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Analysis on the stirling-type pulse tube refrigerator in consideration of dynamics of linear compressor
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Cryogenics 48 (2008) 68–76 www.elsevier.com/locate/cryogenics
Analysis on the stirling-type pulse tube refrigerator in consideration of dynamics of linear compressor Junseok Ko *, Sangkwon Jeong 1 Cryogenic Engineering Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea Received 12 March 2007; received in revised form 5 November 2007; accepted 12 December 2007
Abstract This paper describes the performance analysis of Stirling-type pulse tube refrigerator (PTR) in conjunction with the dynamics of the accompanied linear compressor. The dynamic behavior of the piston in the linear compressor is directly influenced by the load condition of the PTR. In this paper, the dynamic equation of the piston is simultaneously solved with the thermo-hydraulic governing equations of the PTR using linear analysis model and the performance of the PTR is predicted with the accompanied thermal losses. The developed analysis code is verified with the experimental results. The effect of the inertance tube length which plays an important role in the PTR is also specifically investigated from the experimental and simulation results. It clearly shows the effect of the flow impedance of the inertance tube on the dynamic response of the piston as well as the cooling performance of the PTR. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Pulse tube; Inertance tube; Linear compressor
1. Introduction Recent progress in engineering-grade high temperature superconductor (HTS) materials demonstrates more feasibility for practical applications with its easier cooling method than low temperature superconductor (LTS). Those applications include superconducting motor, generator, cable, flywheel energy storage system, magnetic energy storage system, fault current limiter, etc. There is no doubt that an efficient cryocooler with proper auxiliary cooling interface is an indispensable part of realizing those attractive technologies. In fact, the feasibility of successful commercial market penetration of HTS system is deeply connected to efficient cryogenic cooling technology as long as superconducting materials require low temperature. For attractive compact integration of HTS system, an imple*
Corresponding author. Tel.: +82 42 869 3079; fax: +82 42 869 8207. E-mail addresses: [email protected] (J. Ko), [email protected] (S. Jeong). 1 Fax: +82 42 869 3210. 0011-2275/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2007.12.002
mented cryocooler must be efficient and reliable. A Striling cryocooler or Stirling-type PTR (pulse tube refrigerator) is generally suitable with high thermal efficiency for small-size application due to its overall low-volume characteristic. In order to design an efficient Stirling-type PTR, we need to accurately predict its cooling performance along with the accompanied behavior of the linear compressor which drives the cooler. Most previous analytic models on PTR involve the assumption of piston displacement or temporal swept volume as a function of time [1–4]. Then, the thermodynamic performance of the PTR was analyzed according to this hypothesized condition of the compressor. This analysis is useful if the piston movement is perfectly replicated as designed and presumed by adjusting the parameters of the compressor through the tuning process. But, the linear compressor has limitations in the piston displacement due to its inside geometry and the allowable input current due to Joule heating in the coil, and its resonant characteristic varies as the pressure wave is generated at the compression space. These characteristics of a linear
J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
69
Nomenclature A Aff cm d f fosc i Ie km KE Lc L0 mp m_ n P Q_ R Rc T v V W_ x
cross-sectional area (m2) effective area of regenerator (m2) mechanical damping coefficient (kg/s) diameter (m) frequency (s1) oscillating friction factor () current (A) ineffectiveness of regenerator () mechanical spring constant (N/m) motor constant (N/A) electric inductance of coil (H) neutral length of cylinder (m) mass of piston (kg) mass flow rate (kg/s) polytropic constant () pressure (kPa) heat rate (W) gas constant (J/kg-K) electric resistance of coil (X) temperature (K) voltage (V) volume (m3) work rate (W) Piston displacement (m)
Greek symbols / phase angle c specific heat ratio q density l viscosity
compressor may cause the failure of the presumed piston displacement or the poor compression efficiency with the predesigned PTR configuration. For high overall efficiency of a cryocooler, the power transmission efficiency in the PTR as well as in the linear compressor should be maximized simultaneously. For these reasons, the analytic model on PTR with consideration of the dynamics of a linear compressor should be developed for good design of a Stirling-type PTR. In a linear compressor, the linear motion of a piston simultaneously generates the pulsating pressure and the oscillating flow. In contrast to GM-type PTR, a Stirling-type PTR is directly connected to a linear compressor and thus, the generated pressure wave at the compression space strongly affect on the piston dynamics. To fully understand the physical coupling between the piston dynamics and the PTR’s performance, we have to consider those driving and driven mechanical parts of the refrigerator at the same time. This analytic methodology can explain the characteristics of the cryocooler more realistically than the previous models that usually assumed piston displacement, swept volume, or pressure wave at the compression space.
Subscripts ca interface between the compression space and the aftercooler ar interface between the aftercooler and the regenerator rl interface between the regenerator and the coldend heat exchanger le interface between the cold-end heat exchanger and the pulse tube w interface between the pulse tube and the warmend heat exchanger i interface between the warm-end heat exchanger and the inertance tube res interface between the inertance tube and the reservior a aftercooler reg regenerator lhx cold-end heat exchanger pt pulse tube hhx warm-end heat exchanger it inertance tube m mean value p piston r pressure at the reservoir 1 pressure at the compression space 2 pressure at the pulse tube
In this paper, the analytic model is developed with linear analysis in which the waveform of each variable is assumed purely sinusoidal. To verify the analytic model, the Stirling-type PTR has been designed and fabricated using a linear compressor which was originally designed for Stirling cryocooler. Experiments have been performed with three different lengths of inertance tube. With the experimental and analysis results, the thermo-hydraulic characteristics and the cooling performance of a Stirling-type PTR are discussed in relation to the dynamic behavior of a linear compressor. Especially, the role of an inertance tube is deeply discussed. This study is expected to suggest a guideline for better design of highly efficient Stirling-type PTR. 2. Analytic model Fig. 1 is the schematic diagram of a Stirling-type PTR with an inertance tube as a phase control device. The piston of a linear compressor, which is driven by a linear motor, oscillates in the compression space and, then, the motion of the piston generates the pressure wave and the oscillating gas flow to the regenerator, the pulse tube, and the
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compression space
pulse tube
aftercooler . mle
Pm
. mw p2
p1
piston
inertance tube
. mca
. mar regenerator
. mrl
pr . mi
. mres reservoir
cold-end heat exchanger
warm-end heat exchanger
Fig. 1. Schematic diagram of a Stirling-type pulse tube refrigerator.
phase control device. For a given current input to the linear motor, the resultant operating variables, which are the voltage of the linear motor, the piston displacement, the pressure and the mass flow rate at each interface, can be determined from thermo-hydraulic and dynamic relations between the components. 2.1. Governing equations 2.1.1. Linear compressor In a linear compressor, the required voltage for the input current is determined by the electric characteristics of a linear motor and the piston displacement as shown in Eq. (1). The mechanical spring and damping forces, the gas force by the pressure difference between both sides of the piston and the driving force act on the piston. These forces are balanced as represented by Eq. (2). The gas force plays an important role in determining the motion of the piston. The pressure difference between both sides of the piston imposes an additional stiffness and the PV power transmitted to the PTR space can be considered as a damping effect in the dynamic system of the piston. Theses are called as the gas spring and the gas damping effects, respectively. The gas spring effect shifts the resonant frequency to the higher value than the pure natural frequency of the mechanical piston configuration and the gas damping effect resultantly attenuates the piston stroke. In a previous research, the gas effect was described with the gas spring constant and the gas damping coefficient as verified by the experiments [5]. Eq. (3) is the relation among the pressure, the piston displacement and the mass flow rate at the compression space and is derived from the mass and energy conservation equations with the assumption of adiabatic compression and expansion processes and no leakage through the clearance gap [6]. We can also linearize Eq. (3) as Eq. (4) by considering only fundamental terms di dx þ KE dt dt d2 x dx mp 2 ¼ k m x cm þ ðP m P 1 ÞAp þ K E i dt dt Ap ðL0 xÞ dP 1 P 1 Ap dx ¼ m_ ca cRT ca dt RT ca dt Ap L0 dP 1 P m Ap dx ¼ m_ ca cRT ca dt RT ca dt
v ¼ Rc i þ Lc
2.1.2. Regenerator When the oscillating flow passes through the regenerator, the amplitudes of the pressure and the mass flow rate are both attenuated by the flow friction and the compliance effect respectively. These phenomena are governed by the mass and momentum conservation relations which are represented by Eqs. (5) and (6). The pressure drop through the regenerator is basically proportional to the mass flow rate if the compliant volume is not significant. The proportionality can be calculated with the regenerator geometry and the friction factor of the regenerator matrix [7]. In this paper, the discretized grid configuration is used for the regenerator and the inertance tube as shown in Fig. 2. It is assumed that the temperature profile is linear in the regenerator and the #200 twill mesh is packed as a regenerator material. Nam’s experimental result is used as the oscillating friction factor of the #200 twill mesh [7] om_ reg Aff oP reg ¼ ox RT reg ot oP reg ¼ Rreg m_ reg ox
ð5Þ ð6Þ
where Rreg ¼ fosc
jm_ reg j 2 ; qr A2ff dh
f osc ¼ 36:55=Rereg þ 0:16
2.1.3. Pulse tube The relation between the pressure and the mass flow rate in the pulse tube is derived from the mass conservation and the energy balance with the assumption of uniform pressure within the pulse tube as follows [8] m_ le m_ w V pt dP 2 ¼ qc qw cP 2 dt
ð7Þ
2.1.4. Inertance tube and reservoir The inertance tube and the reservoir are to control the phase shift between the pressure and the mass flow rate at the cold-end. The narrow and long tube gives the flow resistance and the inductance to the oscillating flow. As shown in Eqs. (8) and (9), the flow resistance through the inertance tube can be calculated by the oscillating flow model [6]. The relation between the mass flow rate to the reservoir and the reservoir pressure is derived from the
0
ð1Þ
1
2
i-1
i
i+1
n-1
n
ð2Þ 0
ð3Þ ð4Þ
1
2
i-1
i
i+1
n-1
n
n+1
Gas temperature, pressure Mass flow rate
Fig. 2. Grid configuration for regenerator and inertance tube.
J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
mass conservation in the reservoir and represented by Eq. (10). Here, the polytropic constant n is between 1 and c and determined as the compression/expansion process in the reservoir. In this study, the operating frequency is between 30 and 45 Hz and thus, the process in the reservoir would not be exactly the adiabatic process but would be close to it. It is assumed 1.5 of polytropic constant om_ it Ait dP it ¼ ox RT h dt oP it 1 dm_ it ¼ Rit m_ it þ Ait dt ox
ð8Þ
During the operating states of Stirling-type PTR, all physical variables are assumed to have cyclic waveforms for the given sinusoidal current input as represented with Eq. (15). The phase shift of each variable is represented by the phase angle difference from that of the input current I I C p m_ le f Cp p2 p2 m_ le dt dt ffi ð12Þ W_ P dVe ¼ f qm R R q2 Q_ ineff ¼ I e m_ reg ðT a T l Þ ð13Þ
ð9Þ COP ¼
where ait 0:201 Rit ¼ ; ait ¼ 0:1556ðjm_ it jd it =lAit Þ ðjm_ it j=qh d it Ait Þ Ait V res dP r ð10Þ m_ res ¼ nRT h dt
71
Q_ ref W_ in
aðtÞ ffi Am þ A0 cosðxt þ /a Þ
ð14Þ ð15Þ
3. Experiments 3.1. Experimental setup and experiments
2.1.5. Heat exchangers Three heat exchangers (i.e. the aftercooler, the cold-end heat exchanger and the warm-end heat exchanger) are used in a PTR. All heat exchangers are assumed to be isothermal. Then, Eq. (11) is readily derived from the mass conservation relation m_ in m_ out ¼
V dP RT dt
ð11Þ
2.2. Performance analysis and thermal losses In the analysis, the piston displacement, required voltage, pressure and mass flow rates are calculated from the governing equations. And then, the input power, the compression power, the PV power at the cold-end and the thermal losses are evaluated from the calculated results. If there is no thermal loss, the PV power at the cold-end is purely used to generate the refrigeration effect. But, there must be some inherent thermal losses in the real situation. Thus, the remainder of the PV power after being compensated by the thermal losses only produces the actual cooling capacity of the PTR. Five thermal loss mechanisms are considered in this study; the conduction through the regenerator wall, the regenerator matrix and the pulse tube wall, the shuttle heat loss along the pulse tube and the ineffectiveness loss of the regenerator. Among them, the shuttle heat loss is dominant at low frequency region and the ineffectiveness loss is dominant at resonant frequency region. In the evaluation of the cooling capacity of Stirling-type PTR, the PV power at the cold-end and the ineffectiveness loss are mostly important because it is generally driven near its resonant frequency. They are evaluated with Eqs. (12) and (13). Here, I e means the ineffectiveness of the regenerator and is determined from the mass flow rate and the heat transfer characteristics of the regenerator. The COP is calculated using Eq. (14).
We have designed and fabricated the Stirling-type PTR using a double-acting linear compressor which was originally designed for a Stirling cryocooler. Its maximum input power is approximately 200 W. The #200 twill type stainless steel mesh is used as a regenerating material. The cold-end and warm-end heat exchanger is fabricated with the copper foam of 600 lm nominal pore size. Fig. 3a shows the cross-sectional view of the linear compressor and the pressure sensor installation. Fig. 3b and c shows the installation of pressure and temperature sensors for instrumentation in the developed PTR. The specifications and the operating parameters of the linear compressor and fabricated PTR are given in Table 1. All parameters of the linear compressor were obtained from the experimental measurements. Especially, the mechanical damping coefficient is not a constant value but almost reversely proportional to the velocity of the piston. It implies that the mechanical damping mechanism is not sliding friction but Coulomb damping. In this study, the inertance tube with a reservoir is used as a phase control device. If the operating parameters and other geometric configurations are fixed, the length of inertance tube solely determines the flow impedance of the PTR and thus affects on the dynamics of a piston. Experiments were performed with 105, 155 and 200 cm of the inertance tube length to investigate the effect of inertance tube on the dynamic behavior of the linear compressor and the cooling performance of the PTR. 3.2. Actual hardware aspects of the Stirling-type PTR During the experiments, we observed that some physical phenomena were not easily explained by simple models due to the specific configuration of the developed Stirling-type PTR and the inaccuracy of the developed empirical correlations. To consider these actual aspects of the Stirling-type
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a
To regenerator
Copper block
Compressor body
Copper mesh Pressure sensor P1
Tc
Piston
Piston
Compression space
b
Tl
regenerator
Tw
P2
Pr
pulse tube cold-end H.X.
warm-end H.X.
inertance tube
reservoir
c P2 Straightener Tee fitting
from pulse tube
1/8 inch tube
to reservoir
Copper foam
Fig. 3. Schematics and sensor installation of developed Stirling-type PTR (a) cross-sectional view of linear compressor (b) sensor installation in PTR and (c) detail view of warm-end heat exchanger and pressure sensor installation.
Table 1 Specifications and operating parameters Linear compressor mp km cm L0 Dp KE
0.433 kg 2,687 N/m 7:47ðxX 0 Þ0:96 kg/s 0.013 m 0.027 m 7.472 N/A
Geometric parameters Regenerator size Regenerator void fraction Pulse tube Inertance tube Reservoir volume
/19:05 60 mm 0.68 /12:7 60 mm /1=8 inch tube 340 106 m3
Operating parameters Charging pressure Warm-end temperature Cold-end temperature Input current
15 atm 300 K 100 K 8.5 Arms
PTR, we made some corrections in the analysis code on the basis of the measured data from various experiments.
3.2.1. Eddy current The eddy current is always generated in the iron core of a linear compressor. When the AC current is applied to the linear compressor in the absence of a piston, the resultant voltage is determined by only the electro-magnetic characteristics of a linear motor and so represented with Eq. (16). The voltage due to eddy current can be regarded as the additional resistive and inductive voltage and then, the apparent resistance and inductance of a motor coil are the function of the magnitude and frequency of the input current. They are determined from the measured waveforms of the voltage and current in a coil in the absence of the piston v ¼ Rc i þ L c
di di þ veddy ¼ Rapp i þ Lapp dt dt
ð16Þ
3.2.2. Pressure drop effect As shown in Fig. 3a, there is a sudden area change through the flow path between the compression space and the pressure measuring point 1. The measured data of pressure at the point 1 are always smaller than the
J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
required values for satisfying Eq. (2). That means that there exists the flow resistance between the compression space and the pressure measuring point 1 due to sudden area change. There is another undesired flow resistance between the warm-end heat exchanger and the pressure measuring point 2 due to sudden area change as shown in Fig. 3c. The amount of pressure drop as the flow rate was measured from the experiment with the steady flow of gaseous nitrogen. These undesired pressure drop are considered as the flow resistance in the analysis code and its value is determined on the basis of the experimental results. 3.2.3. Empirical correlations Because the flow impedance of the regenerator and inertance tube mainly determine the overall flow impedance of the PTR, the accuracy of the analysis code strongly depends on the empirical correlations for the friction factor and the inertance effect. The friction factor for the regenerator is referred to the experimental results of Nam and Jeong [7]. They measured the friction factor of both steady and oscillating flows with the same regenerator and the ratio of both friction factor slightly varies as Reynolds number. Fig. 4a shows the ratio of the measured friction factor to the calculated value by Nam and Jeong’s correlation for the steady flow. It is
a Ratio of meased data to reference
2.0
1.8
1.6
1.4
1.2
1.0 5
10
15
20
25
30
35
40
45
50
b
1.1
Ratio of values from experiment to values from reference
Reynolds number
1.0 CF for flow resistance IT = 105 cm IT = 155 cm IT = 200 cm CF for inertance effect IT = 105 cm IT = 155 cm IT = 200 cm
0.9 0.8 0.7 0.6 0.5 0.4
30
35
40
45
50
Frequency, Hz
Fig. 4. Correction factor (a) steady flow friction factor ratio of experiment to reference and (b) ratio of experimentally calculated value to reference.
73
observed that Nam and Jeong’s correlation underestimates the pressure drop. It could be due to the difference of the geometric configuration of the regenerator. The calculated ratio is used as the correction factor for the friction factor of the regenerator in this study. Correction factors for the flow resistance and inertance effect of the inertance tube can be derived for the measured data of the pressure at the inlet of inertance tube and at the reservoir. By using Eqs. (8)–(10), we can directly calculate the flow resistance and inertance effect of inertance tube from the measured data of p2 and pr. We obtained the ratio of the calculated value from the experimental results to the calculated value by the suggested relation of the reference. It is used as the correction factor for the flow resistance and inertance effect of the ineratance tube. Fig. 4b shows the used correction factors in the analysis code. The correction factors listed so far in this study would not exactly predict the phenomena, but it can clearly describe the physical situation of the experiment more realistically. 4. Results and discussion The flow impedance of the PTR varies with the inertance tube length, and so does the piston dynamics. The shorter inertance tube has smaller impedance, and the longer inertance tube has larger impedance. If the inertance tube length is infinite, the PTR acts like as a basic type PTR and the phase difference between the pressure and the mass flow rate would be 90° at the cold-end. Using the developed linear analysis code, we investigated the effect of inertance tube length on dynamics of the piston, the thermo-hydraulic behavior of the PTR and the cooling performance of a cryocooler. And the simulation results are compared with the experimental results. The amplitude of the input current is clearly limited for a given linear motor due to the Joule heat generation in a coil. In this study, the input current is 8.5 A in RMS value and the cold-end temperature is 100 K. Fig. 5 shows the frequency response of the piston displacement. The vibration of a piston generates the pressure wave and the oscillating flow in the compression space. The rate of pressurization and oscillating flow varies as the flow impedance of the PTR. The longer inertance tube length tends to suppress the mass flow rate but causes larger pressurization ratio. These phenomena result in stronger gas spring effect and weaker gas damping effect, and thus, the resonant frequency becomes higher and the maximum piston displacement amplitude increases. Fig. 6 shows the results of the resultant voltage and input power. For the longer inertance tube, the voltage amplitude increases and the larger variation of phase shift is observed. It can be explained by the back EMF voltage due to piston movement. As shown in Eq. (1), the voltage is affected by the back EMF even for the same input current. Although the simulation result of the phase shift is slightly higher than the experimental result, the analytic model can
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J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
a
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
5.2 5.0
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
a 38
36
4.8
V0 , V
X0, mm
4.6 4.4
34
32
4.2 4.0
30
3.8 30
35
40
45
25
50
30
Frequency, Hz
b
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
-60 -70
35
40
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
55
50
φv, deg
φx, deg
-100
50
60
b
-80 -90
45
Frequency, Hz
45
40
-110 35
-120 30 25
-130 25
30
35
40
45
30
50
Frequency, Hz Fig. 5. Results of piston displacement (a) amplitude and (b) phase shift.
35
40
45
50
Frequency, Hz
c
190
Input power Win, W
180
predict its behavior. Fig. 6c shows the result of the input power which is calculated from the measured voltage and current. In spite of the same input current, the input power varies as the length of the inertance tube. The longer inertance tube leads the larger stroke of a piston, and thus allows the linear compressor to accept the larger power. Fig. 7 shows the results of the pressure at each measuring point. Generally, the simulation results can explain well the behavior of the experimental results. As predicted above, the analytic model does not precisely predict the pressure variations. This is due to the inaccuracy of the correlations for the oscillating flow friction even though the correction factors are introduced. If the correlation accuracy is improved, the developed analytic model would predict the real phenomena of Stirling-type PTR more accurately. Fig. 8 shows the results of the cooling capacity and COP. For the longer inertance tube, the cooling capacity and COP are maximized at the lower frequency while the piston is resonated at the higher frequency. The simulation results show large variations in its value while the experimental results do not. Although it cannot show good agreement in its accuracy, the analysis can explain reasonably well the behavior of the cooling performance and COP of the PTR.
170 160 Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
150 140 130 25
30
35
40
45
50
Frequency, Hz Fig. 6. Results of voltage and input power (a) amplitude of voltage (b) phase shift of voltage and (c) input power.
The input power, the cooling capacity and the COP are not maximized at the resonant frequency because they are affected by the thermo-hydraulic behavior of the PTR as well as the dynamic response of the linear compressor. If the configuration of the PTR is optimized, they would be maximized at the resonant frequency. In that situation, we can maximize at the resonant frequency the power transfer rate from the electric input power to the PV power at the expansion space. The developed analysis code in this paper can be useful for finding such an optimum design point of Stirling-type PTR.
J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
a
b
220
75 Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
160
210
150
200
P20, kPa
P10, kPa
140 190 180
130
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
170 160 150 25
30
35
40
120
110
45
100 25
50
30
35
Frequency, Hz
40
45
50
Frequency, Hz φp1
-30
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp IT =105 cm Exp IT =155 cm Exp IT =200 cm
φp1
-40
φp1 and φp2, deg
-50
d
7.0
φp2
φp2
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp IT =105 cm Exp IT =155 cm Exp IT =200 cm
-60 -70 -80
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
6.5 6.0 5.5 5.0
r0
-20
P , kPa
c
-90
4.5
-100
4.0
-110 3.5
-120 -130 25
30
35
40
45
50
3.0 25
55
30
35
e
40
45
50
Frequency, Hz
Frequency, Hz 200 150 100
φpr, deg
50 0
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
-50 -100 -150 -200 25
30
35
40
45
50
Frequency, Hz Fig. 7. Results of pressure (a) pressure amplitude at point 1 (b) pressure amplitude at point 2 (c) phase shift of pressure at point 1 and 2 (d) pressure amplitude at reservoir and (e) phase shift of pressure at reservoir.
5. Conclusions The linear analytic model on a Stirling-type PTR is developed in conjunction with the specific dynamics of the linear compressor as well as thermo-hydraulic characteristics of the PTR and it is experimentally verified. The developed analytic model can explain reasonably well the behavior of all physical variables and power transfer of
the Stirling-type PTR. It is expected that more accurate empirical correlations for pressure drop and phase shift of the regenerator and inertance tube will increase the accuracy of the developed analytic model in this study. The effect of inertance tube length on a Stirling-type PTR is investigated. The length of inertance tube is the main parameter to determine the overall flow impedance of the PTR. The longer inertance tube gives the larger flow
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a
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
5
Cooling capacity Qref, W
4
3
2
1
With the simulation and experimental results in this paper, the frequency at which maximum cooling performance occurs, does not coincide with the resonant frequency of the linear compressor. The optimum design of Stirling-type PTR will be obtained by matching these two frequencies and show the improved cooling performance. The PTR analysis considering the dynamics of the linear compressor as well as the thermo-hydraulics of the PTR surely demonstrates a new design approach for development of better Stirling-type PTR. Acknowledgements
0
30
35
40
45
Frequency, Hz
b
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
0.030
0.025
References
COP
0.020
0.015
0.010
0.005
0.000 25
This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the National Research Lab. Program funded by the Ministry of Science and Technology (R0A – 2007 – 000 – 20062 – 0).
30
35
40
45
50
Frequency, Hz Fig. 8. Results of cooling capacity and COP (a) cooling capacity and (b) COP.
impedance to the PTR and thus, imposes the larger gas spring effect and the smaller gas damping effect to the dynamic system of the linear compressor.
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Cryogenics 48 (2008) 68–76 www.elsevier.com/locate/cryogenics
Analysis on the stirling-type pulse tube refrigerator in consideration of dynamics of linear compressor Junseok Ko *, Sangkwon Jeong 1 Cryogenic Engineering Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea Received 12 March 2007; received in revised form 5 November 2007; accepted 12 December 2007
Abstract This paper describes the performance analysis of Stirling-type pulse tube refrigerator (PTR) in conjunction with the dynamics of the accompanied linear compressor. The dynamic behavior of the piston in the linear compressor is directly influenced by the load condition of the PTR. In this paper, the dynamic equation of the piston is simultaneously solved with the thermo-hydraulic governing equations of the PTR using linear analysis model and the performance of the PTR is predicted with the accompanied thermal losses. The developed analysis code is verified with the experimental results. The effect of the inertance tube length which plays an important role in the PTR is also specifically investigated from the experimental and simulation results. It clearly shows the effect of the flow impedance of the inertance tube on the dynamic response of the piston as well as the cooling performance of the PTR. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Pulse tube; Inertance tube; Linear compressor
1. Introduction Recent progress in engineering-grade high temperature superconductor (HTS) materials demonstrates more feasibility for practical applications with its easier cooling method than low temperature superconductor (LTS). Those applications include superconducting motor, generator, cable, flywheel energy storage system, magnetic energy storage system, fault current limiter, etc. There is no doubt that an efficient cryocooler with proper auxiliary cooling interface is an indispensable part of realizing those attractive technologies. In fact, the feasibility of successful commercial market penetration of HTS system is deeply connected to efficient cryogenic cooling technology as long as superconducting materials require low temperature. For attractive compact integration of HTS system, an imple*
Corresponding author. Tel.: +82 42 869 3079; fax: +82 42 869 8207. E-mail addresses: [email protected] (J. Ko), [email protected] (S. Jeong). 1 Fax: +82 42 869 3210. 0011-2275/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2007.12.002
mented cryocooler must be efficient and reliable. A Striling cryocooler or Stirling-type PTR (pulse tube refrigerator) is generally suitable with high thermal efficiency for small-size application due to its overall low-volume characteristic. In order to design an efficient Stirling-type PTR, we need to accurately predict its cooling performance along with the accompanied behavior of the linear compressor which drives the cooler. Most previous analytic models on PTR involve the assumption of piston displacement or temporal swept volume as a function of time [1–4]. Then, the thermodynamic performance of the PTR was analyzed according to this hypothesized condition of the compressor. This analysis is useful if the piston movement is perfectly replicated as designed and presumed by adjusting the parameters of the compressor through the tuning process. But, the linear compressor has limitations in the piston displacement due to its inside geometry and the allowable input current due to Joule heating in the coil, and its resonant characteristic varies as the pressure wave is generated at the compression space. These characteristics of a linear
J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
69
Nomenclature A Aff cm d f fosc i Ie km KE Lc L0 mp m_ n P Q_ R Rc T v V W_ x
cross-sectional area (m2) effective area of regenerator (m2) mechanical damping coefficient (kg/s) diameter (m) frequency (s1) oscillating friction factor () current (A) ineffectiveness of regenerator () mechanical spring constant (N/m) motor constant (N/A) electric inductance of coil (H) neutral length of cylinder (m) mass of piston (kg) mass flow rate (kg/s) polytropic constant () pressure (kPa) heat rate (W) gas constant (J/kg-K) electric resistance of coil (X) temperature (K) voltage (V) volume (m3) work rate (W) Piston displacement (m)
Greek symbols / phase angle c specific heat ratio q density l viscosity
compressor may cause the failure of the presumed piston displacement or the poor compression efficiency with the predesigned PTR configuration. For high overall efficiency of a cryocooler, the power transmission efficiency in the PTR as well as in the linear compressor should be maximized simultaneously. For these reasons, the analytic model on PTR with consideration of the dynamics of a linear compressor should be developed for good design of a Stirling-type PTR. In a linear compressor, the linear motion of a piston simultaneously generates the pulsating pressure and the oscillating flow. In contrast to GM-type PTR, a Stirling-type PTR is directly connected to a linear compressor and thus, the generated pressure wave at the compression space strongly affect on the piston dynamics. To fully understand the physical coupling between the piston dynamics and the PTR’s performance, we have to consider those driving and driven mechanical parts of the refrigerator at the same time. This analytic methodology can explain the characteristics of the cryocooler more realistically than the previous models that usually assumed piston displacement, swept volume, or pressure wave at the compression space.
Subscripts ca interface between the compression space and the aftercooler ar interface between the aftercooler and the regenerator rl interface between the regenerator and the coldend heat exchanger le interface between the cold-end heat exchanger and the pulse tube w interface between the pulse tube and the warmend heat exchanger i interface between the warm-end heat exchanger and the inertance tube res interface between the inertance tube and the reservior a aftercooler reg regenerator lhx cold-end heat exchanger pt pulse tube hhx warm-end heat exchanger it inertance tube m mean value p piston r pressure at the reservoir 1 pressure at the compression space 2 pressure at the pulse tube
In this paper, the analytic model is developed with linear analysis in which the waveform of each variable is assumed purely sinusoidal. To verify the analytic model, the Stirling-type PTR has been designed and fabricated using a linear compressor which was originally designed for Stirling cryocooler. Experiments have been performed with three different lengths of inertance tube. With the experimental and analysis results, the thermo-hydraulic characteristics and the cooling performance of a Stirling-type PTR are discussed in relation to the dynamic behavior of a linear compressor. Especially, the role of an inertance tube is deeply discussed. This study is expected to suggest a guideline for better design of highly efficient Stirling-type PTR. 2. Analytic model Fig. 1 is the schematic diagram of a Stirling-type PTR with an inertance tube as a phase control device. The piston of a linear compressor, which is driven by a linear motor, oscillates in the compression space and, then, the motion of the piston generates the pressure wave and the oscillating gas flow to the regenerator, the pulse tube, and the
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compression space
pulse tube
aftercooler . mle
Pm
. mw p2
p1
piston
inertance tube
. mca
. mar regenerator
. mrl
pr . mi
. mres reservoir
cold-end heat exchanger
warm-end heat exchanger
Fig. 1. Schematic diagram of a Stirling-type pulse tube refrigerator.
phase control device. For a given current input to the linear motor, the resultant operating variables, which are the voltage of the linear motor, the piston displacement, the pressure and the mass flow rate at each interface, can be determined from thermo-hydraulic and dynamic relations between the components. 2.1. Governing equations 2.1.1. Linear compressor In a linear compressor, the required voltage for the input current is determined by the electric characteristics of a linear motor and the piston displacement as shown in Eq. (1). The mechanical spring and damping forces, the gas force by the pressure difference between both sides of the piston and the driving force act on the piston. These forces are balanced as represented by Eq. (2). The gas force plays an important role in determining the motion of the piston. The pressure difference between both sides of the piston imposes an additional stiffness and the PV power transmitted to the PTR space can be considered as a damping effect in the dynamic system of the piston. Theses are called as the gas spring and the gas damping effects, respectively. The gas spring effect shifts the resonant frequency to the higher value than the pure natural frequency of the mechanical piston configuration and the gas damping effect resultantly attenuates the piston stroke. In a previous research, the gas effect was described with the gas spring constant and the gas damping coefficient as verified by the experiments [5]. Eq. (3) is the relation among the pressure, the piston displacement and the mass flow rate at the compression space and is derived from the mass and energy conservation equations with the assumption of adiabatic compression and expansion processes and no leakage through the clearance gap [6]. We can also linearize Eq. (3) as Eq. (4) by considering only fundamental terms di dx þ KE dt dt d2 x dx mp 2 ¼ k m x cm þ ðP m P 1 ÞAp þ K E i dt dt Ap ðL0 xÞ dP 1 P 1 Ap dx ¼ m_ ca cRT ca dt RT ca dt Ap L0 dP 1 P m Ap dx ¼ m_ ca cRT ca dt RT ca dt
v ¼ Rc i þ Lc
2.1.2. Regenerator When the oscillating flow passes through the regenerator, the amplitudes of the pressure and the mass flow rate are both attenuated by the flow friction and the compliance effect respectively. These phenomena are governed by the mass and momentum conservation relations which are represented by Eqs. (5) and (6). The pressure drop through the regenerator is basically proportional to the mass flow rate if the compliant volume is not significant. The proportionality can be calculated with the regenerator geometry and the friction factor of the regenerator matrix [7]. In this paper, the discretized grid configuration is used for the regenerator and the inertance tube as shown in Fig. 2. It is assumed that the temperature profile is linear in the regenerator and the #200 twill mesh is packed as a regenerator material. Nam’s experimental result is used as the oscillating friction factor of the #200 twill mesh [7] om_ reg Aff oP reg ¼ ox RT reg ot oP reg ¼ Rreg m_ reg ox
ð5Þ ð6Þ
where Rreg ¼ fosc
jm_ reg j 2 ; qr A2ff dh
f osc ¼ 36:55=Rereg þ 0:16
2.1.3. Pulse tube The relation between the pressure and the mass flow rate in the pulse tube is derived from the mass conservation and the energy balance with the assumption of uniform pressure within the pulse tube as follows [8] m_ le m_ w V pt dP 2 ¼ qc qw cP 2 dt
ð7Þ
2.1.4. Inertance tube and reservoir The inertance tube and the reservoir are to control the phase shift between the pressure and the mass flow rate at the cold-end. The narrow and long tube gives the flow resistance and the inductance to the oscillating flow. As shown in Eqs. (8) and (9), the flow resistance through the inertance tube can be calculated by the oscillating flow model [6]. The relation between the mass flow rate to the reservoir and the reservoir pressure is derived from the
0
ð1Þ
1
2
i-1
i
i+1
n-1
n
ð2Þ 0
ð3Þ ð4Þ
1
2
i-1
i
i+1
n-1
n
n+1
Gas temperature, pressure Mass flow rate
Fig. 2. Grid configuration for regenerator and inertance tube.
J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
mass conservation in the reservoir and represented by Eq. (10). Here, the polytropic constant n is between 1 and c and determined as the compression/expansion process in the reservoir. In this study, the operating frequency is between 30 and 45 Hz and thus, the process in the reservoir would not be exactly the adiabatic process but would be close to it. It is assumed 1.5 of polytropic constant om_ it Ait dP it ¼ ox RT h dt oP it 1 dm_ it ¼ Rit m_ it þ Ait dt ox
ð8Þ
During the operating states of Stirling-type PTR, all physical variables are assumed to have cyclic waveforms for the given sinusoidal current input as represented with Eq. (15). The phase shift of each variable is represented by the phase angle difference from that of the input current I I C p m_ le f Cp p2 p2 m_ le dt dt ffi ð12Þ W_ P dVe ¼ f qm R R q2 Q_ ineff ¼ I e m_ reg ðT a T l Þ ð13Þ
ð9Þ COP ¼
where ait 0:201 Rit ¼ ; ait ¼ 0:1556ðjm_ it jd it =lAit Þ ðjm_ it j=qh d it Ait Þ Ait V res dP r ð10Þ m_ res ¼ nRT h dt
71
Q_ ref W_ in
aðtÞ ffi Am þ A0 cosðxt þ /a Þ
ð14Þ ð15Þ
3. Experiments 3.1. Experimental setup and experiments
2.1.5. Heat exchangers Three heat exchangers (i.e. the aftercooler, the cold-end heat exchanger and the warm-end heat exchanger) are used in a PTR. All heat exchangers are assumed to be isothermal. Then, Eq. (11) is readily derived from the mass conservation relation m_ in m_ out ¼
V dP RT dt
ð11Þ
2.2. Performance analysis and thermal losses In the analysis, the piston displacement, required voltage, pressure and mass flow rates are calculated from the governing equations. And then, the input power, the compression power, the PV power at the cold-end and the thermal losses are evaluated from the calculated results. If there is no thermal loss, the PV power at the cold-end is purely used to generate the refrigeration effect. But, there must be some inherent thermal losses in the real situation. Thus, the remainder of the PV power after being compensated by the thermal losses only produces the actual cooling capacity of the PTR. Five thermal loss mechanisms are considered in this study; the conduction through the regenerator wall, the regenerator matrix and the pulse tube wall, the shuttle heat loss along the pulse tube and the ineffectiveness loss of the regenerator. Among them, the shuttle heat loss is dominant at low frequency region and the ineffectiveness loss is dominant at resonant frequency region. In the evaluation of the cooling capacity of Stirling-type PTR, the PV power at the cold-end and the ineffectiveness loss are mostly important because it is generally driven near its resonant frequency. They are evaluated with Eqs. (12) and (13). Here, I e means the ineffectiveness of the regenerator and is determined from the mass flow rate and the heat transfer characteristics of the regenerator. The COP is calculated using Eq. (14).
We have designed and fabricated the Stirling-type PTR using a double-acting linear compressor which was originally designed for a Stirling cryocooler. Its maximum input power is approximately 200 W. The #200 twill type stainless steel mesh is used as a regenerating material. The cold-end and warm-end heat exchanger is fabricated with the copper foam of 600 lm nominal pore size. Fig. 3a shows the cross-sectional view of the linear compressor and the pressure sensor installation. Fig. 3b and c shows the installation of pressure and temperature sensors for instrumentation in the developed PTR. The specifications and the operating parameters of the linear compressor and fabricated PTR are given in Table 1. All parameters of the linear compressor were obtained from the experimental measurements. Especially, the mechanical damping coefficient is not a constant value but almost reversely proportional to the velocity of the piston. It implies that the mechanical damping mechanism is not sliding friction but Coulomb damping. In this study, the inertance tube with a reservoir is used as a phase control device. If the operating parameters and other geometric configurations are fixed, the length of inertance tube solely determines the flow impedance of the PTR and thus affects on the dynamics of a piston. Experiments were performed with 105, 155 and 200 cm of the inertance tube length to investigate the effect of inertance tube on the dynamic behavior of the linear compressor and the cooling performance of the PTR. 3.2. Actual hardware aspects of the Stirling-type PTR During the experiments, we observed that some physical phenomena were not easily explained by simple models due to the specific configuration of the developed Stirling-type PTR and the inaccuracy of the developed empirical correlations. To consider these actual aspects of the Stirling-type
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a
To regenerator
Copper block
Compressor body
Copper mesh Pressure sensor P1
Tc
Piston
Piston
Compression space
b
Tl
regenerator
Tw
P2
Pr
pulse tube cold-end H.X.
warm-end H.X.
inertance tube
reservoir
c P2 Straightener Tee fitting
from pulse tube
1/8 inch tube
to reservoir
Copper foam
Fig. 3. Schematics and sensor installation of developed Stirling-type PTR (a) cross-sectional view of linear compressor (b) sensor installation in PTR and (c) detail view of warm-end heat exchanger and pressure sensor installation.
Table 1 Specifications and operating parameters Linear compressor mp km cm L0 Dp KE
0.433 kg 2,687 N/m 7:47ðxX 0 Þ0:96 kg/s 0.013 m 0.027 m 7.472 N/A
Geometric parameters Regenerator size Regenerator void fraction Pulse tube Inertance tube Reservoir volume
/19:05 60 mm 0.68 /12:7 60 mm /1=8 inch tube 340 106 m3
Operating parameters Charging pressure Warm-end temperature Cold-end temperature Input current
15 atm 300 K 100 K 8.5 Arms
PTR, we made some corrections in the analysis code on the basis of the measured data from various experiments.
3.2.1. Eddy current The eddy current is always generated in the iron core of a linear compressor. When the AC current is applied to the linear compressor in the absence of a piston, the resultant voltage is determined by only the electro-magnetic characteristics of a linear motor and so represented with Eq. (16). The voltage due to eddy current can be regarded as the additional resistive and inductive voltage and then, the apparent resistance and inductance of a motor coil are the function of the magnitude and frequency of the input current. They are determined from the measured waveforms of the voltage and current in a coil in the absence of the piston v ¼ Rc i þ L c
di di þ veddy ¼ Rapp i þ Lapp dt dt
ð16Þ
3.2.2. Pressure drop effect As shown in Fig. 3a, there is a sudden area change through the flow path between the compression space and the pressure measuring point 1. The measured data of pressure at the point 1 are always smaller than the
J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
required values for satisfying Eq. (2). That means that there exists the flow resistance between the compression space and the pressure measuring point 1 due to sudden area change. There is another undesired flow resistance between the warm-end heat exchanger and the pressure measuring point 2 due to sudden area change as shown in Fig. 3c. The amount of pressure drop as the flow rate was measured from the experiment with the steady flow of gaseous nitrogen. These undesired pressure drop are considered as the flow resistance in the analysis code and its value is determined on the basis of the experimental results. 3.2.3. Empirical correlations Because the flow impedance of the regenerator and inertance tube mainly determine the overall flow impedance of the PTR, the accuracy of the analysis code strongly depends on the empirical correlations for the friction factor and the inertance effect. The friction factor for the regenerator is referred to the experimental results of Nam and Jeong [7]. They measured the friction factor of both steady and oscillating flows with the same regenerator and the ratio of both friction factor slightly varies as Reynolds number. Fig. 4a shows the ratio of the measured friction factor to the calculated value by Nam and Jeong’s correlation for the steady flow. It is
a Ratio of meased data to reference
2.0
1.8
1.6
1.4
1.2
1.0 5
10
15
20
25
30
35
40
45
50
b
1.1
Ratio of values from experiment to values from reference
Reynolds number
1.0 CF for flow resistance IT = 105 cm IT = 155 cm IT = 200 cm CF for inertance effect IT = 105 cm IT = 155 cm IT = 200 cm
0.9 0.8 0.7 0.6 0.5 0.4
30
35
40
45
50
Frequency, Hz
Fig. 4. Correction factor (a) steady flow friction factor ratio of experiment to reference and (b) ratio of experimentally calculated value to reference.
73
observed that Nam and Jeong’s correlation underestimates the pressure drop. It could be due to the difference of the geometric configuration of the regenerator. The calculated ratio is used as the correction factor for the friction factor of the regenerator in this study. Correction factors for the flow resistance and inertance effect of the inertance tube can be derived for the measured data of the pressure at the inlet of inertance tube and at the reservoir. By using Eqs. (8)–(10), we can directly calculate the flow resistance and inertance effect of inertance tube from the measured data of p2 and pr. We obtained the ratio of the calculated value from the experimental results to the calculated value by the suggested relation of the reference. It is used as the correction factor for the flow resistance and inertance effect of the ineratance tube. Fig. 4b shows the used correction factors in the analysis code. The correction factors listed so far in this study would not exactly predict the phenomena, but it can clearly describe the physical situation of the experiment more realistically. 4. Results and discussion The flow impedance of the PTR varies with the inertance tube length, and so does the piston dynamics. The shorter inertance tube has smaller impedance, and the longer inertance tube has larger impedance. If the inertance tube length is infinite, the PTR acts like as a basic type PTR and the phase difference between the pressure and the mass flow rate would be 90° at the cold-end. Using the developed linear analysis code, we investigated the effect of inertance tube length on dynamics of the piston, the thermo-hydraulic behavior of the PTR and the cooling performance of a cryocooler. And the simulation results are compared with the experimental results. The amplitude of the input current is clearly limited for a given linear motor due to the Joule heat generation in a coil. In this study, the input current is 8.5 A in RMS value and the cold-end temperature is 100 K. Fig. 5 shows the frequency response of the piston displacement. The vibration of a piston generates the pressure wave and the oscillating flow in the compression space. The rate of pressurization and oscillating flow varies as the flow impedance of the PTR. The longer inertance tube length tends to suppress the mass flow rate but causes larger pressurization ratio. These phenomena result in stronger gas spring effect and weaker gas damping effect, and thus, the resonant frequency becomes higher and the maximum piston displacement amplitude increases. Fig. 6 shows the results of the resultant voltage and input power. For the longer inertance tube, the voltage amplitude increases and the larger variation of phase shift is observed. It can be explained by the back EMF voltage due to piston movement. As shown in Eq. (1), the voltage is affected by the back EMF even for the same input current. Although the simulation result of the phase shift is slightly higher than the experimental result, the analytic model can
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a
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
5.2 5.0
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
a 38
36
4.8
V0 , V
X0, mm
4.6 4.4
34
32
4.2 4.0
30
3.8 30
35
40
45
25
50
30
Frequency, Hz
b
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
-60 -70
35
40
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
55
50
φv, deg
φx, deg
-100
50
60
b
-80 -90
45
Frequency, Hz
45
40
-110 35
-120 30 25
-130 25
30
35
40
45
30
50
Frequency, Hz Fig. 5. Results of piston displacement (a) amplitude and (b) phase shift.
35
40
45
50
Frequency, Hz
c
190
Input power Win, W
180
predict its behavior. Fig. 6c shows the result of the input power which is calculated from the measured voltage and current. In spite of the same input current, the input power varies as the length of the inertance tube. The longer inertance tube leads the larger stroke of a piston, and thus allows the linear compressor to accept the larger power. Fig. 7 shows the results of the pressure at each measuring point. Generally, the simulation results can explain well the behavior of the experimental results. As predicted above, the analytic model does not precisely predict the pressure variations. This is due to the inaccuracy of the correlations for the oscillating flow friction even though the correction factors are introduced. If the correlation accuracy is improved, the developed analytic model would predict the real phenomena of Stirling-type PTR more accurately. Fig. 8 shows the results of the cooling capacity and COP. For the longer inertance tube, the cooling capacity and COP are maximized at the lower frequency while the piston is resonated at the higher frequency. The simulation results show large variations in its value while the experimental results do not. Although it cannot show good agreement in its accuracy, the analysis can explain reasonably well the behavior of the cooling performance and COP of the PTR.
170 160 Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
150 140 130 25
30
35
40
45
50
Frequency, Hz Fig. 6. Results of voltage and input power (a) amplitude of voltage (b) phase shift of voltage and (c) input power.
The input power, the cooling capacity and the COP are not maximized at the resonant frequency because they are affected by the thermo-hydraulic behavior of the PTR as well as the dynamic response of the linear compressor. If the configuration of the PTR is optimized, they would be maximized at the resonant frequency. In that situation, we can maximize at the resonant frequency the power transfer rate from the electric input power to the PV power at the expansion space. The developed analysis code in this paper can be useful for finding such an optimum design point of Stirling-type PTR.
J. Ko, S. Jeong / Cryogenics 48 (2008) 68–76
a
b
220
75 Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
160
210
150
200
P20, kPa
P10, kPa
140 190 180
130
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
170 160 150 25
30
35
40
120
110
45
100 25
50
30
35
Frequency, Hz
40
45
50
Frequency, Hz φp1
-30
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp IT =105 cm Exp IT =155 cm Exp IT =200 cm
φp1
-40
φp1 and φp2, deg
-50
d
7.0
φp2
φp2
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp IT =105 cm Exp IT =155 cm Exp IT =200 cm
-60 -70 -80
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
6.5 6.0 5.5 5.0
r0
-20
P , kPa
c
-90
4.5
-100
4.0
-110 3.5
-120 -130 25
30
35
40
45
50
3.0 25
55
30
35
e
40
45
50
Frequency, Hz
Frequency, Hz 200 150 100
φpr, deg
50 0
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
-50 -100 -150 -200 25
30
35
40
45
50
Frequency, Hz Fig. 7. Results of pressure (a) pressure amplitude at point 1 (b) pressure amplitude at point 2 (c) phase shift of pressure at point 1 and 2 (d) pressure amplitude at reservoir and (e) phase shift of pressure at reservoir.
5. Conclusions The linear analytic model on a Stirling-type PTR is developed in conjunction with the specific dynamics of the linear compressor as well as thermo-hydraulic characteristics of the PTR and it is experimentally verified. The developed analytic model can explain reasonably well the behavior of all physical variables and power transfer of
the Stirling-type PTR. It is expected that more accurate empirical correlations for pressure drop and phase shift of the regenerator and inertance tube will increase the accuracy of the developed analytic model in this study. The effect of inertance tube length on a Stirling-type PTR is investigated. The length of inertance tube is the main parameter to determine the overall flow impedance of the PTR. The longer inertance tube gives the larger flow
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a
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
5
Cooling capacity Qref, W
4
3
2
1
With the simulation and experimental results in this paper, the frequency at which maximum cooling performance occurs, does not coincide with the resonant frequency of the linear compressor. The optimum design of Stirling-type PTR will be obtained by matching these two frequencies and show the improved cooling performance. The PTR analysis considering the dynamics of the linear compressor as well as the thermo-hydraulics of the PTR surely demonstrates a new design approach for development of better Stirling-type PTR. Acknowledgements
0
30
35
40
45
Frequency, Hz
b
Simulation IT =105 cm Simulation IT =155 cm Simulation IT =200 cm Exp. IT =105 cm Exp. IT =155 cm Exp. IT =200 cm
0.030
0.025
References
COP
0.020
0.015
0.010
0.005
0.000 25
This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the National Research Lab. Program funded by the Ministry of Science and Technology (R0A – 2007 – 000 – 20062 – 0).
30
35
40
45
50
Frequency, Hz Fig. 8. Results of cooling capacity and COP (a) cooling capacity and (b) COP.
impedance to the PTR and thus, imposes the larger gas spring effect and the smaller gas damping effect to the dynamic system of the linear compressor.
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