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Four operation modes of a pulse tube machine with a step piston compressor
Energy 100 (2016) 190e198
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Four operation modes of a pulse tube machine with a step piston compressor Shaowei Zhu Institute of Refrigeration and Cryogenics, School of Mechanical Engineering, Tongji University, 4800, Cao'an Road, Shanghai, 201804, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 October 2015 Received in revised form 22 December 2015 Accepted 24 January 2016 Available online xxx
An inertance tube pulse tube refrigerator with a step piston compressor is a reversible refrigerator which means that it can be operated as a refrigerator, a cold engine, a heat engine, and a heat pump. We call it as a pulse tube machine. In this paper, a numerical simulation is conducted on an inertance tube pulse tube machine with a step piston compressor. The result shows that the pulse tube machine can work as a refrigerator, a cold engine, a heat engine, and a heat pump, depending on the swept volume ratio of the step piston compressor. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Pulse tube refrigerator Pulse tube engine Cold engine Heat engine Heat pump Cryocooler
1. Introduction
2. Structure
Pulse tube refrigerator can be divided into two types, no work recovery type [1e5] and work recovery type [6e11]. As a work recovery type, step-piston type pulse tube refrigerator [11] is the simplest one because it has only one moving part. It has no any additional parts compared to the inertance tube pulse tube refrigerator except that the ordinary piston and cylinder become step type. Compared to quarter wave pulse tube refrigerator [10], there is only one cold head. Compared to the displacer type pulse tube machine [9] which has two moving parts, the reliability and cost are completely different. So it is a valuable type pulse tube machine which should be studied. As a work recovery type pulse tube machine, it can work as four modeserefrigerator, cold engine, heat engine, and heat pump. Here, the meaning of the cold engine is that it can use the cold energy such as the cold energy of LNG to generate mechanical power. In this paper, its four operation modes are examined by numerical simulation.
Fig. 1 shows a schematic graph of the step piston type pulse tube machine. The after cooler, regenerator, cold head, pulse tube, pulse tube cone, inertance tube, buffer cone, and buffer are connected sequentially. The step piston and step cylinder form the compression space and expansion space. The compression space is connected to the after cooler, the expansion space is connected to the buffer. If we consider the expansion space, buffer and inertance tube as an alternative piston, this machine can be considered as a Stirling machine. The buffer and inertance tube shift the phase of the gas piston in the pulse tube away from the expansion space, then the machine works like a Stirling machine.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.energy.2016.01.066 0360-5442/© 2016 Elsevier Ltd. All rights reserved.
3. Numerical method The numerical method for this paper is the same as that in Ref. [11]. The detail is in Ref. [12]. It is made for the simulation of the double inlet pulse tube refrigerator [3] and further improved for thermal acoustic engine and inertance tube pulse tube refrigerator [5]. It is good for mechanism explaining. Its accuracy is not so high. Roughly speaking, the cooling power from the numerical simulation is about two times of the experimental result. It mainly comes
S. Zhu / Energy 100 (2016) 190e198
T0 Te Vc Vcd Vc0 Vt0
Nomenclature a C f H Hpt Hreg t Pc Pe Q Rc Re
swept volume ratio percentage of Carnot frequency enthalpy flow enthalpy flow of pulse tube enthalpy flow of regenerator time pressure of compression space pressure of expansion space heat pressure ratio of compression space pressure ratio of expansion space
191
room temperature cold head temperature volume of compression space dead volume of compression space swept volume of compression space total swept volume of compression space and expansion space volume of expansion space dead volume of expansion space swept volume of expansion space power of compression space power of expansion space total work
Ve Ves Ve0 Wc We Wt
Fig. 1. Schematic of step piston type pulse tube machine.
from the periodic unsteady fluid flow and heat transfer in the refrigerator and idea compressor model used in the numerical model. Volume of the compression space is expressed by
Vc ¼ Vcd þ 0:5Vc0 ð1 cosð2pftÞÞ
(1)
And the volume of the expansion space by
Re ¼
Pemax Pemin
(6)
Power of the compression space is
I Wc ¼ f
Pc
dVC dt dt
(7)
Power of the expansion space is
Ve ¼ Ved þ 0:5Ve0 ð1 cosð2pftÞÞ
(2)
Total swept volume of the compression space and the expansion space is
Vt0 ¼ Vc0 þ Ve0
I We ¼ f
Pe
dVe dt dt
and total power is
(3)
Swept volume ratio which is the ratio of the swept volume of the expansion space to the swept volume of the compression space and the expansion space is calculated by
Wt ¼ Wc þ We
(4)
Pressure ratio of the compression space can be determined by
P RC ¼ Cmax PCmin And pressure ratio of the expansion space by
(5)
(9)
Heat of the cold head is derived by
Q ¼ Hpt Hreg V a ¼ e0 Vt0
(8)
(10)
Percentage of Carnot when the machine works as a refrigerator is determined by Eq. (11)
Te 100 C ¼ jCOPj T0 Te
(11)
and percentage of Carnot when the machine works as a cold engine by Eq. (12)
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S. Zhu / Energy 100 (2016) 190e198
Table 1 Main parameters of the pulse tube machine. After cooler height, width, length Regenerator Cold head height, width, length Pulse tube Pulse tube cone Inertance tube Buffer cone Buffer Room temperature Cold head temperature Frequency Charge pressure Working medium Pressure ratio at left end of pulse tube
0.33 mm 3770 mm 40 mm Ф120 mm 40 mm, porosity 0.7, mesh wire diameter 0.025 mm 0.33 mm 3770 mm 20 mm Ф50 mm 100 mm 50 mm long Ф30 mm 1150 mm 150 mm long 2 l (Ф100 mm 255 mm) 300 K 112 K 150 Hz 3 MPa Helium 1.2
Fig. 2. a Power of refrigerator and cold engine, b Heat and enthalpy flow of refrigerator and cold engine, c Efficiency of refrigerator and cold engine, d Total swept volume and pressure ratio of refrigerator and cold engine.
S. Zhu / Energy 100 (2016) 190e198
193
Fig. 3. a Pressure of compression space and expansion space of refrigerator, b PV diagrams of compression space and expansion space of refrigerator c Pressure and mass flow rate at cold end of pulse tube of refrigerator, d Equivalent PV diagrams in pulse tube of refrigerator.
a
b
3.4 Pc Pe
Pressure, MPa
3.2
3
2.8
2.6 0
90
180
270
360
Normalized time, degree
c
d
Fig. 4. a Pressure of compression space and expansion space of cold engine, b PV diagrams of compression space and expansion space of cold engine, c Pressure and mass flow rate at cold end of pulse tube of cold engine, d Equivalent PV diagrams in pulse tube of cold engine.
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S. Zhu / Energy 100 (2016) 190e198
a
b
c
Fig. 5. a Total power changes with inertance tube length, b Heat changes with inertance tube length, c Efficiency changes with inertance tube length.
T Te Q 0 C ¼ Wt 100 T
(12)
e
and percentage of Carnot when the machine works as a heat engine by Eq. (13)
Wt Te T0 100 C ¼ Q Te
(13)
Finally percentage of Carnot when the machine works as a heat pump by Eq. (14)
Q COP ¼ W
Te 100 Te T0 t
(14)
Negative power means that power is inputted to the machine, while positive power means that power is generated by the machine. Negative heat of the cold head means that it adsorbs heat, while positive heat of the cold head means that it discharges heat. Positive enthalpy flow means that the direction is
from the regenerator side to the buffer side. Negative enthalpy flow means that the direction is from the buffer side to the regenerator side.
4. Refrigerator and cold engine If the temperature of the cold head is lower than that of the after cooler, this machine can be operated as a refrigerator and a cold engine. When it operates as a refrigerator, power is inputted, and cooling power is received from the cold head, and heat is rejected from the after cooler. When it operates as a cold engine, power is outputted, heat is generated at the cold head, and heat is adsorbed from the after cooler. As a cold engine, it can generate power by using the cold energy of LNG. The main parameters of the pulse tube machine for numerical simulation are listed in Table 1. The pressure ratio at the cold end of the pulse tube is set as 1.2 by changing of total swept volume of the compression space and expansion space during the numerical simulation.
S. Zhu / Energy 100 (2016) 190e198
195
Fig. 6. a Total power changes with pressure ratio, b Heat changes with pressure ratio, c Efficiency changes with pressure ratio.
Fig. 2 shows the compression power, expansion power, input power, heat, efficiency, enthalpy flow of the pulse tube, enthalpy flow of the regenerator, pressure ratio of the compression space and expansion space, total swept volume vs. swept volume ratio. With the swept volume ratio being 0, the swept volume of the expansion space is zero, and the expansion power is zero. The power of the compression space is negative, which means that power is inputted from compression space. It is a typical inertance tube pulse tube refrigerator, and heat is adsorbed from the cold head. With the increasing of the swept volume ratio, the swept volume of the expansion space increases, the expansion power, cooling power and efficiency increase, and the absolute value of compression power and total power increase as well. Due to the expansion power recovering, the absolute value of total power is lower than that of the absolute value of the compression power. There is a peak for each parameter. After the peak point, every parameter decreases to zero. The efficiency increases with the increasing of the swept volume ratio, then reaches to a peak, and decreases again.
When the swept volume ratio is 1, the swept volume of the compression space is zero, the compression power is zero. The power of the expansion space is negative, which means that the power is inputted from the expansion space, heat is discharged from the cold head. There is no power output. With the decreasing of the swept volume ratio, the swept volume of the compression space increases, the power of the compression space, the absolute value of expansion power, the heat and the total power also increase. There is a peak for each parameter. After the peak point, every parameter decreases to zero. When the swept volume ratio is 0.726, the total power is zero, with the decreasing of swept volume ratio, the total power becomes positive, which means that the machine becomes an engine. The efficiency increases with the decrease of the swept volume ratio, reaches the peak, and decreases again. The enthalpy flow of the pulse tube changing with swept volume ratio is similar to that of the expansion power. If there was no inertance tube loss, the enthalpy flow of the pulse tube and the expansion power would be the same. The enthalpy flow of the
196
S. Zhu / Energy 100 (2016) 190e198
Fig. 7. a Power of heat engine and heat pump, b Heat and enthalpy flow of heat engine and heat pump, c Efficiency of heat engine and heat pump, d Total swept volume and pressure ratio of heat engine and heat pump.
regenerator at engine range is very small, which indicates that the matrix wire diameter can be larger. The difference between the enthalpy flow in the pulse tube and in the regenerator is the heat. The pressure ratio at the left end of pulse tube is set as 1.2, then the total swept volume of the compression space and expansion space change with swept volume ratio. There is a peak at the swept volume ratio 0.520. Pressure ratio in the expansion space is lower than 1.2 due to the buffer effect, the pressure ratio in the compression space is shifted away from 1.2 due to the pressure drop of the regenerator. There is a zero point for each parameter. From mathematic view of point, there must be a swept volume ratio at which each parameter becomes zero. Here, we define the zero heat point as critical point that separates the machine as refrigerator mode and engine mode. Fig. 3 shows the pressure at the compression space and expansion space, PV diagrams of the compression space and expansion space, mass flow rate and pressure at the left end of the pulse tube at peak efficiency point of refrigerator when the swept volume ratio is 0.425. Fig. 4 shows the pressure at the compression space and expansion space, PV diagrams of the compression space and expansion space, mass flow rate and pressure at the left end of the pulse tube at peak efficiency point of cold engine when the
swept volume ratio is 0.610. There is a phase angle difference between the pressure in the compression space and expansion space due to the combined effects of the inertance tube and the buffer, so that the PV diagrams in the compression space and expansion space are different though the compression space and expansion space are in phase to let us use only one moving part. Power is inputted in the refrigerator mode and is outputted in the cold engine mode. In the refrigerator mode, the phase angle difference between the mass flow rate and pressure at the left end of the pulse tube is shifted away from zero degree for high regenerator efficiency. In the cold engine mode, it is shifted away from 180 degrees for high regenerator efficiency. The direction of equivalent PV diagrams in the pulse tube shows that the gas at the left end of the pulse tube expands in the refrigerator mode, and is compressed in the cold engine mode. Fig. 5 shows the total power, heat, and efficiency vs. inertance tube length. There is an optimum inertance tube length with which the efficiency gets maximum, and the optimum length for refrigerator is slightly longer than that for the cold engine. The zero heat points slightly changes with the changing of inertance tube length. Fig. 6 shows that the zero heat point does not change with the change of pressure ratio, the heat and total power increase with the
S. Zhu / Energy 100 (2016) 190e198
197
Fig. 8. a Pressure of compression space and expansion space of heat engine, b PV diagrams of compression space and expansion space of heat engine, c Pressure and mass flow rate at cold end of pulse tube of heat engine, d Equivalent PV diagrams in pulse tube of heat engine.
increasing of pressure ratio, the efficiency almost does not change with the changing of the pressure ratio. 5. Heat engine and heat pump Refrigerator and cold engine mode will be changed to the heat engine and heat pump mode if the temperature of the cold head is higher than that of the after cooler. For the heat engine and heat pump mode, the after cooler length is changed to 20 mm and the mesh wire diameter is changed to 0.05 mm, and cold head temperature is changed to 600 K. Fig. 7 shows the compression power, expansion power, total power, heat, efficiency, enthalpy flow of the pulse tube, enthalpy flow of the regenerator, pressure ratio of the compression space and expansion space, total swept volume vs. swept volume ratio. Fig. 7 is similar to Fig. 2. When the swept ratio increases from zero, the difference is that the expansion power is larger than that in refrigerator mode. So with the increasing of the swept volume ratio from zero, the total power changes from negative to positive, which means when there is a pure power output, the machine becomes a heat engine to generate power. With the decreasing of the swept volume ratio from 1, the total power and heat is negative, which means that the machine is a heat pump.
Fig. 8 shows the pressure in the compression space and expansion space, PV diagrams of the compression space and expansion space, mass flow rate and pressure at the left end of the pulse tube at the peak efficiency point of engine when the swept volume ratio is 0.783. Fig. 9 shows the pressure in the compression space and expansion space, PV diagrams of the compression space and expansion space, mass flow rate and pressure at the left end of the pulse tube at the peak efficiency point of heat pump when the swept volume ratio is 0.815. Similar to refrigerator and cold engine modes, there is a phase angle difference between the pressure in the compression space and expansion space in heat engine mode and heat pump mode. The pressure and mass flow rate at the left end of the pulse tube has no big difference compared with heat engine and refrigerator, cold engine and heat pump. In heat engine mode, the direction of equivalent PV diagrams in the pulse tube is the same as that in refrigerator mode. In heat pump mode, the direction of equivalent PV diagrams in the pulse tube is the same as that in cold engine mode. 6. Conclusion The pulse tube machine with step piston compressor can work as refrigerator, cold engine, heat engine, and heat pump modes depending on the swept volume ratio. There is a critical swept
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S. Zhu / Energy 100 (2016) 190e198
a
b
c
d
Fig. 9. a Pressure of compression space and expansion space of heat pump, b PV diagrams of compression space and expansion space of heat pump, c Pressure and mass flow rate at cold end of pulse tube of heat pump, d Equivalent PV diagrams in pulse tube of heat pump.
volume ratio at which the heat of the cold head is zero for a given machine at a given working temperature. The machine works as a refrigerator or heat engine if the swept volume ratio is smaller than the critical swept volume ratio, and it works as a cold engine or heat pump if the swept volume ratio is larger than the critical swept volume ratio. The optimum swept volume ratio for efficiency in each mode is near the critical swept volume ratio. Acknowledgement This work is supported by the National Natural Science Foundation of China (No. 51476117). References [1] Mikulin EI, Tarasov AA, Shkrebyonock MP. Low temperature expansion pulse tubes. Adv Cryog Eng 1984;29:629. [2] Radebaugh 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, Wu PY, Chen ZQ. Double inlet pulse tube refrigerators: an important improvement. Cryogenics 1990;30:514e20. [4] Tominaga A. Phase controls for pulse tube refrigerator of the third generation. Cryog Eng 1992;27(2):147e51. [5] Zhu SW, Shou SL, Yoshimura N, Matsubara Y. Phase shift effect of the long neck tube for the pulse tube refrigerator. In: Cryocooler, vol. 9. New York: Plenum Press; 1997. p. 269e78. [6] Ishizaki Y, Ishizaki E. Prototype of pulse tube refrigerator for practical use. In: Advances in cryogenic engineering, vol. 39B; 1994. p. 1433e9. [7] Radebaugh R. Development of pulse tube refrigerator as an efficient and reliable cryocooler. In: Proc institute of refrigeration, London; 1999e2000. [8] Matsubara Y. Future trend of pulse tube cryocooler research. In: Zhang,Liangzhen Lin Liang, Chen Guobang, editors. Proceedings of the twentieth international cryogenic engineering conference, Beijing, China 2004. Elsevier Ltd.; 2005. p. 189e96. [9] Zhu SW, Nogawa M. Pulse tube stirling machine with warm gas-driven displacer. Cryogenics 2010;50:320e30. [10] Swift GW, Gardner DL, Backhaus SN. Quarter-wave pulse tube. Cryogenics 2011;51:575e83. [11] Zhu SW. Step piston pulse tube refrigerator. Cryogenics 2014;64:63e9. [12] Zhu SW, Matsubara Y. Numerical method of inertance tube pulse tube refrigerator. Cryogenics 2004;44:649e60.
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Four operation modes of a pulse tube machine with a step piston compressor Shaowei Zhu Institute of Refrigeration and Cryogenics, School of Mechanical Engineering, Tongji University, 4800, Cao'an Road, Shanghai, 201804, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 October 2015 Received in revised form 22 December 2015 Accepted 24 January 2016 Available online xxx
An inertance tube pulse tube refrigerator with a step piston compressor is a reversible refrigerator which means that it can be operated as a refrigerator, a cold engine, a heat engine, and a heat pump. We call it as a pulse tube machine. In this paper, a numerical simulation is conducted on an inertance tube pulse tube machine with a step piston compressor. The result shows that the pulse tube machine can work as a refrigerator, a cold engine, a heat engine, and a heat pump, depending on the swept volume ratio of the step piston compressor. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Pulse tube refrigerator Pulse tube engine Cold engine Heat engine Heat pump Cryocooler
1. Introduction
2. Structure
Pulse tube refrigerator can be divided into two types, no work recovery type [1e5] and work recovery type [6e11]. As a work recovery type, step-piston type pulse tube refrigerator [11] is the simplest one because it has only one moving part. It has no any additional parts compared to the inertance tube pulse tube refrigerator except that the ordinary piston and cylinder become step type. Compared to quarter wave pulse tube refrigerator [10], there is only one cold head. Compared to the displacer type pulse tube machine [9] which has two moving parts, the reliability and cost are completely different. So it is a valuable type pulse tube machine which should be studied. As a work recovery type pulse tube machine, it can work as four modeserefrigerator, cold engine, heat engine, and heat pump. Here, the meaning of the cold engine is that it can use the cold energy such as the cold energy of LNG to generate mechanical power. In this paper, its four operation modes are examined by numerical simulation.
Fig. 1 shows a schematic graph of the step piston type pulse tube machine. The after cooler, regenerator, cold head, pulse tube, pulse tube cone, inertance tube, buffer cone, and buffer are connected sequentially. The step piston and step cylinder form the compression space and expansion space. The compression space is connected to the after cooler, the expansion space is connected to the buffer. If we consider the expansion space, buffer and inertance tube as an alternative piston, this machine can be considered as a Stirling machine. The buffer and inertance tube shift the phase of the gas piston in the pulse tube away from the expansion space, then the machine works like a Stirling machine.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.energy.2016.01.066 0360-5442/© 2016 Elsevier Ltd. All rights reserved.
3. Numerical method The numerical method for this paper is the same as that in Ref. [11]. The detail is in Ref. [12]. It is made for the simulation of the double inlet pulse tube refrigerator [3] and further improved for thermal acoustic engine and inertance tube pulse tube refrigerator [5]. It is good for mechanism explaining. Its accuracy is not so high. Roughly speaking, the cooling power from the numerical simulation is about two times of the experimental result. It mainly comes
S. Zhu / Energy 100 (2016) 190e198
T0 Te Vc Vcd Vc0 Vt0
Nomenclature a C f H Hpt Hreg t Pc Pe Q Rc Re
swept volume ratio percentage of Carnot frequency enthalpy flow enthalpy flow of pulse tube enthalpy flow of regenerator time pressure of compression space pressure of expansion space heat pressure ratio of compression space pressure ratio of expansion space
191
room temperature cold head temperature volume of compression space dead volume of compression space swept volume of compression space total swept volume of compression space and expansion space volume of expansion space dead volume of expansion space swept volume of expansion space power of compression space power of expansion space total work
Ve Ves Ve0 Wc We Wt
Fig. 1. Schematic of step piston type pulse tube machine.
from the periodic unsteady fluid flow and heat transfer in the refrigerator and idea compressor model used in the numerical model. Volume of the compression space is expressed by
Vc ¼ Vcd þ 0:5Vc0 ð1 cosð2pftÞÞ
(1)
And the volume of the expansion space by
Re ¼
Pemax Pemin
(6)
Power of the compression space is
I Wc ¼ f
Pc
dVC dt dt
(7)
Power of the expansion space is
Ve ¼ Ved þ 0:5Ve0 ð1 cosð2pftÞÞ
(2)
Total swept volume of the compression space and the expansion space is
Vt0 ¼ Vc0 þ Ve0
I We ¼ f
Pe
dVe dt dt
and total power is
(3)
Swept volume ratio which is the ratio of the swept volume of the expansion space to the swept volume of the compression space and the expansion space is calculated by
Wt ¼ Wc þ We
(4)
Pressure ratio of the compression space can be determined by
P RC ¼ Cmax PCmin And pressure ratio of the expansion space by
(5)
(9)
Heat of the cold head is derived by
Q ¼ Hpt Hreg V a ¼ e0 Vt0
(8)
(10)
Percentage of Carnot when the machine works as a refrigerator is determined by Eq. (11)
Te 100 C ¼ jCOPj T0 Te
(11)
and percentage of Carnot when the machine works as a cold engine by Eq. (12)
192
S. Zhu / Energy 100 (2016) 190e198
Table 1 Main parameters of the pulse tube machine. After cooler height, width, length Regenerator Cold head height, width, length Pulse tube Pulse tube cone Inertance tube Buffer cone Buffer Room temperature Cold head temperature Frequency Charge pressure Working medium Pressure ratio at left end of pulse tube
0.33 mm 3770 mm 40 mm Ф120 mm 40 mm, porosity 0.7, mesh wire diameter 0.025 mm 0.33 mm 3770 mm 20 mm Ф50 mm 100 mm 50 mm long Ф30 mm 1150 mm 150 mm long 2 l (Ф100 mm 255 mm) 300 K 112 K 150 Hz 3 MPa Helium 1.2
Fig. 2. a Power of refrigerator and cold engine, b Heat and enthalpy flow of refrigerator and cold engine, c Efficiency of refrigerator and cold engine, d Total swept volume and pressure ratio of refrigerator and cold engine.
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193
Fig. 3. a Pressure of compression space and expansion space of refrigerator, b PV diagrams of compression space and expansion space of refrigerator c Pressure and mass flow rate at cold end of pulse tube of refrigerator, d Equivalent PV diagrams in pulse tube of refrigerator.
a
b
3.4 Pc Pe
Pressure, MPa
3.2
3
2.8
2.6 0
90
180
270
360
Normalized time, degree
c
d
Fig. 4. a Pressure of compression space and expansion space of cold engine, b PV diagrams of compression space and expansion space of cold engine, c Pressure and mass flow rate at cold end of pulse tube of cold engine, d Equivalent PV diagrams in pulse tube of cold engine.
194
S. Zhu / Energy 100 (2016) 190e198
a
b
c
Fig. 5. a Total power changes with inertance tube length, b Heat changes with inertance tube length, c Efficiency changes with inertance tube length.
T Te Q 0 C ¼ Wt 100 T
(12)
e
and percentage of Carnot when the machine works as a heat engine by Eq. (13)
Wt Te T0 100 C ¼ Q Te
(13)
Finally percentage of Carnot when the machine works as a heat pump by Eq. (14)
Q COP ¼ W
Te 100 Te T0 t
(14)
Negative power means that power is inputted to the machine, while positive power means that power is generated by the machine. Negative heat of the cold head means that it adsorbs heat, while positive heat of the cold head means that it discharges heat. Positive enthalpy flow means that the direction is
from the regenerator side to the buffer side. Negative enthalpy flow means that the direction is from the buffer side to the regenerator side.
4. Refrigerator and cold engine If the temperature of the cold head is lower than that of the after cooler, this machine can be operated as a refrigerator and a cold engine. When it operates as a refrigerator, power is inputted, and cooling power is received from the cold head, and heat is rejected from the after cooler. When it operates as a cold engine, power is outputted, heat is generated at the cold head, and heat is adsorbed from the after cooler. As a cold engine, it can generate power by using the cold energy of LNG. The main parameters of the pulse tube machine for numerical simulation are listed in Table 1. The pressure ratio at the cold end of the pulse tube is set as 1.2 by changing of total swept volume of the compression space and expansion space during the numerical simulation.
S. Zhu / Energy 100 (2016) 190e198
195
Fig. 6. a Total power changes with pressure ratio, b Heat changes with pressure ratio, c Efficiency changes with pressure ratio.
Fig. 2 shows the compression power, expansion power, input power, heat, efficiency, enthalpy flow of the pulse tube, enthalpy flow of the regenerator, pressure ratio of the compression space and expansion space, total swept volume vs. swept volume ratio. With the swept volume ratio being 0, the swept volume of the expansion space is zero, and the expansion power is zero. The power of the compression space is negative, which means that power is inputted from compression space. It is a typical inertance tube pulse tube refrigerator, and heat is adsorbed from the cold head. With the increasing of the swept volume ratio, the swept volume of the expansion space increases, the expansion power, cooling power and efficiency increase, and the absolute value of compression power and total power increase as well. Due to the expansion power recovering, the absolute value of total power is lower than that of the absolute value of the compression power. There is a peak for each parameter. After the peak point, every parameter decreases to zero. The efficiency increases with the increasing of the swept volume ratio, then reaches to a peak, and decreases again.
When the swept volume ratio is 1, the swept volume of the compression space is zero, the compression power is zero. The power of the expansion space is negative, which means that the power is inputted from the expansion space, heat is discharged from the cold head. There is no power output. With the decreasing of the swept volume ratio, the swept volume of the compression space increases, the power of the compression space, the absolute value of expansion power, the heat and the total power also increase. There is a peak for each parameter. After the peak point, every parameter decreases to zero. When the swept volume ratio is 0.726, the total power is zero, with the decreasing of swept volume ratio, the total power becomes positive, which means that the machine becomes an engine. The efficiency increases with the decrease of the swept volume ratio, reaches the peak, and decreases again. The enthalpy flow of the pulse tube changing with swept volume ratio is similar to that of the expansion power. If there was no inertance tube loss, the enthalpy flow of the pulse tube and the expansion power would be the same. The enthalpy flow of the
196
S. Zhu / Energy 100 (2016) 190e198
Fig. 7. a Power of heat engine and heat pump, b Heat and enthalpy flow of heat engine and heat pump, c Efficiency of heat engine and heat pump, d Total swept volume and pressure ratio of heat engine and heat pump.
regenerator at engine range is very small, which indicates that the matrix wire diameter can be larger. The difference between the enthalpy flow in the pulse tube and in the regenerator is the heat. The pressure ratio at the left end of pulse tube is set as 1.2, then the total swept volume of the compression space and expansion space change with swept volume ratio. There is a peak at the swept volume ratio 0.520. Pressure ratio in the expansion space is lower than 1.2 due to the buffer effect, the pressure ratio in the compression space is shifted away from 1.2 due to the pressure drop of the regenerator. There is a zero point for each parameter. From mathematic view of point, there must be a swept volume ratio at which each parameter becomes zero. Here, we define the zero heat point as critical point that separates the machine as refrigerator mode and engine mode. Fig. 3 shows the pressure at the compression space and expansion space, PV diagrams of the compression space and expansion space, mass flow rate and pressure at the left end of the pulse tube at peak efficiency point of refrigerator when the swept volume ratio is 0.425. Fig. 4 shows the pressure at the compression space and expansion space, PV diagrams of the compression space and expansion space, mass flow rate and pressure at the left end of the pulse tube at peak efficiency point of cold engine when the
swept volume ratio is 0.610. There is a phase angle difference between the pressure in the compression space and expansion space due to the combined effects of the inertance tube and the buffer, so that the PV diagrams in the compression space and expansion space are different though the compression space and expansion space are in phase to let us use only one moving part. Power is inputted in the refrigerator mode and is outputted in the cold engine mode. In the refrigerator mode, the phase angle difference between the mass flow rate and pressure at the left end of the pulse tube is shifted away from zero degree for high regenerator efficiency. In the cold engine mode, it is shifted away from 180 degrees for high regenerator efficiency. The direction of equivalent PV diagrams in the pulse tube shows that the gas at the left end of the pulse tube expands in the refrigerator mode, and is compressed in the cold engine mode. Fig. 5 shows the total power, heat, and efficiency vs. inertance tube length. There is an optimum inertance tube length with which the efficiency gets maximum, and the optimum length for refrigerator is slightly longer than that for the cold engine. The zero heat points slightly changes with the changing of inertance tube length. Fig. 6 shows that the zero heat point does not change with the change of pressure ratio, the heat and total power increase with the
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Fig. 8. a Pressure of compression space and expansion space of heat engine, b PV diagrams of compression space and expansion space of heat engine, c Pressure and mass flow rate at cold end of pulse tube of heat engine, d Equivalent PV diagrams in pulse tube of heat engine.
increasing of pressure ratio, the efficiency almost does not change with the changing of the pressure ratio. 5. Heat engine and heat pump Refrigerator and cold engine mode will be changed to the heat engine and heat pump mode if the temperature of the cold head is higher than that of the after cooler. For the heat engine and heat pump mode, the after cooler length is changed to 20 mm and the mesh wire diameter is changed to 0.05 mm, and cold head temperature is changed to 600 K. Fig. 7 shows the compression power, expansion power, total power, heat, efficiency, enthalpy flow of the pulse tube, enthalpy flow of the regenerator, pressure ratio of the compression space and expansion space, total swept volume vs. swept volume ratio. Fig. 7 is similar to Fig. 2. When the swept ratio increases from zero, the difference is that the expansion power is larger than that in refrigerator mode. So with the increasing of the swept volume ratio from zero, the total power changes from negative to positive, which means when there is a pure power output, the machine becomes a heat engine to generate power. With the decreasing of the swept volume ratio from 1, the total power and heat is negative, which means that the machine is a heat pump.
Fig. 8 shows the pressure in the compression space and expansion space, PV diagrams of the compression space and expansion space, mass flow rate and pressure at the left end of the pulse tube at the peak efficiency point of engine when the swept volume ratio is 0.783. Fig. 9 shows the pressure in the compression space and expansion space, PV diagrams of the compression space and expansion space, mass flow rate and pressure at the left end of the pulse tube at the peak efficiency point of heat pump when the swept volume ratio is 0.815. Similar to refrigerator and cold engine modes, there is a phase angle difference between the pressure in the compression space and expansion space in heat engine mode and heat pump mode. The pressure and mass flow rate at the left end of the pulse tube has no big difference compared with heat engine and refrigerator, cold engine and heat pump. In heat engine mode, the direction of equivalent PV diagrams in the pulse tube is the same as that in refrigerator mode. In heat pump mode, the direction of equivalent PV diagrams in the pulse tube is the same as that in cold engine mode. 6. Conclusion The pulse tube machine with step piston compressor can work as refrigerator, cold engine, heat engine, and heat pump modes depending on the swept volume ratio. There is a critical swept
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Fig. 9. a Pressure of compression space and expansion space of heat pump, b PV diagrams of compression space and expansion space of heat pump, c Pressure and mass flow rate at cold end of pulse tube of heat pump, d Equivalent PV diagrams in pulse tube of heat pump.
volume ratio at which the heat of the cold head is zero for a given machine at a given working temperature. The machine works as a refrigerator or heat engine if the swept volume ratio is smaller than the critical swept volume ratio, and it works as a cold engine or heat pump if the swept volume ratio is larger than the critical swept volume ratio. The optimum swept volume ratio for efficiency in each mode is near the critical swept volume ratio. Acknowledgement This work is supported by the National Natural Science Foundation of China (No. 51476117). References [1] Mikulin EI, Tarasov AA, Shkrebyonock MP. Low temperature expansion pulse tubes. Adv Cryog Eng 1984;29:629. [2] Radebaugh 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, Wu PY, Chen ZQ. Double inlet pulse tube refrigerators: an important improvement. Cryogenics 1990;30:514e20. [4] Tominaga A. Phase controls for pulse tube refrigerator of the third generation. Cryog Eng 1992;27(2):147e51. [5] Zhu SW, Shou SL, Yoshimura N, Matsubara Y. Phase shift effect of the long neck tube for the pulse tube refrigerator. In: Cryocooler, vol. 9. New York: Plenum Press; 1997. p. 269e78. [6] Ishizaki Y, Ishizaki E. Prototype of pulse tube refrigerator for practical use. In: Advances in cryogenic engineering, vol. 39B; 1994. p. 1433e9. [7] Radebaugh R. Development of pulse tube refrigerator as an efficient and reliable cryocooler. In: Proc institute of refrigeration, London; 1999e2000. [8] Matsubara Y. Future trend of pulse tube cryocooler research. In: Zhang,Liangzhen Lin Liang, Chen Guobang, editors. Proceedings of the twentieth international cryogenic engineering conference, Beijing, China 2004. Elsevier Ltd.; 2005. p. 189e96. [9] Zhu SW, Nogawa M. Pulse tube stirling machine with warm gas-driven displacer. Cryogenics 2010;50:320e30. [10] Swift GW, Gardner DL, Backhaus SN. Quarter-wave pulse tube. Cryogenics 2011;51:575e83. [11] Zhu SW. Step piston pulse tube refrigerator. Cryogenics 2014;64:63e9. [12] Zhu SW, Matsubara Y. Numerical method of inertance tube pulse tube refrigerator. Cryogenics 2004;44:649e60.