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Influence of resonance tube length on performance of thermoacoustically driven pulse tube refrigerator
Cryogenics 45 (2005) 185–191 www.elsevier.com/locate/cryogenics
Influence of resonance tube length on performance of thermoacoustically driven pulse tube refrigerator K. Tang, G.B. Chen *, T. Jin, R. Bao, B. Kong, L.M. Qiu Cryogenics Laboratory, Zhejiang University, Hangzhou 310027, PR China Received 10 May 2004; received in revised form 30 September 2004; accepted 2 October 2004
Abstract A resonance tube is an important component of a thermoacoustic engine, which has great influence on the performance of the thermoacoustically driven pulse tube refrigerator. A standing wave thermoacoustic engine is simulated with linear thermoacoustics. Computed results show that an appropriate accretion of the resonance tube length may lead to a decrease of the working frequency and an increase of the pressure amplitude, which will improve the match between the thermoacoustic engine and the pulse tube refrigerator. The theoretical prediction is verified by experiments. A refrigeration temperature as low as 88.6 K has been achieved with an optimized length of the resonance tube, helium as working gas, and 2200 W of heating power. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Pulse tube refrigerator; Thermoacoustic engine; Thermoacoustic refrigeration
1. Introduction A thermoacoustically driven pulse tube refrigerator occupies advantages, such as stability, reliability, longevity and so on, due to no moving components. This novel refrigeration system attracts great interest of academic and commercial fields, and considerable efforts have been made to improve its performance [1–5]. A thermoacoustic engine generates acoustic oscillation from heat energy, and delivers acoustic power with a high frequency (about 70 Hz) and a low pressure ratio (1.1 or so) by acoustic wave to a pulse tube refrigerator which conventionally works at a frequency of about 15– 20 Hz and a pressure ratio of 3 [5]. According to our investigations [3,5,6], it was found that the poor match between the thermoacoustic engine and the pulse tube refrigerator would dominantly baffle the performance improvement of the thermoacoustically driven pulse
*
Corresponding author. Tel./fax: +86 571 8795 1771. E-mail address: [email protected] (G.B. Chen).
0011-2275/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2004.10.002
tube refrigerator. In detail, the match is mainly related to the frequency and the pressure amplitude of the acoustic wave, which should meet the requirement of the pulse tube refrigerator. Although a refrigeration temperature of 115.4 K has been achieved in our previous work [5], the working frequency of 70 Hz is still too high and the pressure amplitude of 0.12 MPa is too small for the pulse tube refrigerator to realize a better refrigeration performance. The resonance tube is one of the key components of a thermoacoustic engine. A standing wave thermoacoustic engine is simulated with linear thermoacoustics to study the influence of the resonance tube on the thermoacoustically driven pulse tube refrigeration. Computed results show that an appropriate prolongation of the resonance tube may lead to a decrease of the working frequency and an increase of the pressure amplitude, which will improve the match between the thermoacoustic engine and the pulse tube refrigerator. Experiments of a thermoacoustically driven pulse tube refrigerator were performed to verify the theoretical prediction. Under the conditions of helium as working gas, 2.1 MPa charg-
186
K. Tang et al. / Cryogenics 45 (2005) 185–191
ing pressure and 2000 W heating power, the refrigeration temperature decreases from 112.8 K to 93.2 K, when the resonance tube is prolonged from 4 m to 8 m. The thermoacoustically driven pulse tube refrigeration is obviously enhanced by adjusting the length of the resonance tube. After further optimizing the openings of the orifice and the double inlet valves, a refrigeration temperature as low as 88.6 K has been attained with 8 m resonance tube and 2200 W heating power.
2. Simulation of thermoacoustic engine According to linear thermoacoustics [7,8], the momentum, continuity and energy equations for a short channel are as follows: dp1 ¼ ðixl þ rm ÞU 1 ; ð1Þ dx dU 1 1 ¼ ixc þ ð2Þ p þ eU 1 ; rj 1 dx fj f~ m 1 _ e H 2 2 Re p1 U 1 1 ð1þPrÞ 1f~ ð mÞ dT m ¼ ; 2 q c U j j p 1 m dx ~ 2 Im fj þ Pr f m ðAk þ Asolid k solid Þ 2 2Axð1Pr Þj1fm j ð3Þ l¼
qm 1 Reðfm Þ ; A j1 f m j2
ð4Þ
c¼
A ½1 þ ðc 1Þ Reðfj Þ; cpm
ð5Þ
xqm Im½fm ; A j1 fm j2
ð6Þ
1 c 1 xA Imðfj Þ ¼ ; rj c pm
ð7Þ
rm ¼
e¼
fj fm 1 dT m ; ð1 fm Þð1 PrÞ T m dx
and ksolid are cross section area and thermal conductivity of the solid forming the channel, H_ 2 is total power, i is imaginary unit, Re, Im and superscript mean the real part, the imaginary part and the conjugation of a complex quantity. l, c, rm, rj, and e are inertance, compliance, viscous resistance, thermal-relaxation resistance and proportionality coefficient of a controlled source, respectively, which can be calculated with Eqs. (4)–(8). A standing wave thermoacoustic engine, as shown in Fig. 1, is simulated with Eqs. (1)–(8). The main dimensions of the engine are tabulated in Table 1. As in experiments, working gas is helium, heating power and working pressure are 2000 W and 2.7 MPa, respectively. Screen stack is considered as capillary model. The local flow resistance due to the variation of the flow area is taken into account. In addition, the viscous resistance of the resonance tube is corrected according to the Ref. [8], because the oscillating flow in the resonance tube is in the range of turbulence flow in the computed case. The computed working frequency is presented in Fig. 2. It can be seen that the working frequency decreases with an increase of the resonance tube length. When the resonance tube is prolonged from 3 m to 10 m, the working frequency decreases from 82.8 Hz to 38.1 Hz, by 54.0%. This is helpful for the improvement of the frequency match between the thermoacoustic engine and the pulse tube refrigerator. Fig. 3 shows the relation of the computed velocity amplitude distribution in the thermoacoustic core (composed of a stack, a heater and a water cooler) versus the resonance tube length. 0–0.065 m, 0.065–0.350 m and 0.350–0.414 m of the axial location axes in Fig. 3 stand for the heater, the stack and the water cooler, respectively. We can see that the velocity amplitude increases from the heater to the water cooler with a certain length of the resonance tube, because the heater is closer to the velocity node. On the other hand, when the resonance tube is prolonged, the relative location of the thermoacoustic core shifts nearer to the velocity node,
ð8Þ
where p1 and U1 are pressure and velocity amplitudes, x is angular frequency, qm, Tm, cp, c, k and Pr are mean density, temperature, isobaric specific heat, specific heat ratio, thermal conductivity and Prandtl number of working fluid, respectively, fm and fj are viscous and thermal functions, A is flow area of the channel, Asolid
Fig. 1. Schematic of thermoacoustic engine.
Table 1 Dimensions of thermoacoustic engine
Diameter (mm) Length (mm)
Heater
Stack
Water cooler
Resonance tube
Hot buffer
54 65
56 285
56 64
36 4000–9000 (optional)
– (1.2 L)
K. Tang et al. / Cryogenics 45 (2005) 185–191
187
Pressure amplitude (MPa)
90
Frequency (Hz)
80 70 60 50 40 30
3 4 5 6 7 8 9 10 Length of resonance tube (m)
0.170 0.165 0.160 0.155 0.150 0.145 0.0 9 10 (m) 0.1 Axia 0.2 0.3 7 8 e tube 6 l loc nc atio 0.4 0.5 3 4 5 esona n (m of r h ) t g Len
Fig. 4. Computed pressure amplitude in thermoacoustic core versus length of resonance tube; Axial location: 0–0.065 m, 0.065–0.350 m and 0.350–0.414 m stand for the heater, the stack and the water cooler, respectively.
0.035 0.030 0.025
1.140
0.020
1.135
0.015 0.010 Len 3 4 0.5 gth 0.4 of re5 6 7 0.3 (m) 0.2 son n anc 8 910 0.0 0.1 ocatio e tu ial l be (
m)
Ax
Fig. 3. Computed velocity amplitude in thermoacoustic core versus length of resonance tube; Axial location: 0–0.065 m, 0.065–0.350 m and 0.350–0.414 m stand for the heater, the stack and the water cooler, respectively.
so the velocity amplitude in the thermoacoustic core decreases. The computed pressure amplitude and pressure ratio distributions in the thermoacoustic core are presented in Figs. 4 and 5. Both the pressure amplitude and the pressure ratio decrease from the heater to the water cooler, because the heater is closer to the pressure antinode. In addition, the thermoacoustic core approaches to the pressure antinode with an increase of the resonance tube length, which may lead to increases of both the pressure amplitude and the pressure ratio. However, when the resonance tube is further prolonged, the pressure amplitude and the pressure ratio begin to decrease. The analysis on the acoustic power losses may supply another explanation for the curves in Figs. 4 and 5 from the viewpoint of energy. Fig. 6 shows the computed relative acoustic power losses of components (the ratio of the acoustic power loss in one component to the acoustic power generation from the stack). It can be seen that the acoustic power loss in the thermoacoustic core is much higher than that of any other com-
Pressure ratio
Velocity amplitude (m 3/s)
Fig. 2. Computed working frequency versus length of resonance tube.
0.175
1.130 1.125 1.120 1.115 0.0 ) 0.1 9 10 e (m 7 8 ce tub Axia 0.2 0.3 6 0.4 l loc an 0.5 3 4 5 reson atio n (m of h t ) eng L
Fig. 5. Computed pressure ratio in thermoacoustic core versus length of resonance tube; Axial location: 0–0.065 m, 0.065–0.350 m and 0.350–0.414 m stand for the heater, the stack and the water cooler, respectively.
ponent and may dominate the behavior of the thermoacoustic system, when the resonance tube is short. In this case, a decrease of the acoustic power loss in the thermoacoustic core, induced by the decreases of working frequency and velocity amplitude with an increase of the resonance tube length, may enhance the oscillation in the thermoacoustic system and lead to increases of both the pressure amplitude and the pressure ratio. However, an increase of gas–solid interface, induced by the accretion of the resonance tube length, may result in increases of the viscous and the thermalrelaxation dissipations in the resonance tube. Both the pressure amplitude and the pressure ratio begin to decrease with an increase of the resonance tube length when the acoustic power loss in the resonance tube approaches to or even exceeds that in the thermoacoustic core. The temperature distribution in the thermoacoustic core is computed and presented in Fig. 7. The convective heat transfer in the stack is weakened due to the
K. Tang et al. / Cryogenics 45 (2005) 185–191
Relative acoustic power loss (%)
188
3. Experimental apparatus and results
70 60
1
3.1. Experimental apparatus
50 40 30
2
20 10 0
3 3 4 5 6 7 8 9 10 Length of resonance tube (m)
Fig. 6. Computed relative acoustic power loss versus length of resonance tube; (1) relative acoustic power loss in thermoacoustic core: the ratio of the acoustic power loss in the thermoacoustic core to the acoustic power generation in the stack; (2) relative acoustic power loss in resonance tube: the ratio of the acoustic power loss in the resonance tube to the acoustic power generation in the stack; (3) relative acoustic power loss in hot buffer: the ratio of the acoustic power loss in the hot buffer to the acoustic power generation in the stack.
Temperature (K)
800 700 600 500 400 300 0.0 10 (m) 0.1 8 9 tube Axia 0.20.3 7 e l loc 0.4 5 6 onanc atio 0.5 3 4 s n (m of re ) ength L
Fig. 7. Computed temperature in thermoacoustic core versus length of resonance tube; Axial location: 0–0.065 m, 0.065–0.350 m and 0.350– 0.414 m stand for the heater, the stack and the water cooler, respectively.
decreases of both the working frequency and the velocity amplitude with an accretion of the resonance tube length. Thus, the prolongation of the resonance tube may result in an increase of stackÕs hot end temperature with the same heating power. In a word, the computed results indicate that an appropriate increase of the resonance tube length may lead to a decrease of the working frequency and to increases of the pressure amplitude and the pressure ratio, which provides an approach to improve the match between the thermoacoustic engine and the pulse tube refrigerator. Besides, the hot end temperature of the stack may increase with an accretion of the resonance tube length, which is negative for the utilization of low-grade energy.
The experimental apparatus is composed of a thermoacoustic engine, a pulse tube refrigerator and a measuring system, as shown in Fig. 8. The pulse tube refrigerator is a coaxial single-stage double-inlet one, and its main dimensions are tabulated in Table 2. Temperature measuring locations are shown in Fig. 8. T1 and T4, T2 and T3 locate at the hot and the cold ends of the stacks, each side of the resonance tube, respectively. T1 and T4 are measured with NiCr–NiSi thermocouples, while T2 and T3 with Cu–constantan thermocouples. A Rh–Fe resistance thermometer (with 0.1 K accuracy) is applied to measure the refrigeration temperature at the cold end of the pulse tube. The pressure waveform at the joint of the thermoacoustic engine and the pulse tube refrigerator is measured by a piezoresistive silicon pressure sensor. All temperature and pressure data are collected and recorded by a PC-based digital acquisition system. 3.2. Experimental results and analyses on resonance tube length Experiments are carried out with helium as working gas, 2.1 MPa charging pressure, 2000 W heating power and 4–9 m resonance tube. The openings of the orifice and the double inlet valves in the pulse tube refrigerator are set as 330° and 420°, respectively, which are the optimized values based on our previous work [5]. The relation of the working frequency versus the resonance tube length is shown in Fig. 9. It can be seen that an accretion of the resonance tube length leads to a decrease of the working frequency. When the resonance
Fig. 8. Outline of thermoacoustically driven pulse tube refrigerator.
Table 2 Dimensions of pulse tube refrigerator Pulse tube and regenerator
Matrix
Reservoir
Coaxial, outer diameter of 18.5 mm, length of 110 mm
250 mesh, stainless steel
250 cm3
Frequency (Hz)
70 65
Experiment Computation
60 55 50 45
Pressure amplitude (MPa)
K. Tang et al. / Cryogenics 45 (2005) 185–191
40
0.17 0.16 0.15 0.14 0.13 0.12
4 5 6 7 8 9 Length of resonance tube (m)
tube is prolonged from 4 m to 9 m, the working frequency decreases from 69.9 Hz to 41.1 Hz, by 41.2%. The computed results are also presented in Fig. 9. The computed curve is almost consistent with the experimental data, and the errors are less than 5%. Fig. 10 shows the working pressure of the thermoacoustically driven pulse tube refrigerator. Since the resonance tube is always at room temperature, and the prolongation of the resonance tube results in lower proportion of the high-temperature volume in the whole system. Thus the working pressure decreases with the same charging pressure and heating power. The pressure amplitude and the pressure ratio at the joint are presented in Figs. 11 and 12. When the resonance tube is prolonged from 4 m to 8 m, the pressure amplitude and the pressure ratio increase from 0.116 MPa and 1.086 to 0.150 MPa and 1.121, by 29.3% and 3.2%, respectively. Nevertheless, in the case of the resonance tube length beyond 8 m, the pressure amplitude decreases with an accretion of the resonance tube length, while the pressure ratio still increases, due to the decreasing working pressure of the thermoacous2.85 2.80 2.75 2.70 2.65 2.60 4 5 6 7 8 9 Length of resonance tube (m)
Fig. 10. Working pressure versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
Fig. 11. Pressure amplitude versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
1.14 1.13
Pressure ratio
Fig. 9. Working frequency versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
Working pressure (MPa)
Experiment Computation
0.11
4 5 6 7 8 9 Length of resonance tube (m)
2.55
189
1.12 1.11 1.10 1.09 1.08
Experiment Computation
4 5 6 7 8 9 Length of resonance tube (m)
Fig. 12. Pressure ratio versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
tically driven pulse tube refrigerator, as shown in Fig. 10. We can see that the trends of computed curves are similar to those of the experimental curves in Figs. 11 and 12, but the computed values are markedly higher than the experimental data. There are two conceivable reasons. First, the computed values originate from the simulation only for the thermoacoustic engine, while the experimental data include the influence of the pulse tube refrigerator, which consumes acoustic power to realize refrigeration and attenuates the oscillation. Second, although thermal insulation is carried out for the components at high temperatures in experiments, the heat leak to the surroundings is unavoidable. Thus, the input power utilized effectively by the system is less than the computed heating power value of 2000 W. Total harmonic distortion (THD) [9] of the pressure wave is measured in experiments to analyze the influence of the resonance tube length on the harmonic components. Fig. 13 indicates that THD increases rapidly with an accretion of the resonance tube length, that is, the prolongation of the resonance tube results in the
K. Tang et al. / Cryogenics 45 (2005) 185–191
0.10
THD
0.08 0.06 0.04 0.02 0.00
4 5 6 7 8 9 Length of resonance tube (m)
Fig. 13. THD versus length of resonance tube; THD: total harmonic distortion; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
850
750
105 100 95 90
4 5 6 7 8 9 Length of resonance tube (m)
Fig. 14. Refrigeration temperature versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
420 390
700 360
650 600
330
550
300 4 5 6 7 8 9 Length of resonance tube (m)
Fig. 15. Temperatures of stackÕs hot and cold ends versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
115 110
450 Hot end Cold end
800
Temperature gradient (K/m)
Refrigeration temperature (K)
generation and the growth of harmonic components. In the case of 9 m resonance tube, THD is beyond 0.1, which means that the square root of the summation of squares of harmonic amplitudes is larger than 10% of the fundamental component. The generation and the growth of harmonic components waste the energy of the fundamental component, and attenuate the oscillation. This is another important reason for which the increasing rate of the pressure amplitude diminishes and even becomes negative with a longer resonance tube. The relation of the refrigeration temperature and the resonance tube length is shown in Fig. 14. When the resonance tube is prolonged from 4 m to 8 m, the refrigeration temperature decreases from 112.8 K to 93.2 K and the refrigeration performance is obviously enhanced, since the increases of the pressure amplitude and the pressure ratio with an accretion of the resonance tube length may induce an enhancement of the compression and the expansion processes in the pulse tube, which is beneficial for the pulse tube refrigeration. When the resonance tube is further prolonged to 9 m, the refrigera-
tion temperature begins to rise due to the decrease of the pressure amplitude. Meanwhile, a distortion of the pressure wave with longer high-pressure part and shorter low-pressure part is observed in this case, which may cause an insufficient expansion of the pulse tube refrigeration cycle and becomes another reason for the increase of the refrigeration temperature. Fig. 15 presents the temperatures at both ends of the stack. The cold end temperature of the stack is almost fixed at 330 K with an accretion of the resonance tube length, which indicates that the cold end temperature is almost not affected by the resonance tube length. However, as predicted by the forementioned simulation, the hot end temperature markedly rises with the prolonged resonance tube. The experimental and computed values of the mean temperature gradient over the stack are shown in Fig. 16. It can be seen that the computed curve is similar to the experimental one, but the slope of the computed curve is smaller. It is because that the working pressure of the thermoacoustically driven pulse
Cold end temperature (K)
0.12
Hot end temperature (K)
190
1700 1600
Experiment Computation
1500 1400 1300 1200 1100
4 5 6 7 8 9 Length of resonance tube (m)
Fig. 16. Mean temperature gradient along stack versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
Refrigeration temperature (K)
K. Tang et al. / Cryogenics 45 (2005) 185–191
191
4. Conclusions
300 275 250 225 200 175 150 125 100 75 0
20 40 60 80 100 120 Time (min)
Fig. 17. Cooling-down curve; working gas: helium, charging pressure: 2.1 MPa, heating power: 2200 W, orifice and double inlet of the pulse tube refrigerator: 300° and 405°, respectively.
tube refrigerator decreases with an increase of the resonance tube length, as shown in Fig. 10. 3.3. Cooling process optimization The computed and the experimental results indicate that 8 m is an optimal resonance tube length for the test system. It is worthy of attention that the above-mentioned optimal openings of 330° orifice and 420° double inlet are obtained with 4 m resonance tube [5]. The openings need to be further optimized for the case of 8 m resonance tube, considering the decrease of the working frequency and the increase of the pressure amplitude. Optimizing experiments show that a lower refrigeration temperature can be realized with 300° orifice and 405° double inlet. The experiments of the thermoacoustically driven pulse tube refrigerator are carried out with helium as working gas, 2.1 MPa charging pressure, 2200 W heating power, 8 m resonance tube, 300° orifice and 405° double inlet. A typical cooling-down curve is presented in Fig. 17. When the heaters switched on, the temperatures at the hot ends of the stacks rise rapidly. About 8 min later, the hot end temperatures reach about 335 °C, the self-oscillation takes place in the thermoacoustic system, and the refrigeration temperature begins to drop. About 32 min later after onset, 120 K is obtained. Then the refrigeration temperature keeps on decreasing to 88.6 K. In this case, temperatures of the hot and the cold ends of the stack, working pressure, pressure amplitude, pressure ratio and working frequency are 770 K, 337 K, 2.641 MPa, 0.159 MPa, 1.128 and 44.7 Hz, respectively.
The computed and the experimental results indicate that an appropriate prolongation of the resonance tube may improve the performance of the thermoacoustically driven pulse tube refrigerator. Under the conditions of helium as working gas, 2.1 MPa charging pressure and 2000 W heating power, the refrigeration temperature decreases from 112.8 K to 93.2 K, when the resonance tube is prolonged from 4 m to 8 m. After further optimizing the openings of the orifice and the double inlet valves, a refrigeration temperature as low as 88.6 K has been achieved with 8 m resonance tube and a heating power of 2200 W.
Acknowledgments The project is financially supported by the National Natural Sciences Foundation of China (50376055) and the University Doctoral Subject Special Foundation of China (20010335010).
References [1] Radebaugh R. A review of pulse tube refrigeration. Adv Cryo Eng B 1990;35:1191–205. [2] Godshalk KM, Jin C, et al. Characterization of 350 Hz thermoacoustic driven orifice pulse tube refrigerator with measurements of the phase of the mass flow and pressure. Adv Cryo Eng 1996; 41:1411–8. [3] Jin T, Chen GB, Shen Y. A thermoacoustically driven pulse tube refrigerator capable of working below 120 K. Cryogenics 2001; 41(8):595–601. [4] Arman B, Wollan JJ, Swift GW, et al. Thermoacoustic natural gas liquefiers and recent developments. In: Cryogenics and Refrigeration—Proceedings of ICCRÕ2003. International Academic Publishers; 2003. p. 123–7. [5] Tang K, Chen GB, Kong B. A 115 K thermoacoustically driven pulse tube refrigerator with low onset temperature. Cryogenics 2004;44(5):287–91. [6] Chen GB, Jiang JP, Shi JL, et al. Influence of buffer on resonance frequency of thermoacoustic engine. Cryogenics 2002;42(3–4): 223–7. [7] Swift GW. Thermoacoustic engines. J Acoust Soc Am 1988; 84(4):1145–80. [8] Swift GW. Thermoacoustics: a unifying perspective for some engines and refrigerators. Condensed Matter and Thermal Physics Group, Los Alamos National Laboratory, Fourth draft; 1999. [9] Harmonics, characteristic parameters, methods of study, estimates of existing values in network, WG 36-05, Electra No. 77, 1982.
Influence of resonance tube length on performance of thermoacoustically driven pulse tube refrigerator K. Tang, G.B. Chen *, T. Jin, R. Bao, B. Kong, L.M. Qiu Cryogenics Laboratory, Zhejiang University, Hangzhou 310027, PR China Received 10 May 2004; received in revised form 30 September 2004; accepted 2 October 2004
Abstract A resonance tube is an important component of a thermoacoustic engine, which has great influence on the performance of the thermoacoustically driven pulse tube refrigerator. A standing wave thermoacoustic engine is simulated with linear thermoacoustics. Computed results show that an appropriate accretion of the resonance tube length may lead to a decrease of the working frequency and an increase of the pressure amplitude, which will improve the match between the thermoacoustic engine and the pulse tube refrigerator. The theoretical prediction is verified by experiments. A refrigeration temperature as low as 88.6 K has been achieved with an optimized length of the resonance tube, helium as working gas, and 2200 W of heating power. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Pulse tube refrigerator; Thermoacoustic engine; Thermoacoustic refrigeration
1. Introduction A thermoacoustically driven pulse tube refrigerator occupies advantages, such as stability, reliability, longevity and so on, due to no moving components. This novel refrigeration system attracts great interest of academic and commercial fields, and considerable efforts have been made to improve its performance [1–5]. A thermoacoustic engine generates acoustic oscillation from heat energy, and delivers acoustic power with a high frequency (about 70 Hz) and a low pressure ratio (1.1 or so) by acoustic wave to a pulse tube refrigerator which conventionally works at a frequency of about 15– 20 Hz and a pressure ratio of 3 [5]. According to our investigations [3,5,6], it was found that the poor match between the thermoacoustic engine and the pulse tube refrigerator would dominantly baffle the performance improvement of the thermoacoustically driven pulse
*
Corresponding author. Tel./fax: +86 571 8795 1771. E-mail address: [email protected] (G.B. Chen).
0011-2275/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2004.10.002
tube refrigerator. In detail, the match is mainly related to the frequency and the pressure amplitude of the acoustic wave, which should meet the requirement of the pulse tube refrigerator. Although a refrigeration temperature of 115.4 K has been achieved in our previous work [5], the working frequency of 70 Hz is still too high and the pressure amplitude of 0.12 MPa is too small for the pulse tube refrigerator to realize a better refrigeration performance. The resonance tube is one of the key components of a thermoacoustic engine. A standing wave thermoacoustic engine is simulated with linear thermoacoustics to study the influence of the resonance tube on the thermoacoustically driven pulse tube refrigeration. Computed results show that an appropriate prolongation of the resonance tube may lead to a decrease of the working frequency and an increase of the pressure amplitude, which will improve the match between the thermoacoustic engine and the pulse tube refrigerator. Experiments of a thermoacoustically driven pulse tube refrigerator were performed to verify the theoretical prediction. Under the conditions of helium as working gas, 2.1 MPa charg-
186
K. Tang et al. / Cryogenics 45 (2005) 185–191
ing pressure and 2000 W heating power, the refrigeration temperature decreases from 112.8 K to 93.2 K, when the resonance tube is prolonged from 4 m to 8 m. The thermoacoustically driven pulse tube refrigeration is obviously enhanced by adjusting the length of the resonance tube. After further optimizing the openings of the orifice and the double inlet valves, a refrigeration temperature as low as 88.6 K has been attained with 8 m resonance tube and 2200 W heating power.
2. Simulation of thermoacoustic engine According to linear thermoacoustics [7,8], the momentum, continuity and energy equations for a short channel are as follows: dp1 ¼ ðixl þ rm ÞU 1 ; ð1Þ dx dU 1 1 ¼ ixc þ ð2Þ p þ eU 1 ; rj 1 dx fj f~ m 1 _ e H 2 2 Re p1 U 1 1 ð1þPrÞ 1f~ ð mÞ dT m ¼ ; 2 q c U j j p 1 m dx ~ 2 Im fj þ Pr f m ðAk þ Asolid k solid Þ 2 2Axð1Pr Þj1fm j ð3Þ l¼
qm 1 Reðfm Þ ; A j1 f m j2
ð4Þ
c¼
A ½1 þ ðc 1Þ Reðfj Þ; cpm
ð5Þ
xqm Im½fm ; A j1 fm j2
ð6Þ
1 c 1 xA Imðfj Þ ¼ ; rj c pm
ð7Þ
rm ¼
e¼
fj fm 1 dT m ; ð1 fm Þð1 PrÞ T m dx
and ksolid are cross section area and thermal conductivity of the solid forming the channel, H_ 2 is total power, i is imaginary unit, Re, Im and superscript mean the real part, the imaginary part and the conjugation of a complex quantity. l, c, rm, rj, and e are inertance, compliance, viscous resistance, thermal-relaxation resistance and proportionality coefficient of a controlled source, respectively, which can be calculated with Eqs. (4)–(8). A standing wave thermoacoustic engine, as shown in Fig. 1, is simulated with Eqs. (1)–(8). The main dimensions of the engine are tabulated in Table 1. As in experiments, working gas is helium, heating power and working pressure are 2000 W and 2.7 MPa, respectively. Screen stack is considered as capillary model. The local flow resistance due to the variation of the flow area is taken into account. In addition, the viscous resistance of the resonance tube is corrected according to the Ref. [8], because the oscillating flow in the resonance tube is in the range of turbulence flow in the computed case. The computed working frequency is presented in Fig. 2. It can be seen that the working frequency decreases with an increase of the resonance tube length. When the resonance tube is prolonged from 3 m to 10 m, the working frequency decreases from 82.8 Hz to 38.1 Hz, by 54.0%. This is helpful for the improvement of the frequency match between the thermoacoustic engine and the pulse tube refrigerator. Fig. 3 shows the relation of the computed velocity amplitude distribution in the thermoacoustic core (composed of a stack, a heater and a water cooler) versus the resonance tube length. 0–0.065 m, 0.065–0.350 m and 0.350–0.414 m of the axial location axes in Fig. 3 stand for the heater, the stack and the water cooler, respectively. We can see that the velocity amplitude increases from the heater to the water cooler with a certain length of the resonance tube, because the heater is closer to the velocity node. On the other hand, when the resonance tube is prolonged, the relative location of the thermoacoustic core shifts nearer to the velocity node,
ð8Þ
where p1 and U1 are pressure and velocity amplitudes, x is angular frequency, qm, Tm, cp, c, k and Pr are mean density, temperature, isobaric specific heat, specific heat ratio, thermal conductivity and Prandtl number of working fluid, respectively, fm and fj are viscous and thermal functions, A is flow area of the channel, Asolid
Fig. 1. Schematic of thermoacoustic engine.
Table 1 Dimensions of thermoacoustic engine
Diameter (mm) Length (mm)
Heater
Stack
Water cooler
Resonance tube
Hot buffer
54 65
56 285
56 64
36 4000–9000 (optional)
– (1.2 L)
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187
Pressure amplitude (MPa)
90
Frequency (Hz)
80 70 60 50 40 30
3 4 5 6 7 8 9 10 Length of resonance tube (m)
0.170 0.165 0.160 0.155 0.150 0.145 0.0 9 10 (m) 0.1 Axia 0.2 0.3 7 8 e tube 6 l loc nc atio 0.4 0.5 3 4 5 esona n (m of r h ) t g Len
Fig. 4. Computed pressure amplitude in thermoacoustic core versus length of resonance tube; Axial location: 0–0.065 m, 0.065–0.350 m and 0.350–0.414 m stand for the heater, the stack and the water cooler, respectively.
0.035 0.030 0.025
1.140
0.020
1.135
0.015 0.010 Len 3 4 0.5 gth 0.4 of re5 6 7 0.3 (m) 0.2 son n anc 8 910 0.0 0.1 ocatio e tu ial l be (
m)
Ax
Fig. 3. Computed velocity amplitude in thermoacoustic core versus length of resonance tube; Axial location: 0–0.065 m, 0.065–0.350 m and 0.350–0.414 m stand for the heater, the stack and the water cooler, respectively.
so the velocity amplitude in the thermoacoustic core decreases. The computed pressure amplitude and pressure ratio distributions in the thermoacoustic core are presented in Figs. 4 and 5. Both the pressure amplitude and the pressure ratio decrease from the heater to the water cooler, because the heater is closer to the pressure antinode. In addition, the thermoacoustic core approaches to the pressure antinode with an increase of the resonance tube length, which may lead to increases of both the pressure amplitude and the pressure ratio. However, when the resonance tube is further prolonged, the pressure amplitude and the pressure ratio begin to decrease. The analysis on the acoustic power losses may supply another explanation for the curves in Figs. 4 and 5 from the viewpoint of energy. Fig. 6 shows the computed relative acoustic power losses of components (the ratio of the acoustic power loss in one component to the acoustic power generation from the stack). It can be seen that the acoustic power loss in the thermoacoustic core is much higher than that of any other com-
Pressure ratio
Velocity amplitude (m 3/s)
Fig. 2. Computed working frequency versus length of resonance tube.
0.175
1.130 1.125 1.120 1.115 0.0 ) 0.1 9 10 e (m 7 8 ce tub Axia 0.2 0.3 6 0.4 l loc an 0.5 3 4 5 reson atio n (m of h t ) eng L
Fig. 5. Computed pressure ratio in thermoacoustic core versus length of resonance tube; Axial location: 0–0.065 m, 0.065–0.350 m and 0.350–0.414 m stand for the heater, the stack and the water cooler, respectively.
ponent and may dominate the behavior of the thermoacoustic system, when the resonance tube is short. In this case, a decrease of the acoustic power loss in the thermoacoustic core, induced by the decreases of working frequency and velocity amplitude with an increase of the resonance tube length, may enhance the oscillation in the thermoacoustic system and lead to increases of both the pressure amplitude and the pressure ratio. However, an increase of gas–solid interface, induced by the accretion of the resonance tube length, may result in increases of the viscous and the thermalrelaxation dissipations in the resonance tube. Both the pressure amplitude and the pressure ratio begin to decrease with an increase of the resonance tube length when the acoustic power loss in the resonance tube approaches to or even exceeds that in the thermoacoustic core. The temperature distribution in the thermoacoustic core is computed and presented in Fig. 7. The convective heat transfer in the stack is weakened due to the
K. Tang et al. / Cryogenics 45 (2005) 185–191
Relative acoustic power loss (%)
188
3. Experimental apparatus and results
70 60
1
3.1. Experimental apparatus
50 40 30
2
20 10 0
3 3 4 5 6 7 8 9 10 Length of resonance tube (m)
Fig. 6. Computed relative acoustic power loss versus length of resonance tube; (1) relative acoustic power loss in thermoacoustic core: the ratio of the acoustic power loss in the thermoacoustic core to the acoustic power generation in the stack; (2) relative acoustic power loss in resonance tube: the ratio of the acoustic power loss in the resonance tube to the acoustic power generation in the stack; (3) relative acoustic power loss in hot buffer: the ratio of the acoustic power loss in the hot buffer to the acoustic power generation in the stack.
Temperature (K)
800 700 600 500 400 300 0.0 10 (m) 0.1 8 9 tube Axia 0.20.3 7 e l loc 0.4 5 6 onanc atio 0.5 3 4 s n (m of re ) ength L
Fig. 7. Computed temperature in thermoacoustic core versus length of resonance tube; Axial location: 0–0.065 m, 0.065–0.350 m and 0.350– 0.414 m stand for the heater, the stack and the water cooler, respectively.
decreases of both the working frequency and the velocity amplitude with an accretion of the resonance tube length. Thus, the prolongation of the resonance tube may result in an increase of stackÕs hot end temperature with the same heating power. In a word, the computed results indicate that an appropriate increase of the resonance tube length may lead to a decrease of the working frequency and to increases of the pressure amplitude and the pressure ratio, which provides an approach to improve the match between the thermoacoustic engine and the pulse tube refrigerator. Besides, the hot end temperature of the stack may increase with an accretion of the resonance tube length, which is negative for the utilization of low-grade energy.
The experimental apparatus is composed of a thermoacoustic engine, a pulse tube refrigerator and a measuring system, as shown in Fig. 8. The pulse tube refrigerator is a coaxial single-stage double-inlet one, and its main dimensions are tabulated in Table 2. Temperature measuring locations are shown in Fig. 8. T1 and T4, T2 and T3 locate at the hot and the cold ends of the stacks, each side of the resonance tube, respectively. T1 and T4 are measured with NiCr–NiSi thermocouples, while T2 and T3 with Cu–constantan thermocouples. A Rh–Fe resistance thermometer (with 0.1 K accuracy) is applied to measure the refrigeration temperature at the cold end of the pulse tube. The pressure waveform at the joint of the thermoacoustic engine and the pulse tube refrigerator is measured by a piezoresistive silicon pressure sensor. All temperature and pressure data are collected and recorded by a PC-based digital acquisition system. 3.2. Experimental results and analyses on resonance tube length Experiments are carried out with helium as working gas, 2.1 MPa charging pressure, 2000 W heating power and 4–9 m resonance tube. The openings of the orifice and the double inlet valves in the pulse tube refrigerator are set as 330° and 420°, respectively, which are the optimized values based on our previous work [5]. The relation of the working frequency versus the resonance tube length is shown in Fig. 9. It can be seen that an accretion of the resonance tube length leads to a decrease of the working frequency. When the resonance
Fig. 8. Outline of thermoacoustically driven pulse tube refrigerator.
Table 2 Dimensions of pulse tube refrigerator Pulse tube and regenerator
Matrix
Reservoir
Coaxial, outer diameter of 18.5 mm, length of 110 mm
250 mesh, stainless steel
250 cm3
Frequency (Hz)
70 65
Experiment Computation
60 55 50 45
Pressure amplitude (MPa)
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40
0.17 0.16 0.15 0.14 0.13 0.12
4 5 6 7 8 9 Length of resonance tube (m)
tube is prolonged from 4 m to 9 m, the working frequency decreases from 69.9 Hz to 41.1 Hz, by 41.2%. The computed results are also presented in Fig. 9. The computed curve is almost consistent with the experimental data, and the errors are less than 5%. Fig. 10 shows the working pressure of the thermoacoustically driven pulse tube refrigerator. Since the resonance tube is always at room temperature, and the prolongation of the resonance tube results in lower proportion of the high-temperature volume in the whole system. Thus the working pressure decreases with the same charging pressure and heating power. The pressure amplitude and the pressure ratio at the joint are presented in Figs. 11 and 12. When the resonance tube is prolonged from 4 m to 8 m, the pressure amplitude and the pressure ratio increase from 0.116 MPa and 1.086 to 0.150 MPa and 1.121, by 29.3% and 3.2%, respectively. Nevertheless, in the case of the resonance tube length beyond 8 m, the pressure amplitude decreases with an accretion of the resonance tube length, while the pressure ratio still increases, due to the decreasing working pressure of the thermoacous2.85 2.80 2.75 2.70 2.65 2.60 4 5 6 7 8 9 Length of resonance tube (m)
Fig. 10. Working pressure versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
Fig. 11. Pressure amplitude versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
1.14 1.13
Pressure ratio
Fig. 9. Working frequency versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
Working pressure (MPa)
Experiment Computation
0.11
4 5 6 7 8 9 Length of resonance tube (m)
2.55
189
1.12 1.11 1.10 1.09 1.08
Experiment Computation
4 5 6 7 8 9 Length of resonance tube (m)
Fig. 12. Pressure ratio versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
tically driven pulse tube refrigerator, as shown in Fig. 10. We can see that the trends of computed curves are similar to those of the experimental curves in Figs. 11 and 12, but the computed values are markedly higher than the experimental data. There are two conceivable reasons. First, the computed values originate from the simulation only for the thermoacoustic engine, while the experimental data include the influence of the pulse tube refrigerator, which consumes acoustic power to realize refrigeration and attenuates the oscillation. Second, although thermal insulation is carried out for the components at high temperatures in experiments, the heat leak to the surroundings is unavoidable. Thus, the input power utilized effectively by the system is less than the computed heating power value of 2000 W. Total harmonic distortion (THD) [9] of the pressure wave is measured in experiments to analyze the influence of the resonance tube length on the harmonic components. Fig. 13 indicates that THD increases rapidly with an accretion of the resonance tube length, that is, the prolongation of the resonance tube results in the
K. Tang et al. / Cryogenics 45 (2005) 185–191
0.10
THD
0.08 0.06 0.04 0.02 0.00
4 5 6 7 8 9 Length of resonance tube (m)
Fig. 13. THD versus length of resonance tube; THD: total harmonic distortion; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
850
750
105 100 95 90
4 5 6 7 8 9 Length of resonance tube (m)
Fig. 14. Refrigeration temperature versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
420 390
700 360
650 600
330
550
300 4 5 6 7 8 9 Length of resonance tube (m)
Fig. 15. Temperatures of stackÕs hot and cold ends versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
115 110
450 Hot end Cold end
800
Temperature gradient (K/m)
Refrigeration temperature (K)
generation and the growth of harmonic components. In the case of 9 m resonance tube, THD is beyond 0.1, which means that the square root of the summation of squares of harmonic amplitudes is larger than 10% of the fundamental component. The generation and the growth of harmonic components waste the energy of the fundamental component, and attenuate the oscillation. This is another important reason for which the increasing rate of the pressure amplitude diminishes and even becomes negative with a longer resonance tube. The relation of the refrigeration temperature and the resonance tube length is shown in Fig. 14. When the resonance tube is prolonged from 4 m to 8 m, the refrigeration temperature decreases from 112.8 K to 93.2 K and the refrigeration performance is obviously enhanced, since the increases of the pressure amplitude and the pressure ratio with an accretion of the resonance tube length may induce an enhancement of the compression and the expansion processes in the pulse tube, which is beneficial for the pulse tube refrigeration. When the resonance tube is further prolonged to 9 m, the refrigera-
tion temperature begins to rise due to the decrease of the pressure amplitude. Meanwhile, a distortion of the pressure wave with longer high-pressure part and shorter low-pressure part is observed in this case, which may cause an insufficient expansion of the pulse tube refrigeration cycle and becomes another reason for the increase of the refrigeration temperature. Fig. 15 presents the temperatures at both ends of the stack. The cold end temperature of the stack is almost fixed at 330 K with an accretion of the resonance tube length, which indicates that the cold end temperature is almost not affected by the resonance tube length. However, as predicted by the forementioned simulation, the hot end temperature markedly rises with the prolonged resonance tube. The experimental and computed values of the mean temperature gradient over the stack are shown in Fig. 16. It can be seen that the computed curve is similar to the experimental one, but the slope of the computed curve is smaller. It is because that the working pressure of the thermoacoustically driven pulse
Cold end temperature (K)
0.12
Hot end temperature (K)
190
1700 1600
Experiment Computation
1500 1400 1300 1200 1100
4 5 6 7 8 9 Length of resonance tube (m)
Fig. 16. Mean temperature gradient along stack versus length of resonance tube; working gas: helium, charging pressure: 2.1 MPa, heating power: 2000 W, orifice and double inlet of the pulse tube refrigerator: 330° and 420°, respectively.
Refrigeration temperature (K)
K. Tang et al. / Cryogenics 45 (2005) 185–191
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4. Conclusions
300 275 250 225 200 175 150 125 100 75 0
20 40 60 80 100 120 Time (min)
Fig. 17. Cooling-down curve; working gas: helium, charging pressure: 2.1 MPa, heating power: 2200 W, orifice and double inlet of the pulse tube refrigerator: 300° and 405°, respectively.
tube refrigerator decreases with an increase of the resonance tube length, as shown in Fig. 10. 3.3. Cooling process optimization The computed and the experimental results indicate that 8 m is an optimal resonance tube length for the test system. It is worthy of attention that the above-mentioned optimal openings of 330° orifice and 420° double inlet are obtained with 4 m resonance tube [5]. The openings need to be further optimized for the case of 8 m resonance tube, considering the decrease of the working frequency and the increase of the pressure amplitude. Optimizing experiments show that a lower refrigeration temperature can be realized with 300° orifice and 405° double inlet. The experiments of the thermoacoustically driven pulse tube refrigerator are carried out with helium as working gas, 2.1 MPa charging pressure, 2200 W heating power, 8 m resonance tube, 300° orifice and 405° double inlet. A typical cooling-down curve is presented in Fig. 17. When the heaters switched on, the temperatures at the hot ends of the stacks rise rapidly. About 8 min later, the hot end temperatures reach about 335 °C, the self-oscillation takes place in the thermoacoustic system, and the refrigeration temperature begins to drop. About 32 min later after onset, 120 K is obtained. Then the refrigeration temperature keeps on decreasing to 88.6 K. In this case, temperatures of the hot and the cold ends of the stack, working pressure, pressure amplitude, pressure ratio and working frequency are 770 K, 337 K, 2.641 MPa, 0.159 MPa, 1.128 and 44.7 Hz, respectively.
The computed and the experimental results indicate that an appropriate prolongation of the resonance tube may improve the performance of the thermoacoustically driven pulse tube refrigerator. Under the conditions of helium as working gas, 2.1 MPa charging pressure and 2000 W heating power, the refrigeration temperature decreases from 112.8 K to 93.2 K, when the resonance tube is prolonged from 4 m to 8 m. After further optimizing the openings of the orifice and the double inlet valves, a refrigeration temperature as low as 88.6 K has been achieved with 8 m resonance tube and a heating power of 2200 W.
Acknowledgments The project is financially supported by the National Natural Sciences Foundation of China (50376055) and the University Doctoral Subject Special Foundation of China (20010335010).
References [1] Radebaugh R. A review of pulse tube refrigeration. Adv Cryo Eng B 1990;35:1191–205. [2] Godshalk KM, Jin C, et al. Characterization of 350 Hz thermoacoustic driven orifice pulse tube refrigerator with measurements of the phase of the mass flow and pressure. Adv Cryo Eng 1996; 41:1411–8. [3] Jin T, Chen GB, Shen Y. A thermoacoustically driven pulse tube refrigerator capable of working below 120 K. Cryogenics 2001; 41(8):595–601. [4] Arman B, Wollan JJ, Swift GW, et al. Thermoacoustic natural gas liquefiers and recent developments. In: Cryogenics and Refrigeration—Proceedings of ICCRÕ2003. International Academic Publishers; 2003. p. 123–7. [5] Tang K, Chen GB, Kong B. A 115 K thermoacoustically driven pulse tube refrigerator with low onset temperature. Cryogenics 2004;44(5):287–91. [6] Chen GB, Jiang JP, Shi JL, et al. Influence of buffer on resonance frequency of thermoacoustic engine. Cryogenics 2002;42(3–4): 223–7. [7] Swift GW. Thermoacoustic engines. J Acoust Soc Am 1988; 84(4):1145–80. [8] Swift GW. Thermoacoustics: a unifying perspective for some engines and refrigerators. Condensed Matter and Thermal Physics Group, Los Alamos National Laboratory, Fourth draft; 1999. [9] Harmonics, characteristic parameters, methods of study, estimates of existing values in network, WG 36-05, Electra No. 77, 1982.