q (kJ/kg)
GP5
GP4
GP3
GP2
GP1
GP7
GP14
GP15
BC CD DA AB DAB BCD BCDAB
0.62 31.47 29.52 65.07 35.55 32.09 3.46
30.45 71.68 23.25 69.54 46.29 41.23 5.06
60.64 106.23 54.66 107.20 52.54 45.59 6.95
121.31 169.20 61.61 118.22 56.61 47.90 8.71
127.64 194.00 61.12 137.33 76.21 66.36 9.85
3.62 1.10 6.05 11.35 5.30 4.72 0.58
5.09 4.99 5.33 5.21 10.54 10.08 0.46
0.96 13.91 3.41 10.41 13.82 12.95 0.87
Table 6 Heat exchange in the four processes of gas parcel 4 under different hC. Q (kJ/kg)
30°
0°
30°
60°
83°
BC CD DA AB BCD DAB BCDAB
35.32 70.14 27.59 68.40 34.82 40.81 5.99
30.45 71.68 23.25 69.54 41.23 46.29 5.06
38.13 63.08 35.85 63.68 24.95 27.83 2.88
44.89 54.53 49.78 61.15 9.64 11.37 1.73
67.05 64.61 82.47 80.37 2.43 2.09 0.34
3.2. Gas parcels inside the regenerator and pulse tube Fig. 6 shows the T-s diagrams of gas parcels 1–5 in the regenerator. Table 5 gives the amount of their heat exchange in the four processes. All the gas parcels absorb heat at the cold side and release heat to the hot side, pumping heat from the CHX to the aftercooler in a relay manner. As shown in Table 5, the gas parcels consume a net amount of work and release a net amount of heat in a cycle, which means that the regenerator not only performs as a heat accumulator but also a heat-work convertor. In contrast, as shown in Table 5, the gas parcels 7, 14 and 15 along the pulse tube transfer heat from the hot side to the cold side in a relay manner. Therefore, as analyzed above, in the PTC with hC = 0° the heat transfer direction by the oscillating gas parcels is from the pulse tube side to the regenerator side. It should be noted that a larger amount of heat is rejected at the aftercooler (as shown in Table 5) since in a real regenerator there is entropy flow inside caused by heat pumping, in order to balance the entropy flow from the low temperature side, more heat must be rejected at the high temperature side. 4. The effects of hC on the thermodynamic cycle of the gas parcels Fig. 7. The p-v (a) and T-s (b) diagrams of gas parcel 5 (hC = 83°).
As shown in Table 3, GP 5 on the regenerator side absorbs heat while in the CHX, and GP 7 on the pulse tube side releases heat to the CHX. However, the heat absorbed (QBCD) by GP 5 is much larger than the heat released (QDAB) by GP 7. Table 4 further shows the amount of heat they exchange during a cycle. As can be seen in Table 4, the heat absorbed (QBCD) by the gas parcels on the regenerator side is much larger than the heat released (QDAB) by the gas parcels on the pulse tube side, and there is net cooling power at the CHX. This is because the gas parcels in the regenerator exchange heat almost isothermally while the gas parcels in the pulse tube traverse their cycle near adiabatically. In the pulse tube the boundary effect is more obvious than that in the porous regenerator. The temperature is more inhomogeneous near the wall [10], so the gas parcel (GP 13) near the pulse tube wall loses more heat to the cold side in a cycle, as shown in Table 4.
As is commonly known, the cold end phase angle hC plays an essential role in the cooling performance of PTCs for it macroscopically affects the PV power at the cold end. It is necessary to further investigate the effects of hC on the gas parcels’ thermodynamic behavior and reveal its working mechanism macroscopically. The periodic amount of heat exchanged by gas parcel 4 (GP 4) in the regenerator under various typical cold end phase angles is shown in Table 6. Note that when hC decreases from 0° to 30°, the heat absorbed by GP 4 in process BCD decreases, as does the heat released in process DAB. This is because, as shown in Fig. 2, when hC = 0° the process BCD is completely an expansion process and has the largest potential to absorb heat, while the process DAB is completely a compression process and has the largest potential to release heat. When hC < 0°, the process CD will contain part of the early compression period, while the process AB will contain part of the early expansion period, so the heat absorbed in process BCD and the heat released in process DAB will both decrease. In the range of 30° < hC < 0°, the process BCD still
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Table 7 Heat exchange in the four processes of gas parcel 5 and 7 under different hC. Q (kJ/kg)
30° GP 5
0° GP 7
BC 4.50 3.52 CD 26.2 0.97 DA 28.26 3.01 AB 62.95 8.03 BCD 30.66 4.50 DAB 34.70 5.02 DQc 25.64 DQc = QBCD of gas parcel 5 + QDAB of gas parcel
30°
60°
83°
GP 5
GP 7
GP 5
GP 7
GP 5
GP 7
GP 5
GP 7
0.62 31.47 29.52 65.07 32.09 35.55 26.79 7
3.62 1.10 6.05 11.35 4.72 5.30
15.16 30.38 37.00 54.31 15.22 12.31 10.7
2.94 1.1 6.73 11.25 4.04 4.52
26.61 28.45 40.36 43.57 1.84 3.21 0.75
0.17 1.00 9.68 10.78 0.83 1.09
53.50 49.92 65.11 61.89 3.58 3.23 0.37
2.90 1.17 15.09 11.14 4.07 3.95
Table 8 Work and heat exchange in the four processes of gas parcels at cold end (hC = 83°, unit: kJ/kg). Process
BC CD DA AB BCD DAB BCDAB
Fig. 8. The p-v (a) and T-s (b) diagrams of gas parcel 7 (hC = 83°).
contains most of the expansion period and the process DAB contains most of the compression period. Therefore, GP 4 still absorbs heat in process BCD and releases heat in process DAB, and the heat transfer direction remains the same. Note that compare to GP4 of 0°, although GP 4 of 30° releases more net heat (QBCDAB) in a cycle, it pumps less heat from the cold side to the hot side as it absorbs less heat in process BCD. As shown in Table 6, when hC increases from 0° to 83° (the basic PTC), the net heat absorbed in process BCD and the net heat released in process DAB by GP 4 both decrease as hC increases. In fact, when hC = 83°, a net amount of heat is released in process BCD and absorbed in process DAB, This is because when hC increases from 0°, process BC will contain more and more of the
GP 5
GP 7
W
Q
W
Q
39.28 36.14 46.12 43.33 3.14 2.79 0.35
53.50 49.92 65.11 61.89 3.58 3.23 0.35
20.47 16.49 21.30 17.44 3.98 3.86 0.12
2.90 1.17 15.09 11.14 4.07 3.95 0.12
late compression period while process DA will contain more and more of the late expansion period, as shown in Fig. 2. Thus, the net heat absorbed in process BCD and the net heat released in process DAB will both decrease, and when hC is large enough, the direction of heat transfer for these two processes may reverse. The periodic amount of heat exchange for gas parcels 5 and 7 at the two sides of the CHX under different values of hC is given in Table 7. Note that the effect of different values of hC on the periodic heat transfer for GP 5 and 7 is similar with that of GP 4. As stated above, the cold end phase angle affects the cooling performance by determining the direction and the amount of heat transfer for the gas parcels in the PTC. For the phase shifting PTCs (30° < hC < 60°), the heat is transferred from the pulse tube side to the regenerator side and the gas parcels in the regenerator can produce cooling by pumping heat at the CHX. While the gas parcels on the pulse tube side (moving between the CHX and the pulse tube or moving completely inside the pulse tube) bring a heat loss to the CHX, even though they expand and cool down when moving in the pulse tube. Nevertheless there is net cooling effect DQc in the CHX, as shown in Table 7. In fact all the regenerative cryocoolers with phase shifting have the same feature that the gases absorb heat at the cold end and ‘‘transfer” the heat one by one through the regenerator to the aftercooler and release a larger amount of compression heat there. That is how the heat at the CHX is pumped away and rejected to the ambient. For the basic PTC (hC = 83°), the heat transfer direction changes to be from the regenerator side to the pulse tube side, the gas parcels in the pulse tube produce cooling by heat pumping, and the gas parcels moving between the regenerator and the CHX brings a heat loss to the CHX. Figs. 7 and 8 show the T-s and p-v diagrams of GP 5 and 7 in the basic PTC, and the amount of their heat and work exchange in a cycle is given in Table 8. As can be observed, the heat absorbed (QDAB) by GP 7 in the CHX is larger than the heat released (QBCD) by GP 5 there, so there is net cooling effect. As shown in Table 7, DQc (per unit mass) is maximum when hC = 0° and the gas parcel in the regenerator has the largest heat pumping ability, which is consistent with the phase shifting theory [3]. Also, DQc is minimum in the basic PTC, that is when hC = 83°. In
X. Zhi et al. / International Journal of Heat and Mass Transfer 103 (2016) 382–389
this case, only a small part of the gases in the pulse tube, those in the near wall area can pump heat effectively like all of the gases in the regenerator. This is the fundamental reason why the cooling performance of the phase shifting PTC is much better than that of the basic PTC. It should be noted that for a PTC the refrigeration efficiency is not only related to the cooling power in the CHX, but also the PV power loss in the regenerator. The previous study shows that when hC is around 30°, and the mass flow rate and the pressure are in phase at the middle of the regenerator, then the amplitude of the mass flow in the regenerator is minimized and the pressure drop is small, which guarantees a high efficiency in the PTC [17]. 5. Conclusions The refrigeration mechanism of the pulse tube cryocooler (PTC) under different cold end phase angles is studied by investigating the thermodynamic cycles of the gas parcels in its key components. The mechanism by which the cooling power is generated at the cold end is revealed by comparing the heat transfer characteristics of the gas parcels from both sides of the cold end heat exchanger (CHX). The results show that the cold end phase angle determines whether the regenerator or the pulse tube produces refrigeration via heat pumping since the phase angle affects the direction and magnitude of the heat transfer of the gas during a cycle. In the basic PTC, where hC is close to 90°, the heat is transferred from the regenerator side to the pulse tube side, and the gas in the pulse tube produces refrigeration by pumping heat from the CHX. In the phase shifting PTC, where hC is normally between 30° and 60°, the heat is transferred from the pulse tube side to regenerator side, the gas moving inside the regenerator helps to pump heat from the CHX to the aftercooler in a relay manner. In contrast, the gas completely moving in the pulse tube or moving between the CHX and the pulse tube transfers heat to the cold end as a heat loss, demonstrating that not all the gases expanding in the pulse tube can produce cooling power at the CHX. Since the heat pumping mechanism of the gas in the pulse tube is limited, enhancing the heat pumping effect of the gas in the regenerator by phase shifting improves the cooling performance, and represents the fundamen-
389
tal reason why the cooling performance of the phase shifting PTC is much better than that of the basic PTC. Acknowledgements The project is supported by National Funds for Distinguished Young Scientists of China (Contract No. 50825601) and the Cheung Kong Scholars Program of China. References [1] R. Radebaugh, Cryocoolers: the state of the art and recent developments, J. Phys. Condens. Matter 21 (2009) 164219. [2] W. Gifford, R. Longsworth, Surface heat pumping, Adv. Cryo. Eng. 11 (1966) 171–181. [3] R. Radebaugh, J. Zimmerman, D.R. Smith, et al., A comparison of three types pulse tube refrigerators: new methods for reaching 60 K, Advances in Cryogenic Engineering, vol. 31, Plenum Press, New York, 1986, pp. 779–789. [4] W.E. Gifford, R.C. Longsworth, Pulse tube refrigeration progress, Adv. Cryog. Eng. 10 (1965) 69–79. [5] J. Liang, A. Ravex, P. Rolland, Study on pulse tube refrigeration part 1: thermodynamic non-symmetry effect, Cryogenics 36 (2) (1996) 87–93. [6] J. Liang, Thermodynamic cycles in oscillating flow regenerators, J. Appl. Phys. 82 (9) (1997) 4159–4163. [7] E.C. Luo, W. Dai, Z. Wu, et al., Meso-scope thermodynamic theory for cyclic flow engines, Cryog. Eng. 1 (2004) 1–11 [in Chinese]. [8] J.H. Xiao, The thermoacoustic effect and thermoacoustic theory for the regenerative cryocooler (heat engine), Chinese Academy of Sciences, Beijing, 1991, pp. 75–85 [in Chinese]. [9] G.W. Swift, Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators, Acoustical Society of America, Sewickley, PA, 2002. [10] W. Liang, A.T.A.M. de Waele, A new type of streaming in pulse tubes, Cryogenics 47 (2007) 468–473. [11] L. Chen, Y. Zhang, E. Luo, et al., CFD analysis of thermodynamic cycles in a pulse tube refrigerator, Cryogenics 50 (2010) 743–749. [12] L.M. Qiu, X.Q. Zhi, Z.H. Gan, et al., Function of gas parcels in the pulse tube, Int. J. Refrig. 38 (2014) 358–366. [13] Fluent INC. Fluent 6.3 User’s Guide, 2003. [14] I.F. Macdonald, M.S. El-Sayed, K. Mow, et al., Flow through porous media – the Ergun equation revisited, Ind. Eng. Chem. Fundam. 18 (3) (1979) 199–208. [15] E. Fried, I.E. Idelchik, Flow resistance: a design guide for engineers, Hemisphere (1989). [16] Q. Cao, L.M. Qiu, Z.H. Gan, et al., Energy analysis of a single stage stirling pulse tube cryocooler, J. Eng. Thermophys. 32 (6) (2011) 905–906 [in Chinese]. [17] R. Radebaugh, Thermodynamics of Regenerative Refrigerators. Generation of Low Temperature and It’s Applications, 2003, pp. 1–20.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Refrigeration mechanism of the gas parcels in pulse tube cryocoolers under different phase angles Xiaoqin Zhi a, Limin Qiu a,⇑, John M. Pfotenhauer b, Zhihua Gan a, Yan Yan c a
Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, China College of Mechanic Engineering, University of Wisconsin, Madison 53706, USA c School of Mechanical and Engineering, Southeast University, Nanjing 21000, China b
a r t i c l e
i n f o
Article history: Received 8 April 2016 Received in revised form 21 June 2016 Accepted 12 July 2016
Keywords: Pulse tube cryocooler Heat transfer Refrigeration mechanism Thermodynamic cycles Phase angle
a b s t r a c t The thermodynamic behavior of gas parcels in the key parts of the pulse tube cryocooler (PTC) is studied. The refrigeration mechanism of PTCs under different cold end phase angles is revealed by analyzing and comparing the heat transfer characteristics of the gas parcels at both sides of the cold end heat exchanger. Results show that the cold end phase angle, an important parameter of the PTC, determines the cooling performance of the PTC by affecting the heat transfer characteristics of the gas parcels in a cycle. In PTCs with a cold end phase angle normally between 30° and 60°, the gases pump heat from the cold end heat exchanger to the aftercooler all the way through the regenerator. In contrast, for the basic PTC with a cold end phase angle close to 90°, the heat transfer direction is reversed, the gases pump heat from the cold end heat exchanger to the hot end heat exchanger all the way through the pulse tube. This study unifies the various perspectives on the PTC’s refrigeration mechanism by revealing the inherent effect of the cold end phase angle, indicating that enhancing the heat pumping function of the gas in the regenerator is an essential way to improve the cooling performance. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The pulse tube cryocooler (PTC) with no moving parts at the cold chamber has attracted increasing attention due to its promising applications in the military, space and superconductor fields, each having rigorous requirements of compactness and high reliability for the cooling system. The recently published literature shows that the PTC is receiving more interest as a cryocooler than the traditional J-T, Stirling and G-M cryocoolers. It is likely to replace other cryocoolers in more applications because of its simplified structure and reliable performance [1]. However, the simplified structure of the PTC leads to complex working mechanisms due to the phase shifting effect of the gas inside. As a member of the regenerative cryocooler family, the PTC has a ‘‘gas piston” inside the pulse tube that causes gas expansion and decreasing temperatures at the cold end like the piston expander in the Stirling and G-M cryocoolers. Therefore, the early studies of the PTC mainly focused on the pulse tube component, for example, the surface heat pumping theory [2], the phase shifting theory [3], the adiabatic expansion principle, and the thermodynamic non-symmetry analysis [4,5]. These studies consider that it is the ⇑ Corresponding author. Fax: +86 571 87952793. E-mail address: [email protected] (L. Qiu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.07.032 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
gas in the pulse tube that produces cooling by expanding and absorbing heat at the cold end heat exchanger. However, with the development of the thermoacoustic theory, people realized that the regenerator is not only a heat exchanger, but also a work-heat converter, working either as an engine or a refrigerator under different thermoacoustic fields. Since then, theoretical investigations have been carried out on the regenerator based on classic thermodynamic principles [6,7] and thermoacoustic theory [8,9], these demonstrating that the gas in the regenerator can produce cooling by pumping heat away from the cold end heat exchanger. The contradictory nature of the above studies, as to whether the pulse tube or the regenerator produce cooling at the cold end is mainly caused by the over-simplification associated with the one dimensional, ideal regenerator with no dead volume, and the ideal isothermal or adiabatic analyses. These inappropriate methods of analysis may lead to the limitations and uncertain authenticity of the results obtained. Also some results analyses are shown in Euler way of focusing on the time-averaged parameters at fixed cross section, which is not so visual to reveal the periodical behavior of the oscillating gases. In a PTC, the various gas parcels each experience differing thermodynamic processes through periodic oscillation, and the inhomogeneous characteristics of energy, mass transfer and temperature distribution, especially inside the pulse tube, cannot
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be ignored [10]. It is helpful to reveal the refrigeration mechanism of the PTC by studying the gas parcels’ thermodynamic behavior through the use of a multi-dimensional, quantitative approach such as is possible at a micro scale in a Lagrangian way. Such research has been realized in the regenerator and the pulse tube by CFD simulation methods [11,12]. The cooling power is actually the result of a cooperative interaction between the different parts in a PTC. However, most of the previous investigations studying the refrigeration mechanism are carried out on a single component (either the regenerator or the pulse tube) under different working conditions, and these are unable to explain how the net cooling effect generated in the cold end heat exchanger is a result of the combined work of the gas in the regenerator and pulse tube. Additionally, although it is an essential parameter of the oscillating flow, no research has yet investigated how the cold end phase angle affects the thermodynamic cycles of the gas in the regenerator at a micro scale, connecting that behavior to the macro-scale cooling performance. In this study, the refrigeration mechanism of the PTC is investigated by analyzing the thermodynamic behavior of the gas parcels in its key locations. The heat transfer characteristics of the gas parcels in the pulse tube, regenerator and cold end heat exchanger are compared in order to explain the net cooling effect that is produced. Furthermore, the effects of different cold end phase angles on the gas parcels’ periodic heat transfer characteristics are studied at a micro-scale to reveal its influence on the cooling performance.
@ ! !! ! /qf u þ r /qf u u ¼ /rp þ r /s u þ SxðyÞ @t @ ! ð/qf Ef þ ð1 /Þqs Es Þ þ r u ðqf Ef þ pÞ @t h i ! ¼ r ð/kf þ ð1 /Þks ÞrT þ ð/s u Þ Ef ¼ Hf þ SxðyÞ ¼
u2 p ; 2 qf
Es ¼ C v s T
2.1. Mathematic model and boundary conditions The structure of the two-dimentional, axis-symmetric single stage PTC is shown in Fig. 1. The details are given in Table 1. The PTC model contains two kinds of flow areas, the porous media area as the aftercooler a, regenerator b, cold end heat exchanger (CHX) c and hot end heat exchanger (HHX) e; and the non-porous media area as the pulse tube d. For all the flow areas, the mass, momentum and energy equations can be described as follows [13]:
@ ! ð/qf Þ þ r /qf u ¼ 0 @t
ð1Þ
ð3Þ ð4Þ
1 !! f q juju 2Dh r f
ð5Þ
The explanations of each item in the above equations are given in reference [12,13]. For the pulse tube d, the porosityu = 1, and the momentum source term Sx(y) = 0; for the areas a, b, c, e, their porosity is given in Table 1, the momentum source term is calculated by Eq. (5), in which fr is the flow resistance coefficient of the porous media. In the regenerator and the hot end heat exchanger, which are packed with stacked screens, the term fr is treated as isotropic and it can be calculated by the following empirical formula [14]:
f r ¼ 129=Re þ 2:91Re0:103
ð6Þ
For the areas a, c which are slit type heat exchangers, the term fr can be calculated as [15]:
f r ¼ 0:11ð/=dh þ 68=ReÞ
2. Numerical model of the single stage PTCs
ð2Þ
0:25
ð7Þ
In order to study the influence of the cold end phase angle hC, its value is determined for both the phase shifting type and the basic type of PTCs. The cold end phase angle is defined as the phase angle between the mass flow oscillation and the pressure oscillation at the cold end. It is positive when the mass flow oscillation leads the pressure oscillation. For the phase shifting PTCs, the mass flow _ in and the pressure wave pout are respectively used as the rate m inlet and outlet boundary conditions:
m_in ¼ ma sinð2pft þ hh p=180Þ
ð8Þ
pout ¼ pa sinð2pftÞ
ð9Þ
Fig. 1. Sketch of the pulse tube cryocooler and the typical gas parcels at different positions.
Table 1 Details for different parts of the pulse tube cryocooler. Items
Porosity
Material
Boundary conditions
Aftercooler a Regenerator b Cold end heat exchanger (CHX) c Pulse tube d Hot end heat exchanger (HHX) e
0.2 0.72 0.24 1 0.78
Slit type, copper SS304 mesh screen Slit type, copper Stainless steel Copper mesh screen
Isothermal, 280 K Adiabatic 80 K (160 K for basic PTC) Adiabatic Isothermal, 280 K
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where ma (4 g/s) and pa (1.6 bar) are the amplitudes of the oscillating mass flow rate and the pressure wave (with an average pressure of 2 MPa), f (40 Hz) is the operating frequency, and hH is the phase angle between m_in and pout at the hot end of the regenerator. For the basic PTC where the hot end heat exchanger e is closed, only the mass flow rate is used as the inlet boundary condition. Various values for the cold end phase angle hC can be obtained by using different values of hH. Here the input values of hH used for phase shifting type PTCs are 30°, 55°, 72° and 81°, the corresponding values of hC obtained based on hH are 30°, 0°, 30° and 60°. For the basic PTC the value of hC directly calculated from the model is 83°. In all PTC models, the temperature of the aftercooler and the HHX are fixed at 280 K, while the temperature for the CHX is fixed at 80 K for the phase shifting PTCs and 160 K for the basic PTC. The heat transfer between the wall and the gas inside the pulse tube is considered and the wall thickness is included in the model. The walls of the heat exchangers are set as isothermal while that of other components is set as adiabatic. For the heat exchangers, the heat capacity and thermal conductivity are set to the respective constant values corresponding to the local temperatures; for the regenerator they are treated as piecewise linear functions of temperature. The ideal gas version of helium-4 is used as the working fluid. Because the pressure change is small, the viscosity, heat capacity and thermal conductivity of the helium-4 are also treated as piecewise linear functions of temperature. In all the models, the charge pressure is 2.0 MPa, the operating frequency is 40 Hz, and each cycle is divided to 100 time steps in the computation. Table 2 shows a comparison of this work to the previous study of the same PTC (with the same operating condition and geometric parameters) [16]. The internal fluid properties of the PTC obtained here agree with those obtained by the common cryocooler design software SAGE. Also, the cooling power, in this case is close to that obtained in a previous experiment (8.8 W at 80 K) [17]. 2.2. Data processing method for visualizing the thermodynamic cycle of the gas parcel CFD simulations based on the Euler algorithm cannot provide the thermodynamic cycles of the gases directly. However, due to the instantaneous results available with CFD, the thermodynamic cycles of individual gas parcels can be extracted according to the Lagrange perspective by tracking their motion at each time step. For example, at moment n, the gas parcel’s location is Xn, and its velocity is Un. After one time step Dt, the gas parcel’s location Xn+1 at the next moment n + 1 is calculated as follows:
X nþ1 ¼ X n þ
ðU nþ1 þ U n Þ Dt 2
period, for example from moment i to j, can be calculated as follows:
Z
j
W ij ¼ i
Z Q ij ¼
pdv
n¼j
1X ðT n þ T nþ1 Þðsnþ1 sn Þ 2 n¼i
ð11Þ
n¼j
j
Tds i
1X ðp þ pnþ1 Þðv nþ1 v n Þ 2 n¼i n
ð12Þ
The typical gas parcel in each of the key sections of PTC are analyzed, and their equilibrium positions are shown in Fig. 1. In order to avoid calculating results at the junction between the exchangers and its nearby parts, the equilibrium position of gas parcels 1, 5, 7, 8, 9, 10, 12, 13 and 15 is about 1 mm away from the heat exchangers, which is similar to the treatment in a previous study [12]. Their movement in a cycle can be divided into two processes: the BCD process, in which the gas parcel moves at the right side of the equilibrium position; and the DAB process, in which it moves at the left side. In the pulse tube, the BCD process belongs to the hot side and the DAB process belongs to the cold side. The opposite situation exists in the regenerator due to the reverse temperature gradient compared to the pulse tube. In each cycle, the gas parcels move left and right around the equilibrium position and exchange energy with the surroundings, resulting in a heat and work transfer between the cold side and hot side.
3. Thermodynamic cycle of the gas parcels under hC = 0° In a PTC, where the displacement of the gas lags the mass flow rate by 90°, we can infer the associated pressure values for the typical positions A, B, C, D of a gas parcel oscillation for different values of hC, as shown in Fig. 2. A phase angle of hC = 0° (pressure and mass flow rate are in phase) is a common value in a PTC. When hC = 0°, as shown in Fig. 2, the section BCD represents a complete expansion process because the pressure drops throughout this section, while the DAB section is associated completely with a compression process because the pressure increases throughout this section.
ð10Þ
where an average velocity from two adjacent moments is used in order to track the gas parcels accurately. From Eq. (10), the gas parcel’s locations at each time step can be obtained, so its corresponding pressure p, specific volume v, temperature T and specific entropy s at each time step can be known to get the p-v and T-s diagrams in a cycle. In the p-v and T-s diagrams, the amount of heat and power exchange between the gas parcels during any time
Table 2 Results comparison between previous SAGE simulation and this work. Items
Previous simulation
This work
Cold end mass flow amplitude (g/s) Cold end pressure ratio Cold end phase angle (°) Cold end PV power (W) Cold end enthalpy flow (W) Cooling power (W)
3.53 1.190 29.1 26.7 18.29 9.56
3.32 1.181 30 23.9 16.2 9.12
Fig. 2. Relationship between pressure wave and mass flow rate under different hC.
X. Zhi et al. / International Journal of Heat and Mass Transfer 103 (2016) 382–389
385
Fig. 3. The p-v (a) and T-s (b) diagram of gas parcel 5.
Fig. 4. The p-v (a) and T-s (b) diagram of gas parcel 6.
Table 3 Work and heat exchange in the four processes of gas parcel 5,6 and 7 (hC = 0°, unit: kJ/kg).
heat isothermally by expanding. In process BC, GP 5 absorbs a net amount of heat and produces a net amount work (as shown in Table 3). In process CD, GP 5 continues to expand and absorbs heat in the CHX. After it leaves the CHX, it moves toward the hot side in the regenerator where the local temperature is increasing, so gas parcel 5 absorbs a large amount of heat from the regenerator and its temperature increases, even though it is expanding. Therefore, in process BCD, GP 5 expands and absorbs a net amount of heat at the cold side. In process DA, as GP 5 moves towards the hot side of the regenerator, the compression period starts. Because its temperature at the early part of the compression period is lower than the temperature of the regenerator, it continues to absorb heat, and its specific entropy increases. Later, it releases heat because the compression process elevates the gas temperature above the local matrix temperature in the regenerator. In process DA, GP 5 consumes a net amount of work and absorbs a net amount of heat. In process AB, GP 5 moves back to the equilibrium position, and continues to be compressed. Since the local temperature of the regenerator is decreasing, gas parcel 5 releases a large amount of heat to the surroundings. The heat it releases is larger than the work it consumes (as shown in Table 3), so its temperature decreases. In process DAB, GP 5 consumes a net amount of work and releases a net amount of heat at the hot side. Therefore, as stated above, GP 5 expands and absorbs heat through the cold portion of its cycle in the CHX, and is compressed, releasing heat during the hot portion of its cycle. For the complete cycle, a net amount of work is consumed as the cost of transferring heat from the cold side to hot side.
Process
BC CD DA AB BCD DAB BCDAB
GP 5
GP 6
GP 7
W
Q
W
Q
W
Q
9.31 23.35 8.29 27.82 32.66 36.11 3.45
0.62 31.47 29.52 65.07 32.09 35.55 3.46
15.87 11.84 16.00 11.71 27.71 27.71 0.00
15.87 11.84 16.00 11.71 27.71 27.71 0.00
11.05 7.15 7.44 11.35 18.20 18.79 0.58
3.23 1.48 6.04 11.35 4.72 5.30 0.58
3.1. The gas parcels at the CHX Figs. 3–5 show the T-s and p-v diagrams of gas parcel 5, 6, 7, and the corresponding amounts of heat and work exchanged in processes BC, CD, DA and AB in a cycle are shown in Table 3. Gas parcel 5 (GP 5) moves between the regenerator and the CHX. In process BC, as shown in Fig. 3, GP 5 moves from the equilibrium position to the CHX, and its pressure drops. The initial rate of temperature drop during the early expanding period is lower than the decrease of temperature in the regenerator matrix in the direction of the CHX (i.e. the regenerator temperature gradient), therefore, before the gas parcel enters the CHX, it releases some heat, and its specific entropy and volume decrease. Upon entering the CHX, its temperature remains at 80 K and it absorbs
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Fig. 6. T-s diagram of gas parcels 1 to 5 in the regenerator.
Fig. 5. The p-v (a) and T-s (b) diagram of gas parcel 7.
Gas parcel 6 (GP 6) moves completely within the CHX, experiences isothermal compression and expansion, and produces no net energy transfer within a cycle. In process BCD it expands and absorbs heat, and in process DAB it is compressed and releases the same amount heat there. Therefore, gas parcels in the CHX transfer heat from the pulse tube side to the regenerator side equally, as shown in Fig. 4 and Table 3. Gas parcel 7 (GP 7) moves between the CHX and pulse tube. In process BCD, GP 7 moves within the pulse tube, and as shown in Fig. 5, it expands throughout this period and absorbs heat. In process DAB, GP 7 moves into the CHX, and experiences a compression process. At the beginning, its temperature is lower than the CHX because it is expanding, so when it enters the CHX, it first absorbs heat from the CHX. But later, since its temperature remains 80 K as it is compressed, it releases heat back to the CHX. In process DAB,
the gas parcel releases a net amount of heat to the CHX and consumes a net amount of work, as shown in Table 3. Thus, GP 7 does not produce cooling at the CHX even though its temperature decreases by expanding in the pulse tube. Rather, it transfers heat loss from the pulse tube to the CHX, which is in keeping with the previous study [12]. The total heat released to the CHX (QDAB) is the sum of the heat absorbed in process BCD (QBCD) and the net work consumed in the cycle (WBCDAB). In an ideal pulse tube where the gas expands adiabatically, there is no heat absorbed in process BCD. However, there is always a net amount of work consumed due to the transition between the adiabatic and isothermal boundary. Therefore, there is always some heat loss caused by the gas as it moves between the pulse tube and the CHX, which isn’t revealed in the existing studies. It should be noted here that the GP 7 only moves between the CHX and the pulse tube, so it cannot produce cooling at the CHX as it is compressed in the CHX. In some cases where the CHX is short enough and the mass flow rate is large, there are gas parcels moving between the pulse tube and the regenerator by crossing the CHX. These gases move across three components in a cycle and do not individually strictly belong to any single component. They act as a bridge between the gases at the regenerator side and the pulse tube side. During a cycle, these gas parcels mainly expand in the pulse tube, and then absorb a large amount of heat in the CHX after expanding; later they mainly compress in the regenerator and reject a large amount of heat there. As a result, these gas parcels still produce net cooling at the CHX. Therefore, it is not true that all the gases expanding in the pulse tube will produce a heat loss rather than a cooling effect at the CHX. However, it is true of the gas parcels discussed above, that is those gas parcels on the pulse tube side that are moving between the CHX and the pulse tube alone, and those moving completely within the pulse tube.
Table 4 Heat exchange in the four processes of gas parcels at cold end (hC = 0°). Q (kJ/kg)
BC CD DA AB BCD DAB BCDAB
Regenerator side
Pulse tube side
GP 5
GP 8
GP 9
GP 10
GP 7
GP 11
GP 12
GP 13
0.62 31.47 29.52 65.07 32.09 35.55 3.46
0.68 31.12 29.79 65.13 31.80 35.35 3.55
0.70 31.07 29.77 65.24 31.77 35.47 3.70
0.74 31.35 29.39 65.21 32.09 35.82 3.73
3.62 1.10 6.05 11.35 4.72 5.30 0.58
3.33 3.23 4.10 11.20 6.61 7.10 0.53
3.12 4.21 3.33 11.20 7.33 7.87 0.54
6.07 10.23 7.20 9.70 16.30 16.90 0.60
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X. Zhi et al. / International Journal of Heat and Mass Transfer 103 (2016) 382–389 Table 5 Heat exchange amount of gas parcels along the regenerator and pulse tube (hC = 0°). Q (kJ/kg)
GP5
GP4
GP3
GP2
GP1
GP7
GP14
GP15
BC CD DA AB DAB BCD BCDAB
0.62 31.47 29.52 65.07 35.55 32.09 3.46
30.45 71.68 23.25 69.54 46.29 41.23 5.06
60.64 106.23 54.66 107.20 52.54 45.59 6.95
121.31 169.20 61.61 118.22 56.61 47.90 8.71
127.64 194.00 61.12 137.33 76.21 66.36 9.85
3.62 1.10 6.05 11.35 5.30 4.72 0.58
5.09 4.99 5.33 5.21 10.54 10.08 0.46
0.96 13.91 3.41 10.41 13.82 12.95 0.87
Table 6 Heat exchange in the four processes of gas parcel 4 under different hC. Q (kJ/kg)
30°
0°
30°
60°
83°
BC CD DA AB BCD DAB BCDAB
35.32 70.14 27.59 68.40 34.82 40.81 5.99
30.45 71.68 23.25 69.54 41.23 46.29 5.06
38.13 63.08 35.85 63.68 24.95 27.83 2.88
44.89 54.53 49.78 61.15 9.64 11.37 1.73
67.05 64.61 82.47 80.37 2.43 2.09 0.34
3.2. Gas parcels inside the regenerator and pulse tube Fig. 6 shows the T-s diagrams of gas parcels 1–5 in the regenerator. Table 5 gives the amount of their heat exchange in the four processes. All the gas parcels absorb heat at the cold side and release heat to the hot side, pumping heat from the CHX to the aftercooler in a relay manner. As shown in Table 5, the gas parcels consume a net amount of work and release a net amount of heat in a cycle, which means that the regenerator not only performs as a heat accumulator but also a heat-work convertor. In contrast, as shown in Table 5, the gas parcels 7, 14 and 15 along the pulse tube transfer heat from the hot side to the cold side in a relay manner. Therefore, as analyzed above, in the PTC with hC = 0° the heat transfer direction by the oscillating gas parcels is from the pulse tube side to the regenerator side. It should be noted that a larger amount of heat is rejected at the aftercooler (as shown in Table 5) since in a real regenerator there is entropy flow inside caused by heat pumping, in order to balance the entropy flow from the low temperature side, more heat must be rejected at the high temperature side. 4. The effects of hC on the thermodynamic cycle of the gas parcels Fig. 7. The p-v (a) and T-s (b) diagrams of gas parcel 5 (hC = 83°).
As shown in Table 3, GP 5 on the regenerator side absorbs heat while in the CHX, and GP 7 on the pulse tube side releases heat to the CHX. However, the heat absorbed (QBCD) by GP 5 is much larger than the heat released (QDAB) by GP 7. Table 4 further shows the amount of heat they exchange during a cycle. As can be seen in Table 4, the heat absorbed (QBCD) by the gas parcels on the regenerator side is much larger than the heat released (QDAB) by the gas parcels on the pulse tube side, and there is net cooling power at the CHX. This is because the gas parcels in the regenerator exchange heat almost isothermally while the gas parcels in the pulse tube traverse their cycle near adiabatically. In the pulse tube the boundary effect is more obvious than that in the porous regenerator. The temperature is more inhomogeneous near the wall [10], so the gas parcel (GP 13) near the pulse tube wall loses more heat to the cold side in a cycle, as shown in Table 4.
As is commonly known, the cold end phase angle hC plays an essential role in the cooling performance of PTCs for it macroscopically affects the PV power at the cold end. It is necessary to further investigate the effects of hC on the gas parcels’ thermodynamic behavior and reveal its working mechanism macroscopically. The periodic amount of heat exchanged by gas parcel 4 (GP 4) in the regenerator under various typical cold end phase angles is shown in Table 6. Note that when hC decreases from 0° to 30°, the heat absorbed by GP 4 in process BCD decreases, as does the heat released in process DAB. This is because, as shown in Fig. 2, when hC = 0° the process BCD is completely an expansion process and has the largest potential to absorb heat, while the process DAB is completely a compression process and has the largest potential to release heat. When hC < 0°, the process CD will contain part of the early compression period, while the process AB will contain part of the early expansion period, so the heat absorbed in process BCD and the heat released in process DAB will both decrease. In the range of 30° < hC < 0°, the process BCD still
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Table 7 Heat exchange in the four processes of gas parcel 5 and 7 under different hC. Q (kJ/kg)
30° GP 5
0° GP 7
BC 4.50 3.52 CD 26.2 0.97 DA 28.26 3.01 AB 62.95 8.03 BCD 30.66 4.50 DAB 34.70 5.02 DQc 25.64 DQc = QBCD of gas parcel 5 + QDAB of gas parcel
30°
60°
83°
GP 5
GP 7
GP 5
GP 7
GP 5
GP 7
GP 5
GP 7
0.62 31.47 29.52 65.07 32.09 35.55 26.79 7
3.62 1.10 6.05 11.35 4.72 5.30
15.16 30.38 37.00 54.31 15.22 12.31 10.7
2.94 1.1 6.73 11.25 4.04 4.52
26.61 28.45 40.36 43.57 1.84 3.21 0.75
0.17 1.00 9.68 10.78 0.83 1.09
53.50 49.92 65.11 61.89 3.58 3.23 0.37
2.90 1.17 15.09 11.14 4.07 3.95
Table 8 Work and heat exchange in the four processes of gas parcels at cold end (hC = 83°, unit: kJ/kg). Process
BC CD DA AB BCD DAB BCDAB
Fig. 8. The p-v (a) and T-s (b) diagrams of gas parcel 7 (hC = 83°).
contains most of the expansion period and the process DAB contains most of the compression period. Therefore, GP 4 still absorbs heat in process BCD and releases heat in process DAB, and the heat transfer direction remains the same. Note that compare to GP4 of 0°, although GP 4 of 30° releases more net heat (QBCDAB) in a cycle, it pumps less heat from the cold side to the hot side as it absorbs less heat in process BCD. As shown in Table 6, when hC increases from 0° to 83° (the basic PTC), the net heat absorbed in process BCD and the net heat released in process DAB by GP 4 both decrease as hC increases. In fact, when hC = 83°, a net amount of heat is released in process BCD and absorbed in process DAB, This is because when hC increases from 0°, process BC will contain more and more of the
GP 5
GP 7
W
Q
W
Q
39.28 36.14 46.12 43.33 3.14 2.79 0.35
53.50 49.92 65.11 61.89 3.58 3.23 0.35
20.47 16.49 21.30 17.44 3.98 3.86 0.12
2.90 1.17 15.09 11.14 4.07 3.95 0.12
late compression period while process DA will contain more and more of the late expansion period, as shown in Fig. 2. Thus, the net heat absorbed in process BCD and the net heat released in process DAB will both decrease, and when hC is large enough, the direction of heat transfer for these two processes may reverse. The periodic amount of heat exchange for gas parcels 5 and 7 at the two sides of the CHX under different values of hC is given in Table 7. Note that the effect of different values of hC on the periodic heat transfer for GP 5 and 7 is similar with that of GP 4. As stated above, the cold end phase angle affects the cooling performance by determining the direction and the amount of heat transfer for the gas parcels in the PTC. For the phase shifting PTCs (30° < hC < 60°), the heat is transferred from the pulse tube side to the regenerator side and the gas parcels in the regenerator can produce cooling by pumping heat at the CHX. While the gas parcels on the pulse tube side (moving between the CHX and the pulse tube or moving completely inside the pulse tube) bring a heat loss to the CHX, even though they expand and cool down when moving in the pulse tube. Nevertheless there is net cooling effect DQc in the CHX, as shown in Table 7. In fact all the regenerative cryocoolers with phase shifting have the same feature that the gases absorb heat at the cold end and ‘‘transfer” the heat one by one through the regenerator to the aftercooler and release a larger amount of compression heat there. That is how the heat at the CHX is pumped away and rejected to the ambient. For the basic PTC (hC = 83°), the heat transfer direction changes to be from the regenerator side to the pulse tube side, the gas parcels in the pulse tube produce cooling by heat pumping, and the gas parcels moving between the regenerator and the CHX brings a heat loss to the CHX. Figs. 7 and 8 show the T-s and p-v diagrams of GP 5 and 7 in the basic PTC, and the amount of their heat and work exchange in a cycle is given in Table 8. As can be observed, the heat absorbed (QDAB) by GP 7 in the CHX is larger than the heat released (QBCD) by GP 5 there, so there is net cooling effect. As shown in Table 7, DQc (per unit mass) is maximum when hC = 0° and the gas parcel in the regenerator has the largest heat pumping ability, which is consistent with the phase shifting theory [3]. Also, DQc is minimum in the basic PTC, that is when hC = 83°. In
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this case, only a small part of the gases in the pulse tube, those in the near wall area can pump heat effectively like all of the gases in the regenerator. This is the fundamental reason why the cooling performance of the phase shifting PTC is much better than that of the basic PTC. It should be noted that for a PTC the refrigeration efficiency is not only related to the cooling power in the CHX, but also the PV power loss in the regenerator. The previous study shows that when hC is around 30°, and the mass flow rate and the pressure are in phase at the middle of the regenerator, then the amplitude of the mass flow in the regenerator is minimized and the pressure drop is small, which guarantees a high efficiency in the PTC [17]. 5. Conclusions The refrigeration mechanism of the pulse tube cryocooler (PTC) under different cold end phase angles is studied by investigating the thermodynamic cycles of the gas parcels in its key components. The mechanism by which the cooling power is generated at the cold end is revealed by comparing the heat transfer characteristics of the gas parcels from both sides of the cold end heat exchanger (CHX). The results show that the cold end phase angle determines whether the regenerator or the pulse tube produces refrigeration via heat pumping since the phase angle affects the direction and magnitude of the heat transfer of the gas during a cycle. In the basic PTC, where hC is close to 90°, the heat is transferred from the regenerator side to the pulse tube side, and the gas in the pulse tube produces refrigeration by pumping heat from the CHX. In the phase shifting PTC, where hC is normally between 30° and 60°, the heat is transferred from the pulse tube side to regenerator side, the gas moving inside the regenerator helps to pump heat from the CHX to the aftercooler in a relay manner. In contrast, the gas completely moving in the pulse tube or moving between the CHX and the pulse tube transfers heat to the cold end as a heat loss, demonstrating that not all the gases expanding in the pulse tube can produce cooling power at the CHX. Since the heat pumping mechanism of the gas in the pulse tube is limited, enhancing the heat pumping effect of the gas in the regenerator by phase shifting improves the cooling performance, and represents the fundamen-
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tal reason why the cooling performance of the phase shifting PTC is much better than that of the basic PTC. Acknowledgements The project is supported by National Funds for Distinguished Young Scientists of China (Contract No. 50825601) and the Cheung Kong Scholars Program of China. References [1] R. Radebaugh, Cryocoolers: the state of the art and recent developments, J. Phys. Condens. Matter 21 (2009) 164219. [2] W. Gifford, R. Longsworth, Surface heat pumping, Adv. Cryo. Eng. 11 (1966) 171–181. [3] R. Radebaugh, J. Zimmerman, D.R. Smith, et al., A comparison of three types pulse tube refrigerators: new methods for reaching 60 K, Advances in Cryogenic Engineering, vol. 31, Plenum Press, New York, 1986, pp. 779–789. [4] W.E. Gifford, R.C. Longsworth, Pulse tube refrigeration progress, Adv. Cryog. Eng. 10 (1965) 69–79. [5] J. Liang, A. Ravex, P. Rolland, Study on pulse tube refrigeration part 1: thermodynamic non-symmetry effect, Cryogenics 36 (2) (1996) 87–93. [6] J. Liang, Thermodynamic cycles in oscillating flow regenerators, J. Appl. Phys. 82 (9) (1997) 4159–4163. [7] E.C. Luo, W. Dai, Z. Wu, et al., Meso-scope thermodynamic theory for cyclic flow engines, Cryog. Eng. 1 (2004) 1–11 [in Chinese]. [8] J.H. Xiao, The thermoacoustic effect and thermoacoustic theory for the regenerative cryocooler (heat engine), Chinese Academy of Sciences, Beijing, 1991, pp. 75–85 [in Chinese]. [9] G.W. Swift, Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators, Acoustical Society of America, Sewickley, PA, 2002. [10] W. Liang, A.T.A.M. de Waele, A new type of streaming in pulse tubes, Cryogenics 47 (2007) 468–473. [11] L. Chen, Y. Zhang, E. Luo, et al., CFD analysis of thermodynamic cycles in a pulse tube refrigerator, Cryogenics 50 (2010) 743–749. [12] L.M. Qiu, X.Q. Zhi, Z.H. Gan, et al., Function of gas parcels in the pulse tube, Int. J. Refrig. 38 (2014) 358–366. [13] Fluent INC. Fluent 6.3 User’s Guide, 2003. [14] I.F. Macdonald, M.S. El-Sayed, K. Mow, et al., Flow through porous media – the Ergun equation revisited, Ind. Eng. Chem. Fundam. 18 (3) (1979) 199–208. [15] E. Fried, I.E. Idelchik, Flow resistance: a design guide for engineers, Hemisphere (1989). [16] Q. Cao, L.M. Qiu, Z.H. Gan, et al., Energy analysis of a single stage stirling pulse tube cryocooler, J. Eng. Thermophys. 32 (6) (2011) 905–906 [in Chinese]. [17] R. Radebaugh, Thermodynamics of Regenerative Refrigerators. Generation of Low Temperature and It’s Applications, 2003, pp. 1–20.