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Numerical simulation of a step-piston type series two-stage pulse tube refrigerator
Cryogenics 47 (2007) 483–489 www.elsevier.com/locate/cryogenics
Numerical simulation of a step-piston type series two-stage pulse tube refrigerator Shaowei Zhu *, Masafumi Nogawa, Tatsuo Inoue Corporate R&D Sector, Aisin Seiki Co. Ltd., 2-1, Asahi-machi, Kariya, Aichi 448-8650, Japan Received 29 May 2006; received in revised form 26 April 2007; accepted 27 April 2007
Abstract A two-stage pulse tube refrigerator has a great advantage in that there are no moving parts at low temperatures. The problem is low theoretical efficiency. In an ordinary two-stage pulse tube refrigerator, the expansion work of the first stage pulse tube is rather large, but is changed to heat. The theoretical efficiency is lower than that of a Stirling refrigerator. A series two-stage pulse tube refrigerator was introduced for solving this problem. The hot end of the regenerator of the second stage is connected to the hot end of the first stage pulse tube. The expansion work in the first stage pulse tube is part of the input work of the second stage, therefore the efficiency is increased. In a simulation result for a step-piston type two-stage series pulse tube refrigerator, the efficiency is increased by 13.8%. 2007 Published by Elsevier Ltd. Keywords: Pulse tube refrigerator; Refrigerator; Numerical simulation
1. Introduction A two-stage pulse tube refrigerator has a great advantage in that there are no moving parts at low temperatures. The reliability is high and the vibration is low. The problem is low theoretical efficiency. In an ordinary pulse tube refrigerator, such as orifice [1,2], double inlet [3] and inertance tube pulse tube refrigerators [4,5], the expansion work in the pulse tube is rather large compared to the input work, but cannot be recovered. The expansion work is changed to heat by an irreversible process through the orifice or the inertance tube. In a two-stage pulse tube refrigerator [6], the expansion work of the second stage pulse tube is rather small compare to the input work because the refrigeration temperature is very low, such as the range from 4 K to 20 K, and the expansion work of the first stage pulse tube is rather large compared to the input work because the refrigeration temperature is higher such as the range of 60–80 K. Compared to a Stirling refrigerator,
*
Corresponding author. Tel.: +81 566 24 9360; fax: +81 566 24 9391. E-mail address: [email protected] (S. Zhu).
0011-2275/$ - see front matter 2007 Published by Elsevier Ltd. doi:10.1016/j.cryogenics.2007.04.008
this is the main disadvantage of the pulse tube refrigerator. In Ref. [7], a series connected two-stage pulse tube refrigerator which is a modification of series pulse tube refrigerator [8] was introduced. There is a heat bridge connecting the cold head of the first stage and the heat exchanger around the middle of the second refrigerator, and the hot end of the first stage pulse tube is connected to the hot end of the second stage regenerator. Then the expansion work of the first stage is part of the work input of the second stage. The efficiency is increased compared to that in Ref. [6]. In this paper, a step-piston type series two-stage pulse tube refrigerator is investigated based on a numerical simulation. 2. Series two-stage pulse tube refrigerators Fig. 1 shows the schematic of a step-piston type, a bypass type, and a GM type series two-stage pulse tube refrigerator. In Fig. 1a, parts 1–5 belong to the first stage, and parts 6–11 belong to the second stage. The hot end of the first pulse tube PT1 is connected to the hot end of the
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S. Zhu et al. / Cryogenics 47 (2007) 483–489
Nomenclature A As AL CV CP CS CO E h H m_ m_ O P PB PPT2 R s t T
cross sectional gas flow area cross sectional matrix area heat transfer area of matrix per meter constant volume specific heat of gas constant pressure specific heat of gas specific heat of matrix orifice opening factor exergy flow heat transfer coefficient enthalpy flow mass flow rate mass flow rate through orifice pressure buffer pressure pressure in pulse tube gas constant entropy time temperature of gas
regenerator R21. The heat bridge connects the first cold head and the second middle cooler. Buffer and orifice are for the second stage. The orifice also can be an inertance tube or capillary. The volumes of step compressors 1–3 change periodically to let the gas pressure oscillate. The working gas in the step compressor 1 flows into the cold end of the pulse tube PT1 through the first after cooler, the regenerator R11, and the first cold head. At the cold end of the pulse tube PT1, the gas expands to low pressure and low temperature, and then returns to the step compressor 1. The cooling power is given through the first cold head when the expanded gas returns back to the step compressor 1. A part of the gas at the hot end of the pulse tube PT1 flows to the cold end of the pulse tube PT2 through the first pulse tube cooler, regenerator R21, second middle cooler, regenerator R22, and second cold head. At the cold end of the pulse tube PT2, the gas expands to low pressure and low temperature, and return. The cooling power is given through the second cold head when the expanded low temperature gas passes it. During the process mentioned above, the gas in the step compressor 2 also flows in and out of the hot end of the pulse tube PT1 for controlling the phase angle difference between the pressure wave and mass flow rate at the cold end of the pulse tube PT1 to improve the performance of the regenerator R11. The gas from the step compressor 3 also flows in and out of the hot end of the pulse tube PT2 for controlling the phase angle difference between the pressure wave and mass flow rate at the cold end of the pulse tube PT2 to increase the performance of the regenerator R22 and R21. The cooling powers of the first stage and the second stage are adjusted by the orifice opening factor and volume ratio of each step compressors.
TS TH TC1 TC2 V Vid Vi Wi W x
temperature of matrix room temperature first stage refrigeration temperature second stage refrigeration temperature volume dead volume of step compressor i, i = 1, 2, 3 swept volume of step compressor i, i = 1, 2, 3 compression work of step compressor i, i = 1, 2, 3 compression work coordinate
Greek letters q density of gas qS density of matrix s period x angle frequency g efficiency
In Fig. 1a, the expansion work of the first stage pulse tube PT1 and the input work of step compressor 2 become the work input of the second stage. The heat bridge cools the gas between the regenerators R21 and R22. It has a function to decrease the enthalpy loss of the regenerator R22 for increasing the efficiency of the second stage. The input work of the step compressor 3 is lost and the expansion work of PT2 is lost by the irreversible process of the orifice. In Fig. 1a, if 12 is an inertance tube and the expansion work of the second stage is enough to let the gas in inertance tube oscillate or the inertance tube and the buffer are at low temperatures, the step compressor 3 is not necessary. In double inlet pulse tube refrigerators, there are two kinds of structure, step-piston type and bypass type [3]. They have the same function of adjusting the phase angle of the pressure wave and the mass flow rate for obtaining a higher regenerator efficiency. The step-piston type is a little complex. The step compressors 2 and 3 can be replaced by first bypass and second bypass, respectively, as shown in Fig. 1b. There is a possibility that DC gas flow [3,9] is generated by the bypasses. As for the DC gas flow, the integration of one cycle of mass flow rate is not zero. It means that there is a steady gas flow through the bypass from the regenerator to the pulse tube or from the pulse tube to the regenerator. Excessive DC gas flow is considered as a source of trouble in double inlet type pulse tube refrigerators. We can use an asymmetric bypass such as subsonic nozzle or two needle valves located face to face or back to back on the bypass lines in order to cancel the DC gas flow. Fig. 1c is a schematic of GM type series pulse tube refrigerator. A GM compressor with valves replaces the
S. Zhu et al. / Cryogenics 47 (2007) 483–489
15 16
1
2
17
3
4
5
6
7 8
9
485
13
10 11 12
14
20 19
18 1
2
3
4
5
6 7
8 9
10
13
11 12
14
21 26
1
2
22 3
15 4
5
23 6 7
24 16 8
9
10
13
11 12
27 25
14
Fig. 1. (a) Step-piston type series two-stage pulse tube refrigerator. (b) Bypass type series two-stage pulse tube refrigerator. (c) GM type series two-stage pulse tube refrigerator. (1) First after cooler; (2) first regenerator R11; (3) first cold head; (4) first pulse tube PT1; (5) first pulse tube cooler; (6) second regenerator R21; (7) second middle cooler; (8) second regenerator R22; (9) second cold head; (10) second pulse tube PT2; (11) second stage pulse tube cooler; (12) inertance tube (or orifice); (13) buffer; (14) heat bridge; (15) step compressor 1; (16) step compressor 2; (17) step compressor 3; (18) compressor; (19) first bypass; (20) second bypass; (21) buffer; (22) on/off valve; (23) on/off valve; (24) buffer; (25) low pressure valve; (26) high pressure valve and (27) GM type compressor.
compressor in Fig. 1b. There are two buffers through on/off valves on the bypass line or the hot end of the first stage pulse tube for obtaining better phase controlling. In an ordinary two-stage pulse tube refrigerator [7], the expansion work of the first stage and second stage is changed to heat. The work flow through the bypasses is also changed to heat. In the series two-stage pulse tube refrigerator, the expansion work of the first stage and the work flow through the bypass or step-piston is the work source in the second stage. So the higher efficiency can be obtained compared to the ordinary two-stage pulse tube refrigerator. Based on the same idea, the multistage construction consisting of more than two-stages can be realized.
In the case of Fig. 1a, the numerical simulation was done by a nodal analysis method based on Ref. [5]. The main assumptions in the numerical simulation is that there is no pressure drop, that the system is one dimensional, and that the working gas is as an ideal gas. The governing equations are as follows: Energy equation for gas oðC V qT Þ o _ Þ þ hAL ðT T S Þ ¼ 0 þ ðC P mT ot ox
ð1Þ
ð2Þ
Continuity equation A
oq om_ þ ¼0 ot ox
Ideal gas state equation P ¼ qRT Mass flow rate through orifice ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C O ðP B P PT2 ÞP PT2 m_ O ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C O ðP PT2 P B ÞP B Enthalpy flow I 1 _ p T dt mC H¼ s
3. Governing equations
A
Energy equation for matrix oT S þ hAL ðT S T Þ ¼ 0 AS C S ot
Exergy flow I 1 _ dt E ¼ H TH ms s
ð3Þ
ð4Þ
ð5Þ
ð6Þ
ð7Þ
The cooling power of the first stage is obtained as follows: Q1 ¼ H PT1 H R11 H R21 þ H R22 ð8Þ The cooling power of the second stage is obtained as follows:
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Q2 ¼ H PT2 H R22
S. Zhu et al. / Cryogenics 47 (2007) 483–489
ð9Þ
2.3 PT2
The equivalent PV diagram in the pulse tube definition is the same with that in Ref. [5]. The expansion work in pulse tubes is the equivalent PV work. The volume of the step compressors 1–3 is
2.1
ð10Þ
Pressure, MPa
V i ¼ V id þ 0:5V i0 ð1 þ cosðxtÞÞ i ¼ 1; 2; 3
PT1
2.2
The compression work of the step compressors 1–3 is I 1 P dV i i ¼ 1; 2; 3 Wi ¼ ð11Þ s
2
1.9
The compression work is the total of the step compressors 1–3 Efficiency is T H T C1 T H T C2 g ¼ Q1 þ Q2 W T C1 T C2
1.8
ð12Þ
1.7 0
0.2
ð13Þ
0.4 0.6 Normalized length of pulse tube
0.8
1
2.3
0.015 1
Table 1 Calculation condition Room temperature First stage temperature Second stage temperature Average pressure Frequency First compressor swept volume Second compressor swept volume Third compressor swept volume Regenerator R11 Regenerator R21 Regenerator R22 First stage pulse tube Second stage pulse tube Orifice opening factor Buffer volume
300 K 80 K 20 K 2 MPa 50 Hz 27.5 cm3 20 cm3 2.5 cm3 B40 · 50 mm B30 · 50 mm B24 · 50 mm B14 · 100 mm B9 · 100 mm 16E10 250 cm3
2
0.01
2.2
3 0.005
2.1
0
2
-0.005
1.9
-0.01
1.8
-0.015 0
90
180 Phase angle, degree
1.7 360
270
300 Step compressor 2 input
Ex H
Exergy flow and enthalpy flow, W
The working medium is helium gas. The main parameters for calculation are in Table 1. The first stage temperature is 80 K, the second stage temperature is 20 K, and the room temperature is 300 K. With these parameters in Table 1, the first stage cooling power, the second stage cooling power, input power, and expansion work of the first stage pulse tube, are given as 45.7 W, 8.9 W, 491.4 W, 67.6 W, respectively. The first stage expansion work is 13.8% of the input work. In the ordinary two-stage pulse tube refrigerator, the expansion work is changed to heat. In this simulation condition, the 13.8% expansion work means that the efficiency is increased by 13.8% because it is used in the second stage. Fig. 2a shows the PV diagrams in the first stage pulse tube PT1 and the second stage pulse tube PT2, and there is a reasonable distance between the PV diagrams at the cold end and the hot end of the pulse tubes. So the shuttle loss in the pulse tubes should be a reasonably small value. Fig. 2b shows that the mass flow rate at the cold end of the first stage pulse tube and the second stage pulse tube,
Mass flow rate at cold end of pulse tube, kg/s
4. Numerical results
Pressure, MPa
W ¼ W1þW2þW3
200
Step compressor 3 input
100
0 0
100
200 300 400 X postion along refrigerator, mm
500
Fig. 2. (a) Equivalent PV diagrams. (b) Mass flow rate and pressure: (1) Mass flow rate at the cold end of the first stage pulse tube. (2) Mass flow rate at the cold end of the second stage pulse tube. (3) Pressure. (c) Exergy flow and enthalpy flow long refrigerator.
80
800
70
700
60
600
50
500
400
40 1st stage cooling power 30
300
2nd stage cooling power
Compression work, W
487
1st stage expansion work 20
200 Compression work
10
100
0 8
10
12 14 16 18 20 Normalized orifice opening factor
22
0 24
60
1.3
50
1.25
Efficiency Pressure ratio
30
1.15
20
1.1
10
1.05
0 8
10
12 14 16 18 20 Normalized orifice opening factor
22
Pressure, ratio
1.2
40 Efficiency, %
and the pressure wave. The direction of mass flow rate is from left to right in Fig. 1a. When the time is equal to zero in Eq. (9), it is zero degrees. The mass flow rate at the cold end of the first stage pulse tube and pressure wave is in phase. The phase of the pressure wave is ahead of mass flow rate at the cold end of the second stage pulse tube. Therefore the regenerators are under reasonable working conditions. Fig. 2c shows the exergy flow and enthalpy flow along the refrigerator. From 0–230 mm, it is first after cooler, first regenerator R21, first cold head, and the first pulse tube PT1. From 230–455 mm, it is the regenerator R21, second middle cooler, R22, second cold head, PT2, and second stage pulse tube cooler. The enthalpy is constant in regenerators and pulse tubes, is not constant in heat exchangers. At left end, 270 W input work of step compressor 1 is added. The exergy flow is gradually decreased due to the irreversible loss in the after cooler and the regenerator. The large decrease of exergy flow near the left end of PT1 is mainly from the first cold head output of cooling power. The remaining exergy flow through the pulse tube PT1 with 196.6 W input work of the step compressor 2 goes into the second stage. There is a little jump of exergy flow around 250 mm where it is second middle cooler due to the cooling by the first stage. In second middle cooler, the enthalpy flow is decreased due to the pre-cooling effect. The exergy flow decreases due to heat transfer and the second stage cooling power output. It should be pointed out that a part of the expansion work does not become the exergy flow for the second stage because there is an irreversible loss due to the temperature difference in the heat exchanger at the hot end of the pulse tube PT1. Fig. 2c shows that there is a jump of exergy flow at the right end. It is the 25.6 W input work of the step compressor 3 which is 5% of the total input work. It is compressor work loss. If the inertance tube is used to replace the orifice, the input work of the step compressor 3 is largely decreased. If the inertance tube and buffer are cooled by the first stage, the step compressor 3 is not necessary. The system becomes very simple.
Cooling power and expansion work, W
S. Zhu et al. / Cryogenics 47 (2007) 483–489
1 24
18 16
4.1. Effect of orifice In the series two-stage pulse tube refrigerator, the orifice is a common part. Orifice opening factor will influence on the cooling power of the first stage and second stage. Fig. 3 shows the calculation results. The parameters are the same with those in Table 1. The orifice opening factor is normalized by 1E10. Fig. 3a shows the cooling power, expansion work and compression work vs. the orifice opening factor. There is an optimum orifice opening factor for obtaining the higher input power, the higher expansion work in the first stage pulse tube, and the higher cooling power at the first stage and the second stage. Fig. 3b shows the efficiency and pressure ratio vs. the orifice opening factor. There is an optimum orifice opening
Regenerator enthalpy flow, W
14 12 10 8 6 R11 R21
4
R22 2 0 8
10
12 14 16 18 20 Normalized orifice opening factor
22
24
Fig. 3. (a) Cooling power, expansion work and compression work vs. orifice opening factor. (b) Efficiency and pressure ratio vs. orifice opening factor. (c) Regenerator loss vs. orifice opening factor.
S. Zhu et al. / Cryogenics 47 (2007) 483–489
4.3. Effect of swept volume of step compressor 3 The swept volume of the third compressor also influences the cooling performance at the first stage and second stage. 5 shows the calculation results. The parameters are the same with those in Table 1. The total swept volume is
90
900
80
800
70
700
60
600
50
500
40
400 1st stage cooling power
30
2nd stage cooling power
300
20
1st stage expansion work
200
Compression work, W
1000
Compression work 10
100
0
0 0
10 20 30 40 50 60 Swept volume ratio of step compressor 2,%
70
60
1.3
50
1.25
40
1.2
30
1.15 Efficiency Pressure ratio
20
1.1
10
1.05
0
Pressure ratio
The swept volume of the step compressor 2 is not only for the adjustment of the phase angle between mass flow rate and pressure wave for the first stage pulse tube, but also for the part of the work source of the second stage. It has a strong influence on the first stage and second stage. Fig. 4 shows results of the calculation. The parameters are same as those in Table 1. The total swept volume is kept at 50 cm3, the swept volume of step compressor 3 is not changed, and the swept volume of the step compressor 2 is changed. So the increase of the swept volume of step compressor 2 means the decrease of the swept volume of step compressor 1. The swept volume ratio of step compressor 2 is defined as the swept volume of step compressor 2 over the total volume of the step compressors 1–3. The small swept volume ratio means the small swept volume of step compressor 2 and small input work to the second stage. Fig. 4a shows the cooling power, expansion work and compression work vs. the swept volume ratio of step compressor 2. There is an optimum swept volume ratio of the step compressor 2 for obtaining the higher cooling power in the first stage. The input power of the compressor and the second stage cooling power increase with the increase of the swept volume ratio of step compressor 2. The expansion work of the first stage pulse tube decreases with the increase of the swept volume ratio of the step compressor 2, which means that the series two-stage pulse tube refrigerator effect is small when the first stage input power is too small. Fig. 4b shows the efficiency and pressure ratio vs. the swept volume ratio of step compressor 2. There is an optimum swept volume ratio of the step compressor 2 for obtaining higher efficiency. The pressure ratio increases with the increase of the swept volume ratio of step compressor 2. Fig. 4c shows the regenerator enthalpy flow vs. the swept volume ratio of step compressor 2. The regenerator loss of R11 decreases with the increase of the swept volume ratio of the step compressor 2 and the regenerator losses of R21 and R22 increases with the increase of the swept volume ratio of the step compressor 2.
Cooling power and expansion work, W
4.2. Effect of swept volume of step compressor 2
100
Efficiency, %
factor for obtaining the higher efficiency. The pressure ratio decreases with the increase of the orifice opening factor. Fig. 3c shows the regenerator enthalpy flow vs. the orifice opening factor. The regenerator enthalpy flow is the regenerator loss. It increases with the increase of the orifice opening factor.
1 0
10
20
30
40
50
60
70
Swept volume ratio of step compressor 2, % 35
30
Regenerator enthalpy flow, W
488
R11 R21
25
R22
20
15
10
5
0 0
10 20 30 40 50 60 Swept volume ratio of step compressor 2,%
70
Fig. 4. (a) Cooling power, expansion work and compression work vs. swept volume ratio of step compressor 2. (b) Efficiency and pressure ratio vs. swept volume ratio of step compressor 2. (c) Regenerator loss vs. swept volume ratio of step compressor 2.
800
70
700
60
600
50
500
40
400
1st stage cooling power 2nd stage cooling power
30
300
1st stage expansion work 20
200
compression work
10
100
0
0
-10
-100 0
Efficiency, %
Compression work, W
80
2
4 6 8 10 12 14 Swept volume ratio of step compressor 3, %
16
60
1.6
50
1.5
40
1.4
30
1.3
20
Pressure ratio
Cooling power of first and second stage, W
S. Zhu et al. / Cryogenics 47 (2007) 483–489
1.2
489
kept at 50 cm3, the swept volume of the step compressor 2 is not changed, and the swept volume of the step compressor 3 is changed. So the increase of the swept volume of the step compressor 3 means the decrease of the swept volume of the step compressor 1. The swept volume ratio of the step compressor 3 is defined as the swept volume of the step compressor 3 over the total volume of the step compressors 1–3. The small swept volume ratio means that there is a small swept volume for step compressor 3. Fig. 5a shows the cooling power, expansion work and compression work vs. the swept volume ratio of step compressor 3. Step compressor 3 has an optimum swept volume ratio for achieving the higher cooling power at the first stage and second stage, compression work, and the expansion work of the first pulse tube increase with the increase of the swept volume ratio of the step compressor 3. Fig. 5b shows the efficiency and pressure ratio vs. the swept volume ratio of step compressor 3. There is an optimum swept volume ratio of the step compressor 3 for achieving the higher efficiency. The pressure ratio increases with the increase of the swept volume ratio of step compressor 3. Fig. 5c shows the regenerator enthalpy flow vs. the swept volume ratio of step compressor 3. There is an optimum swept volume ratio of the step compressor 3 for achieving the lower regenerator loss of R22. The regenerator loss of R21 and R22 decreases with the increase of the swept volume ratio of step compressor 3.
Efficiency
5. Conclusion
Pressure ratio 10
1.1
0
1 0
2
4 6 8 10 12 14 Swept volume ratio of step compressor 3,%
16
18
Numerical simulation was done to analyze the behavior of the step-piston type series two-stage pulse tube refrigerator. In a numerical simulation condition, the efficiency increases by 13.8%. References
16
Regenerator enthalpy flow, W
14 12 10 8 R11
6
R21 4
R22
2 0 0
2
4 6 8 10 12 14 Swpet volume ratio of step compressor 3, %
16
Fig. 5. (a) Cooling power, expansion work and compression power vs. swept volume ratio of step compressor 3. (b) Efficiency and pressure ratio vs. swept volume ratio of step compressor 3. (c) Regenerator loss vs. swept volume ratio of step compressor 3.
[1] Mikulin EI, Tarasov AA, Shkrebyonock MP. Low temperature expansion pulse tubes. Adv Cryog Eng 1984;29:629. [2] Radubaugh R, Zimmerman J, Smith DR, Louie B. A comparison of three types of pulse tube refrigerators: new method for reaching 60 K. Adv Cryog Eng 1986;31:779. [3] Zhu SW. Theory of pulse tube refrigerator and its important improvement: double inlet pulse tube refrigerator. Ph.D. thesis. Xi’an Jiaotong University; 1990. [4] Tominaga A. Phase controls for pulse tube refrigerator of the third generation. Cryog eng 1992;27(2):147–51. [5] Zhu SW, Matsubara Y. Numerical method of inertance tube pulse tube refrigerator. Cryogenics 2004;44:649–60. [6] Matsubara Y, Gao JL. Novel configuration of three stage pulse tube refrigerator for temperatures below 4 K. Cryogenics 1994;34(4): 259–62. [7] Zhu SW, Nogawa M, Kawano S, Inoue T. Pulse tube refrigerator. Japan Patent, Publication number 2004-347175. [8] Zhu SW, Kawano S, Nogawa S, Inoue T. Pulse tube refrigerator. US Patent, US 6,389,819. [9] Duband L, Charles I, Ravex A, Miquet L, Jewell C. Experimental results on inertance and permanent flow in pulse tube cooler. Cryocooler 1999;10:281–90.
Numerical simulation of a step-piston type series two-stage pulse tube refrigerator Shaowei Zhu *, Masafumi Nogawa, Tatsuo Inoue Corporate R&D Sector, Aisin Seiki Co. Ltd., 2-1, Asahi-machi, Kariya, Aichi 448-8650, Japan Received 29 May 2006; received in revised form 26 April 2007; accepted 27 April 2007
Abstract A two-stage pulse tube refrigerator has a great advantage in that there are no moving parts at low temperatures. The problem is low theoretical efficiency. In an ordinary two-stage pulse tube refrigerator, the expansion work of the first stage pulse tube is rather large, but is changed to heat. The theoretical efficiency is lower than that of a Stirling refrigerator. A series two-stage pulse tube refrigerator was introduced for solving this problem. The hot end of the regenerator of the second stage is connected to the hot end of the first stage pulse tube. The expansion work in the first stage pulse tube is part of the input work of the second stage, therefore the efficiency is increased. In a simulation result for a step-piston type two-stage series pulse tube refrigerator, the efficiency is increased by 13.8%. 2007 Published by Elsevier Ltd. Keywords: Pulse tube refrigerator; Refrigerator; Numerical simulation
1. Introduction A two-stage pulse tube refrigerator has a great advantage in that there are no moving parts at low temperatures. The reliability is high and the vibration is low. The problem is low theoretical efficiency. In an ordinary pulse tube refrigerator, such as orifice [1,2], double inlet [3] and inertance tube pulse tube refrigerators [4,5], the expansion work in the pulse tube is rather large compared to the input work, but cannot be recovered. The expansion work is changed to heat by an irreversible process through the orifice or the inertance tube. In a two-stage pulse tube refrigerator [6], the expansion work of the second stage pulse tube is rather small compare to the input work because the refrigeration temperature is very low, such as the range from 4 K to 20 K, and the expansion work of the first stage pulse tube is rather large compared to the input work because the refrigeration temperature is higher such as the range of 60–80 K. Compared to a Stirling refrigerator,
*
Corresponding author. Tel.: +81 566 24 9360; fax: +81 566 24 9391. E-mail address: [email protected] (S. Zhu).
0011-2275/$ - see front matter 2007 Published by Elsevier Ltd. doi:10.1016/j.cryogenics.2007.04.008
this is the main disadvantage of the pulse tube refrigerator. In Ref. [7], a series connected two-stage pulse tube refrigerator which is a modification of series pulse tube refrigerator [8] was introduced. There is a heat bridge connecting the cold head of the first stage and the heat exchanger around the middle of the second refrigerator, and the hot end of the first stage pulse tube is connected to the hot end of the second stage regenerator. Then the expansion work of the first stage is part of the work input of the second stage. The efficiency is increased compared to that in Ref. [6]. In this paper, a step-piston type series two-stage pulse tube refrigerator is investigated based on a numerical simulation. 2. Series two-stage pulse tube refrigerators Fig. 1 shows the schematic of a step-piston type, a bypass type, and a GM type series two-stage pulse tube refrigerator. In Fig. 1a, parts 1–5 belong to the first stage, and parts 6–11 belong to the second stage. The hot end of the first pulse tube PT1 is connected to the hot end of the
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S. Zhu et al. / Cryogenics 47 (2007) 483–489
Nomenclature A As AL CV CP CS CO E h H m_ m_ O P PB PPT2 R s t T
cross sectional gas flow area cross sectional matrix area heat transfer area of matrix per meter constant volume specific heat of gas constant pressure specific heat of gas specific heat of matrix orifice opening factor exergy flow heat transfer coefficient enthalpy flow mass flow rate mass flow rate through orifice pressure buffer pressure pressure in pulse tube gas constant entropy time temperature of gas
regenerator R21. The heat bridge connects the first cold head and the second middle cooler. Buffer and orifice are for the second stage. The orifice also can be an inertance tube or capillary. The volumes of step compressors 1–3 change periodically to let the gas pressure oscillate. The working gas in the step compressor 1 flows into the cold end of the pulse tube PT1 through the first after cooler, the regenerator R11, and the first cold head. At the cold end of the pulse tube PT1, the gas expands to low pressure and low temperature, and then returns to the step compressor 1. The cooling power is given through the first cold head when the expanded gas returns back to the step compressor 1. A part of the gas at the hot end of the pulse tube PT1 flows to the cold end of the pulse tube PT2 through the first pulse tube cooler, regenerator R21, second middle cooler, regenerator R22, and second cold head. At the cold end of the pulse tube PT2, the gas expands to low pressure and low temperature, and return. The cooling power is given through the second cold head when the expanded low temperature gas passes it. During the process mentioned above, the gas in the step compressor 2 also flows in and out of the hot end of the pulse tube PT1 for controlling the phase angle difference between the pressure wave and mass flow rate at the cold end of the pulse tube PT1 to improve the performance of the regenerator R11. The gas from the step compressor 3 also flows in and out of the hot end of the pulse tube PT2 for controlling the phase angle difference between the pressure wave and mass flow rate at the cold end of the pulse tube PT2 to increase the performance of the regenerator R22 and R21. The cooling powers of the first stage and the second stage are adjusted by the orifice opening factor and volume ratio of each step compressors.
TS TH TC1 TC2 V Vid Vi Wi W x
temperature of matrix room temperature first stage refrigeration temperature second stage refrigeration temperature volume dead volume of step compressor i, i = 1, 2, 3 swept volume of step compressor i, i = 1, 2, 3 compression work of step compressor i, i = 1, 2, 3 compression work coordinate
Greek letters q density of gas qS density of matrix s period x angle frequency g efficiency
In Fig. 1a, the expansion work of the first stage pulse tube PT1 and the input work of step compressor 2 become the work input of the second stage. The heat bridge cools the gas between the regenerators R21 and R22. It has a function to decrease the enthalpy loss of the regenerator R22 for increasing the efficiency of the second stage. The input work of the step compressor 3 is lost and the expansion work of PT2 is lost by the irreversible process of the orifice. In Fig. 1a, if 12 is an inertance tube and the expansion work of the second stage is enough to let the gas in inertance tube oscillate or the inertance tube and the buffer are at low temperatures, the step compressor 3 is not necessary. In double inlet pulse tube refrigerators, there are two kinds of structure, step-piston type and bypass type [3]. They have the same function of adjusting the phase angle of the pressure wave and the mass flow rate for obtaining a higher regenerator efficiency. The step-piston type is a little complex. The step compressors 2 and 3 can be replaced by first bypass and second bypass, respectively, as shown in Fig. 1b. There is a possibility that DC gas flow [3,9] is generated by the bypasses. As for the DC gas flow, the integration of one cycle of mass flow rate is not zero. It means that there is a steady gas flow through the bypass from the regenerator to the pulse tube or from the pulse tube to the regenerator. Excessive DC gas flow is considered as a source of trouble in double inlet type pulse tube refrigerators. We can use an asymmetric bypass such as subsonic nozzle or two needle valves located face to face or back to back on the bypass lines in order to cancel the DC gas flow. Fig. 1c is a schematic of GM type series pulse tube refrigerator. A GM compressor with valves replaces the
S. Zhu et al. / Cryogenics 47 (2007) 483–489
15 16
1
2
17
3
4
5
6
7 8
9
485
13
10 11 12
14
20 19
18 1
2
3
4
5
6 7
8 9
10
13
11 12
14
21 26
1
2
22 3
15 4
5
23 6 7
24 16 8
9
10
13
11 12
27 25
14
Fig. 1. (a) Step-piston type series two-stage pulse tube refrigerator. (b) Bypass type series two-stage pulse tube refrigerator. (c) GM type series two-stage pulse tube refrigerator. (1) First after cooler; (2) first regenerator R11; (3) first cold head; (4) first pulse tube PT1; (5) first pulse tube cooler; (6) second regenerator R21; (7) second middle cooler; (8) second regenerator R22; (9) second cold head; (10) second pulse tube PT2; (11) second stage pulse tube cooler; (12) inertance tube (or orifice); (13) buffer; (14) heat bridge; (15) step compressor 1; (16) step compressor 2; (17) step compressor 3; (18) compressor; (19) first bypass; (20) second bypass; (21) buffer; (22) on/off valve; (23) on/off valve; (24) buffer; (25) low pressure valve; (26) high pressure valve and (27) GM type compressor.
compressor in Fig. 1b. There are two buffers through on/off valves on the bypass line or the hot end of the first stage pulse tube for obtaining better phase controlling. In an ordinary two-stage pulse tube refrigerator [7], the expansion work of the first stage and second stage is changed to heat. The work flow through the bypasses is also changed to heat. In the series two-stage pulse tube refrigerator, the expansion work of the first stage and the work flow through the bypass or step-piston is the work source in the second stage. So the higher efficiency can be obtained compared to the ordinary two-stage pulse tube refrigerator. Based on the same idea, the multistage construction consisting of more than two-stages can be realized.
In the case of Fig. 1a, the numerical simulation was done by a nodal analysis method based on Ref. [5]. The main assumptions in the numerical simulation is that there is no pressure drop, that the system is one dimensional, and that the working gas is as an ideal gas. The governing equations are as follows: Energy equation for gas oðC V qT Þ o _ Þ þ hAL ðT T S Þ ¼ 0 þ ðC P mT ot ox
ð1Þ
ð2Þ
Continuity equation A
oq om_ þ ¼0 ot ox
Ideal gas state equation P ¼ qRT Mass flow rate through orifice ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C O ðP B P PT2 ÞP PT2 m_ O ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C O ðP PT2 P B ÞP B Enthalpy flow I 1 _ p T dt mC H¼ s
3. Governing equations
A
Energy equation for matrix oT S þ hAL ðT S T Þ ¼ 0 AS C S ot
Exergy flow I 1 _ dt E ¼ H TH ms s
ð3Þ
ð4Þ
ð5Þ
ð6Þ
ð7Þ
The cooling power of the first stage is obtained as follows: Q1 ¼ H PT1 H R11 H R21 þ H R22 ð8Þ The cooling power of the second stage is obtained as follows:
486
Q2 ¼ H PT2 H R22
S. Zhu et al. / Cryogenics 47 (2007) 483–489
ð9Þ
2.3 PT2
The equivalent PV diagram in the pulse tube definition is the same with that in Ref. [5]. The expansion work in pulse tubes is the equivalent PV work. The volume of the step compressors 1–3 is
2.1
ð10Þ
Pressure, MPa
V i ¼ V id þ 0:5V i0 ð1 þ cosðxtÞÞ i ¼ 1; 2; 3
PT1
2.2
The compression work of the step compressors 1–3 is I 1 P dV i i ¼ 1; 2; 3 Wi ¼ ð11Þ s
2
1.9
The compression work is the total of the step compressors 1–3 Efficiency is T H T C1 T H T C2 g ¼ Q1 þ Q2 W T C1 T C2
1.8
ð12Þ
1.7 0
0.2
ð13Þ
0.4 0.6 Normalized length of pulse tube
0.8
1
2.3
0.015 1
Table 1 Calculation condition Room temperature First stage temperature Second stage temperature Average pressure Frequency First compressor swept volume Second compressor swept volume Third compressor swept volume Regenerator R11 Regenerator R21 Regenerator R22 First stage pulse tube Second stage pulse tube Orifice opening factor Buffer volume
300 K 80 K 20 K 2 MPa 50 Hz 27.5 cm3 20 cm3 2.5 cm3 B40 · 50 mm B30 · 50 mm B24 · 50 mm B14 · 100 mm B9 · 100 mm 16E10 250 cm3
2
0.01
2.2
3 0.005
2.1
0
2
-0.005
1.9
-0.01
1.8
-0.015 0
90
180 Phase angle, degree
1.7 360
270
300 Step compressor 2 input
Ex H
Exergy flow and enthalpy flow, W
The working medium is helium gas. The main parameters for calculation are in Table 1. The first stage temperature is 80 K, the second stage temperature is 20 K, and the room temperature is 300 K. With these parameters in Table 1, the first stage cooling power, the second stage cooling power, input power, and expansion work of the first stage pulse tube, are given as 45.7 W, 8.9 W, 491.4 W, 67.6 W, respectively. The first stage expansion work is 13.8% of the input work. In the ordinary two-stage pulse tube refrigerator, the expansion work is changed to heat. In this simulation condition, the 13.8% expansion work means that the efficiency is increased by 13.8% because it is used in the second stage. Fig. 2a shows the PV diagrams in the first stage pulse tube PT1 and the second stage pulse tube PT2, and there is a reasonable distance between the PV diagrams at the cold end and the hot end of the pulse tubes. So the shuttle loss in the pulse tubes should be a reasonably small value. Fig. 2b shows that the mass flow rate at the cold end of the first stage pulse tube and the second stage pulse tube,
Mass flow rate at cold end of pulse tube, kg/s
4. Numerical results
Pressure, MPa
W ¼ W1þW2þW3
200
Step compressor 3 input
100
0 0
100
200 300 400 X postion along refrigerator, mm
500
Fig. 2. (a) Equivalent PV diagrams. (b) Mass flow rate and pressure: (1) Mass flow rate at the cold end of the first stage pulse tube. (2) Mass flow rate at the cold end of the second stage pulse tube. (3) Pressure. (c) Exergy flow and enthalpy flow long refrigerator.
80
800
70
700
60
600
50
500
400
40 1st stage cooling power 30
300
2nd stage cooling power
Compression work, W
487
1st stage expansion work 20
200 Compression work
10
100
0 8
10
12 14 16 18 20 Normalized orifice opening factor
22
0 24
60
1.3
50
1.25
Efficiency Pressure ratio
30
1.15
20
1.1
10
1.05
0 8
10
12 14 16 18 20 Normalized orifice opening factor
22
Pressure, ratio
1.2
40 Efficiency, %
and the pressure wave. The direction of mass flow rate is from left to right in Fig. 1a. When the time is equal to zero in Eq. (9), it is zero degrees. The mass flow rate at the cold end of the first stage pulse tube and pressure wave is in phase. The phase of the pressure wave is ahead of mass flow rate at the cold end of the second stage pulse tube. Therefore the regenerators are under reasonable working conditions. Fig. 2c shows the exergy flow and enthalpy flow along the refrigerator. From 0–230 mm, it is first after cooler, first regenerator R21, first cold head, and the first pulse tube PT1. From 230–455 mm, it is the regenerator R21, second middle cooler, R22, second cold head, PT2, and second stage pulse tube cooler. The enthalpy is constant in regenerators and pulse tubes, is not constant in heat exchangers. At left end, 270 W input work of step compressor 1 is added. The exergy flow is gradually decreased due to the irreversible loss in the after cooler and the regenerator. The large decrease of exergy flow near the left end of PT1 is mainly from the first cold head output of cooling power. The remaining exergy flow through the pulse tube PT1 with 196.6 W input work of the step compressor 2 goes into the second stage. There is a little jump of exergy flow around 250 mm where it is second middle cooler due to the cooling by the first stage. In second middle cooler, the enthalpy flow is decreased due to the pre-cooling effect. The exergy flow decreases due to heat transfer and the second stage cooling power output. It should be pointed out that a part of the expansion work does not become the exergy flow for the second stage because there is an irreversible loss due to the temperature difference in the heat exchanger at the hot end of the pulse tube PT1. Fig. 2c shows that there is a jump of exergy flow at the right end. It is the 25.6 W input work of the step compressor 3 which is 5% of the total input work. It is compressor work loss. If the inertance tube is used to replace the orifice, the input work of the step compressor 3 is largely decreased. If the inertance tube and buffer are cooled by the first stage, the step compressor 3 is not necessary. The system becomes very simple.
Cooling power and expansion work, W
S. Zhu et al. / Cryogenics 47 (2007) 483–489
1 24
18 16
4.1. Effect of orifice In the series two-stage pulse tube refrigerator, the orifice is a common part. Orifice opening factor will influence on the cooling power of the first stage and second stage. Fig. 3 shows the calculation results. The parameters are the same with those in Table 1. The orifice opening factor is normalized by 1E10. Fig. 3a shows the cooling power, expansion work and compression work vs. the orifice opening factor. There is an optimum orifice opening factor for obtaining the higher input power, the higher expansion work in the first stage pulse tube, and the higher cooling power at the first stage and the second stage. Fig. 3b shows the efficiency and pressure ratio vs. the orifice opening factor. There is an optimum orifice opening
Regenerator enthalpy flow, W
14 12 10 8 6 R11 R21
4
R22 2 0 8
10
12 14 16 18 20 Normalized orifice opening factor
22
24
Fig. 3. (a) Cooling power, expansion work and compression work vs. orifice opening factor. (b) Efficiency and pressure ratio vs. orifice opening factor. (c) Regenerator loss vs. orifice opening factor.
S. Zhu et al. / Cryogenics 47 (2007) 483–489
4.3. Effect of swept volume of step compressor 3 The swept volume of the third compressor also influences the cooling performance at the first stage and second stage. 5 shows the calculation results. The parameters are the same with those in Table 1. The total swept volume is
90
900
80
800
70
700
60
600
50
500
40
400 1st stage cooling power
30
2nd stage cooling power
300
20
1st stage expansion work
200
Compression work, W
1000
Compression work 10
100
0
0 0
10 20 30 40 50 60 Swept volume ratio of step compressor 2,%
70
60
1.3
50
1.25
40
1.2
30
1.15 Efficiency Pressure ratio
20
1.1
10
1.05
0
Pressure ratio
The swept volume of the step compressor 2 is not only for the adjustment of the phase angle between mass flow rate and pressure wave for the first stage pulse tube, but also for the part of the work source of the second stage. It has a strong influence on the first stage and second stage. Fig. 4 shows results of the calculation. The parameters are same as those in Table 1. The total swept volume is kept at 50 cm3, the swept volume of step compressor 3 is not changed, and the swept volume of the step compressor 2 is changed. So the increase of the swept volume of step compressor 2 means the decrease of the swept volume of step compressor 1. The swept volume ratio of step compressor 2 is defined as the swept volume of step compressor 2 over the total volume of the step compressors 1–3. The small swept volume ratio means the small swept volume of step compressor 2 and small input work to the second stage. Fig. 4a shows the cooling power, expansion work and compression work vs. the swept volume ratio of step compressor 2. There is an optimum swept volume ratio of the step compressor 2 for obtaining the higher cooling power in the first stage. The input power of the compressor and the second stage cooling power increase with the increase of the swept volume ratio of step compressor 2. The expansion work of the first stage pulse tube decreases with the increase of the swept volume ratio of the step compressor 2, which means that the series two-stage pulse tube refrigerator effect is small when the first stage input power is too small. Fig. 4b shows the efficiency and pressure ratio vs. the swept volume ratio of step compressor 2. There is an optimum swept volume ratio of the step compressor 2 for obtaining higher efficiency. The pressure ratio increases with the increase of the swept volume ratio of step compressor 2. Fig. 4c shows the regenerator enthalpy flow vs. the swept volume ratio of step compressor 2. The regenerator loss of R11 decreases with the increase of the swept volume ratio of the step compressor 2 and the regenerator losses of R21 and R22 increases with the increase of the swept volume ratio of the step compressor 2.
Cooling power and expansion work, W
4.2. Effect of swept volume of step compressor 2
100
Efficiency, %
factor for obtaining the higher efficiency. The pressure ratio decreases with the increase of the orifice opening factor. Fig. 3c shows the regenerator enthalpy flow vs. the orifice opening factor. The regenerator enthalpy flow is the regenerator loss. It increases with the increase of the orifice opening factor.
1 0
10
20
30
40
50
60
70
Swept volume ratio of step compressor 2, % 35
30
Regenerator enthalpy flow, W
488
R11 R21
25
R22
20
15
10
5
0 0
10 20 30 40 50 60 Swept volume ratio of step compressor 2,%
70
Fig. 4. (a) Cooling power, expansion work and compression work vs. swept volume ratio of step compressor 2. (b) Efficiency and pressure ratio vs. swept volume ratio of step compressor 2. (c) Regenerator loss vs. swept volume ratio of step compressor 2.
800
70
700
60
600
50
500
40
400
1st stage cooling power 2nd stage cooling power
30
300
1st stage expansion work 20
200
compression work
10
100
0
0
-10
-100 0
Efficiency, %
Compression work, W
80
2
4 6 8 10 12 14 Swept volume ratio of step compressor 3, %
16
60
1.6
50
1.5
40
1.4
30
1.3
20
Pressure ratio
Cooling power of first and second stage, W
S. Zhu et al. / Cryogenics 47 (2007) 483–489
1.2
489
kept at 50 cm3, the swept volume of the step compressor 2 is not changed, and the swept volume of the step compressor 3 is changed. So the increase of the swept volume of the step compressor 3 means the decrease of the swept volume of the step compressor 1. The swept volume ratio of the step compressor 3 is defined as the swept volume of the step compressor 3 over the total volume of the step compressors 1–3. The small swept volume ratio means that there is a small swept volume for step compressor 3. Fig. 5a shows the cooling power, expansion work and compression work vs. the swept volume ratio of step compressor 3. Step compressor 3 has an optimum swept volume ratio for achieving the higher cooling power at the first stage and second stage, compression work, and the expansion work of the first pulse tube increase with the increase of the swept volume ratio of the step compressor 3. Fig. 5b shows the efficiency and pressure ratio vs. the swept volume ratio of step compressor 3. There is an optimum swept volume ratio of the step compressor 3 for achieving the higher efficiency. The pressure ratio increases with the increase of the swept volume ratio of step compressor 3. Fig. 5c shows the regenerator enthalpy flow vs. the swept volume ratio of step compressor 3. There is an optimum swept volume ratio of the step compressor 3 for achieving the lower regenerator loss of R22. The regenerator loss of R21 and R22 decreases with the increase of the swept volume ratio of step compressor 3.
Efficiency
5. Conclusion
Pressure ratio 10
1.1
0
1 0
2
4 6 8 10 12 14 Swept volume ratio of step compressor 3,%
16
18
Numerical simulation was done to analyze the behavior of the step-piston type series two-stage pulse tube refrigerator. In a numerical simulation condition, the efficiency increases by 13.8%. References
16
Regenerator enthalpy flow, W
14 12 10 8 R11
6
R21 4
R22
2 0 0
2
4 6 8 10 12 14 Swpet volume ratio of step compressor 3, %
16
Fig. 5. (a) Cooling power, expansion work and compression power vs. swept volume ratio of step compressor 3. (b) Efficiency and pressure ratio vs. swept volume ratio of step compressor 3. (c) Regenerator loss vs. swept volume ratio of step compressor 3.
[1] Mikulin EI, Tarasov AA, Shkrebyonock MP. Low temperature expansion pulse tubes. Adv Cryog Eng 1984;29:629. [2] Radubaugh R, Zimmerman J, Smith DR, Louie B. A comparison of three types of pulse tube refrigerators: new method for reaching 60 K. Adv Cryog Eng 1986;31:779. [3] Zhu SW. Theory of pulse tube refrigerator and its important improvement: double inlet pulse tube refrigerator. Ph.D. thesis. Xi’an Jiaotong University; 1990. [4] Tominaga A. Phase controls for pulse tube refrigerator of the third generation. Cryog eng 1992;27(2):147–51. [5] Zhu SW, Matsubara Y. Numerical method of inertance tube pulse tube refrigerator. Cryogenics 2004;44:649–60. [6] Matsubara Y, Gao JL. Novel configuration of three stage pulse tube refrigerator for temperatures below 4 K. Cryogenics 1994;34(4): 259–62. [7] Zhu SW, Nogawa M, Kawano S, Inoue T. Pulse tube refrigerator. Japan Patent, Publication number 2004-347175. [8] Zhu SW, Kawano S, Nogawa S, Inoue T. Pulse tube refrigerator. US Patent, US 6,389,819. [9] Duband L, Charles I, Ravex A, Miquet L, Jewell C. Experimental results on inertance and permanent flow in pulse tube cooler. Cryocooler 1999;10:281–90.