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A cascade-looped thermoacoustic driven cryocooler with different-diameter resonance tubes. Part I: Theoretical analysis of thermodynamic performance and characteristics
Energy 181 (2019) 943e953
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A cascade-looped thermoacoustic driven cryocooler with differentdiameter resonance tubes. Part I: Theoretical analysis of thermodynamic performance and characteristics Jingyuan Xu a, b, Jianying Hu a, *, Ercang Luo a, **, Limin Zhang a, Wei Dai a a b
Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing, 100190, China Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 January 2019 Received in revised form 30 April 2019 Accepted 2 June 2019 Available online 3 June 2019
A looped-cascade thermoacoustic driven cryocooler is proposed and theoretically studied in this paper. This system is capable of achieving ideal acoustic fields by employing different-diameter resonance tubes. It overcomes the limitations of current configurations and possesses the advantages of high efficiency, large capacity and compact size. First, power matching between the engine-stage number and the working temperatures is investigated. Theoretical results show that either too few or too many engine stages induce negative effects: the former results in low efficiency and the latter results in overhigh heating temperatures. Then, thermodynamic characteristics of the three-stage system are presented. Simulation results show that a cooling power of 1.17 kW and an overall relative Carnot efficiency of 15% can be achieved at 110 K, which is superior to the performance of the existing looped configurations. The distributions of key parameters are also presented for a better understanding of the energy conversion process. Finally, the effects of the crucial parametersdresonance-tube area-ratio and engineregenerator area-ratiodare presented. Simulation results show that having an either too low or too high area-ratio has a significant negative effect on system performance due to improper phase relations and low acoustic impedance. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Thermoacoustic cooling Thermoacoustic Cryocooler Cascade looped
1. Introduction Thermoacoustic heat engines (TAHE) use thermal energy to generate acoustic power (mechanical power) and have therefore become a popular topic in research relating to new sustainable energy conversion technology. Using the pressure wave generated in the TAHE to drive the pulse tube cryocooler (PTC), the thermoacoustically-driven pulse tube cryocooler (TDPTC) is capable of converting externally added heat into cooling power and provides heat source flexibility, structural simplicity and high reliability without any moving parts. TDPTC is considered to be a promising candidate to meet the demands of small-scale liquefaction natural gas (LNG) plants in remote areas: part of the natural gas (NG) is burned as a heat source of the TAHE and the remaining NG is liquefied by the PTC, which therefore does not require electrical
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (J. Hu), [email protected] (E. Luo). https://doi.org/10.1016/j.energy.2019.06.009 0360-5442/© 2019 Elsevier Ltd. All rights reserved.
power. In the past twenty years, several attempts have been made to use TDPTC for NG liquefaction, and great progress has been achieved. In 1997, the Los Alamos National Laboratory, the Cryoenco company and NIST jointly developed a TDPTC for NG liquefaction for the first time. This system employed a standing-wave TAHE to drive an orifice PTC by using NG energy as a heat source. The system provided a cooling power of 2.1 kW at 125 K (i.e. a liquefaction capacity of 140 gallons per day), which was capable of liquefying 40% of the NG while burning 60% LNG [1]. Since then, by employing three parallel PTCs and a Swift's standingeandetraveling TAHE [2], a TDPTC with the aim of achieving more than three times liquefaction capacity has been developed. The highest cooling power reported to date has been 3.8 kW at a cooling temperature of 150 K [3]. Despite the achievements of these TDPTCs, two key defects still exist in these systems which limits their further applications: a large-volume standing-wave resonance tube is required in the systems, resulting in uncompact size and low power density, and the acoustic fields of resonance tube are dominated by a standing wave, leading to low efficiency in power transmission.
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Inspired by the multi-cylinder Stirling engine, the looped multistage TAHE can provide a feasible solution for these problems. The configuration incorporates multiple engine stages in a loop using slim resonance tubes, which are characterized by compact size, large power density and high potential efficiency. De Blok developed a four-stage looped multi-stage TAHE for the first time [4]. Experimental results showed that the low onset temperature of 299 K was achieved by using argon as the working gas, which promoted the development of thermoacoustic technology in the utilization of low-grade thermal energy. Since then, many research groups have been devoted to developments of a looped multi-stage TAHE [5e9], using it as the PTC drive. In 2012, a looped multi-stage thermoacoustically-driven cryocooler (LMSTDC) was developed by De Blok. It has an asymmetric configuration that consisted of three engine stages and a cooler stage with several identical resonance tubes. With 1300 W input power at a heating temperature of 484 K, it obtained a cooling power of 95.1 W at 230 K [10]. Based on the system, Zhang et al. then performed a detailed numerical investigation of onset characteristics and performance analysis [11]. Later, Jin et al. investigated a one-stage LMSTDC. Powered by heat at 483 K-523 K, the numerical results show that the overall energy efficiency exceeded 13% with a cooling temperature of 270 K [12]. Recently, Luo et al. developed a LMSTDC with a direct-coupling configuration for room temperature cooling. The cooling capacity of 4.5 kW and a COP of 0.19 were obtained with the heating temperature of 613 K and the cooling temperature of 283 K [13]. In contrast to the afore-mentioned systems aimed at roomtemperature refrigeration, Luo et al. proposed a bypass-type
LMSTDC for NG liquefaction. The configuration included multiple symmetrical engine-stages in a loop with each branch-connected PTC. Experimental results show that a cooling power of 1.2 kW and an overall relative Carnot efficiency of 8% were experimentally achieved in the three-stage system at 130 K [14,15]. This paper introduces a cascade-looped LMSTDC with differentdiameter resonance tubes, operating in natural gas liquefaction temperature ranges. The proposed configuration avoids structural complexity, can cascade amplified acoustic power, and has wellcoupled acoustic impedance. Section 2 shows the system configuration and matching principle. Section 3 introduces the simulation model. Section 4 presents the results and discussion, including thermodynamic characteristics analysis, the effects of the resonance tube area-ratio, and the effects of the regenerator area-ratio. Section 5 presents conclusions the conclusions. 2. Configuration and matching principle 2.1. Configuration Fig. 1 shows the schematic diagram of the proposed cascadelooped LMSTDC. In this configuration, multiple engine stages and a cooler stage are connected in a loop by different-diameter resonance tubes. There is no theoretical limit on the number of engine stages, which refers to the stage number of the system. Each engine consists of a regenerator (REG) sandwiched between a main ambient heat exchanger (MAHX) and a high-temperature heat exchanger (HHX), a thermal buffer tube (TBT) and a secondary
Fig. 1. Schematic diagram of the looped multi-stage thermoacoustically-driven cryocooler. The green arrows indicate the acoustic power flow. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
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ambient heat exchanger (SAHX). The cooler consists of a REG sandwiched between a MAHX and a cold-temperature heat exchanger (CHX), a pulse tube (PT) and a secondary ambient heat exchanger (SAHX). The diameter of the RTs increases with the increasing engine stage. An elastic membrane at the inlet of the 1st engine suppresses the DC flow [16]. In the system, the TAHE is capable of converting thermal energy into acoustic power, and the PTC is used to convert acoustic power into cooling capacity. The working gas within the RT acts like a gastype expansion piston for the upstream energy-conversion stage as well as a gas-type compression piston for the downstream stage. RTs therefore not only play a crucial role in transmitting acoustic power, but also in matching the acoustic fields of each energyconversion stage. As for the proposed asymmetric configuration, the same-diameter RTs are not conducive to providing desired acoustic fields, owing to the inconsistent acoustic impedance at each stage. For this reason, different-diameter RTs are employed to match the acoustic impedance of the system. The working principle of the cooling system is as follows: each HHX is heated by a hightemperature heating source and the MAHX is cooled by chilled water, which results in a temperature gradient across the engine REG. When the axial temperature gradient exceeds a critical value, a self-excited thermoacoustic oscillation begins. The acoustic power flow is shown as green arrows in Fig. 1. The acoustic power is successively amplified multiple times before it is consumed by the cooler. The remaining acoustic power is recovered by the 1st engine and then the circulation is repeated.
transmitted by the RTs, and consumed in the PTC. The power amplification capability of the TAHE is mainly determined by the engine-stage number and the heating temperature; the power consumption capability of the PTC is related to the cooling temperature. The power amplification capability should be well matched with the consumption capability in the system. If power amplification capability is far less than power consumption capability, the system can even be unable to obtain cooling power at NG temperature ranges. Therefore, power matching of the system is quite crucial to the overall system performance. Based on the power matching principle, the relationships between the engine-stage number N, the heating temperature Th and the cooling temperature Tc are investigated. As shown in Fig. 2, the looped configuration is spread to a straight-line configuration, i.e., the left-most end actually coincides with the right-most end. We define an acoustic power ratio ε as a ratio of acoustic power at the outlet to that at the inlet. In the calculation, N is typically chosen as 1 to 6. Th is set to be the same for each engine stage; thus the acoustic power ratio of each engine εeng is the same. Furthermore, owing to the similar power transmission characteristics, the acoustic power ratio of each resonance tube εRT is set to be same. The acoustic power at the most-right end Wc0 is:
2.2. Power matching principle
Nþ1 εN eng εRT εcooler ¼ 1
In a LMSTDC system, acoustic power is amplified in the TAHE,
0
0
Nþ1 Wc ¼ εN eng εRT εcooler W0
(1)
εcooler is the acoustic power ratio of the cooler unit. When Wc0 ¼ W00 , the system is considered to be power-matched, which can be expressed as:
(2)
The relationships between acoustic power ratio and the working
Fig. 2. Acoustic power at each energy-conversion stage. T0 is 293 K. WN-10 and WN is the acoustic power at the inlet and outlet of #N engine stage, WN0 and Wc is the acoustic power at the inlet and outlet of the cooler. The left-most end coincides with the right-most end in realistic system.
Fig. 3. Numerical models for engine and cooler unit, and calculations results of relationship between working temperature and acoustic power ratio. Solid points are numerical simulation data, the lines and equations are fitted, T0 ¼ 293 K.
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temperatures for an engine- or a cooler-unit are separately calculated. The numerical model and calculation results are shown in Fig. 3. Each model consists of an energy-conversion unit sandwiched by two pistons. For different dimensions of the energyconversion unit, the input power is directly proportional to the diameter, while the acoustic power ratio almost remains the same. The input power is therefore kept constant under the optimal dimension of the unit. The piston parameters are optimized to offer optimal acoustic impendence for the energy conversion process. The acoustic power ratio is taken when the maximum output acoustic power or the maximum cooling power is achieved for the engine- or cooler-unit. According to calculation results in Fig. 3, εeng and εcooler are:
εeng ¼ 0:002378 jTh j þ 0:03
(3)
εcooler ¼ 0:002156 jTc j þ 0:0272
(4)
where jThj and jTcj are numerical values of heating temperature and cooling temperature. For example, with heating temperatures of 900 K, 750 K and 600 K, εeng are 2.17, 1.81 and 1.45; With cooling temperatures of 110 K, 170 K and 230 K, εcooler are 0.26, 0.39 and 0.52. Incorporating Equations (3) and (4) into Equation (2), the relationships between the engine-stage number and working temperatures at the power matching condition can be found as presented in Fig. 4. Here, εRT is set to be 0.8 for all the RTs. For a fixed engine-stage number, the heating temperature decreases when the cooling temperature increases. For example, in a three-stage system, the heating temperatures are 1001 K, 830 K and 690 K when the cooling temperatures are 70 K, 130 K and 230 K. For a fixed cooling temperature, the heating temperature decreases with more engine stages. Taking a closer look at the cooling temperature of 130 K, the heating temperatures are about 1800 K, 1060 K, 830 K, 740 K, 686 K and 657 K when the engine-stage number increases from 1 to 6. This indicates that the engine-stage number can not be small in the case of an over-high heating temperature. The power matching principles provide crucial guidance for the design of the system and choice of working temperatures. In power-matched conditions, the influence of the engine-stage number on cooling efficiency is investigated at 130 K. For the Nstage engine, the input acoustic power WN-10, the out acoustic power WN, and input heating power Qh-N can be expressed by:
Fig. 4. Relationship between engine-stage number and working temperatures at the power matching condition. The contour lines are Tc. εRT ¼ 0.8.
0
N1 W 0N1 ¼ εN1 eng εRT W 0
(5)
0
N1 WN ¼ εN eng εRT W 0
(6)
. 0 N1 QhN ¼ WN W N1 heng ¼ εN1 εeng 1 W 00 heng eng εRT (7) where heng is the engine efficiency. The overall input heating power Qh is:
Qh ¼ Qh1 þ Qh2 þ … þ QhN # " N1 0 ð1εRT Þεeng ð1ðεeng εRT Þ N N1 þ εeng εRT 1 W 0 1εeng εRT ¼
(8)
heng
the cooling power Qc is
0 0 N Qc ¼ WN Wc hcooler ¼ W0 εN eng εRT ð1 εcooler Þhcooler
(9)
where hcooler is the cooler efficiency. The overall relative Carnot efficiency hoverall is defined by:
hoverall ¼
Qc
T0 Tc
1
Qh 1 TT0h
εeng N εRT Nð1 εcooler Þhcooler heng 1 εeng εRT ¼ N 1 εeng εRT εeng 1 T0 1 Tc 1 TT0h
(10)
We calculate heng and hcooler based on the numerical models in Fig. 3 hcooler is calculated as 0.4 when Tc is 130 K. The relationship between heng and Th is:
heng ¼ 0:00178 Th 8:19 107 Th j2 0:414
(11)
Taking Th of 900 K, 650 K and 500 K as examples, heng is 0.52, 0.39 and 0.22. According to the results in Fig. 3, when N increases from 2 to 6 at Tc of 130 K, the corresponding optimal Th is 1060 K, 830 K, 740 K, 686 K and 657 K. Incorporating the above results into Equation (10), the overall Carnot efficiency hoverall of different engine-stage number N in power-matched conditions is shown in Fig. 5. The efficiency decreases with more engine stages. As for εRT ¼ 0.7, the system efficiency varies from 19% to 10.5% when N increases from 2 to 6. In addition, the results show that the resonance-tube efficiency is crucial to the overall efficiency. Based on above analysis, in the NG liquefaction temperature range, the engine-stage number should be carefully chosen in terms of both input heating power and system efficiency. An engine-stage number that is too large or small will have a negative impact: the former induces high heating temperature; the latter causes low system efficiency. Based on a compromise, a three-stage system was chosen as a typical example in the following study.
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where Nu, k, Sx and Tw eT are the Nusselt number, gas conductivity, wetted perimeter and the temperature difference between the negative z surface and section-average respectively. Term q is formulated in terms of effective gas conductivity ke,
q ¼ ke
vT A vx
(17)
In a screen-type regenerator, f and Nu are given by
Fig. 5. Dependence of overall Carnot efficiency on the engine-stage number at the power-matched condition. Tc ¼ 130 K, T0 ¼ 293 K.
f ¼ 129=Re þ 2:91R0:103 e
(18)
Nu ¼ 1 þ 0:99P 0:66 b1:79 e
(19)
where Re is Reynolds number, Pe is Peclet number, and b is porosity. In a plated-fin type heat exchanger in the turbulent case, f and Nu are given by
f ¼ 0:11ðε=dh þ 68=Re Þ0:25
(20)
3. Simulation model
P 0:33 Nu ¼ 0:035R0:75 e r
(21)
Numerical simulations were implemented in the Sage program [17]. This program is a graphical interface that supports simulation and optimization of an underlying class of engineering models, such as the Stirling-cycle, pulse-tube and low-T cooler. The model class contains many model instances. Each of them is a particular collection of component building blocks. To form a complete system model, all the model instances need to be incorporated, connected and assembled in the conservation of mass, momentum and energy, with particular input of the parameter values. Then the well-established model can be solved, mapped or optimized. Sage program is widely used for thermoacoustic machines [13e15,18,19] and pulse tube cryocoolers [20e22]. In the gas domain, the continuity, momentum and energy equations are [17],
where ε is average height of surface irregularities, Pr is Prandtl number. The acoustic power is defined as
W¼
1 jP jjU jcos q 2 1 1
(22)
where U1 is volume flow rate, q is the phase difference between the pressure wave and the volume flow rate. The acoustic impedance Za is given by
Za ¼
P1 P P ¼ Re 1 þ iIm 1 U1 U1 U1
(23)
vrA vruA þ ¼0 vt vx
(12)
The relative Carnot efficiency of the TAHE, hTAHE, the relative Carnot efficiency of the PTC, hPTC, and the relative Carnot efficiency of the overall system, hoverall, are defined by:
vruA vuruA vP1 þ þ A FA ¼ 0 vt vx vx
(13)
hTAHE ¼
Win Wout ; P Qh 1 TT0h
(24)
3
vreA vA v þ P1 þ ðureA þ uP1 A þ qÞ Qw ¼ 0 vt vt vx
where P1, u, A and r are the pressure wave, mean-flow velocity, cross section area and gas density respectively. e is mass-specific total gas energy with e ¼ kþu2/2 (k is mass-specific internal gas energy). F amounts to the viscous pressure gradient, Qw is the heat flow per unit length through the negative z surface due to film heat transfer, and q is the gas axial-conductive heat flux. Terms F, Qw and q are calculated as empirical functions. The purpose of these gasdynamic equations is to determine three implicit solution variables of r, ruA and re. For a heat exchanger or regenerator with hydraulic diameter dh and length L, term F is formulated in terms of the Darcy friction factor f and total local loss coefficient K,
F ¼ ðf =dh þ K=LÞrujuj=2
(15)
For a matrix gas domains, term Qw is formulated as,
Qw ¼ hSx ðTw TÞ ¼ Nu ðk=dh ÞSx ðTw TÞ
(14)
(16)
hPTC ¼
T0 Tc
Qc
1 (25)
Win Wout
hoveral ¼
Qc
T0 Tc
1
; P Qh 1 TT0h
(26)
3
respectively, where Qc is the cooling power, Qh is the input heat power for each engine stage, Win is the acoustic power input to the cooler, and Wout is the acoustic power output at the first RT. In the calculation, the working gas was 7 MPa helium, the heating temperature was 923 K for the three HHXs, the cooling temperature was 110 K, and the ambient temperature was 293 K. The optimization target was to achieve the highest overall efficiency with 1 kW in the NG liquefaction temperature range. The optimized dimensions of the main components are listed in Table 1. Each engine stage is identical here. It is noted that the RT diameters are 15 mm, 18 mm, 22 mm, 27 mm for the first, second, third and
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Table 1 Optimized dimensions of the three-stage LTWTDC system. All unitless dimensions are in millimeters (mm).
Engine
Cooler
RT
Component
Diameter
Length
Other dimensions
MAHX REG HHX TBT SAHX MAHX REG CHX PT SAHX 1st 2nd 3rd 4th
80
60 70 80 150 40 64 70 30 150 9 2400
Shell-tube type, 20% in porosity, 2 in hydraulic diameter 80% in porosity, 150-mesh, 52 mm in wire diameter Plated-fin type, 37% in porosity, 1 in channel width 5 in wall thickness Shell-tube type, 43% in porosity, 3 in hydraulic diameter Plated-fin type, 26% in porosity, 0.4 in channel width 70% in porosity, 300-mesh, 31 mm in wire diameter Plated-fin type, 18% in porosity, 0.25 in channel width 2.2 in wall thickness 40-mech cooper mesh 1.5 in Wall thickness
110
55 15 18 22 27
forth RT, respectively. 4. Results and discussion 4.1. Thermodynamic characteristics analysis 4.1.1. System performance analysis Table 2lists the thermodynamic performance of the system. From the calculation, the frequency of 55 Hz was found to be the fundamental feature of the self-excited oscillation. The input power increased as the engine stage increased: the power was 3051 W for the first engine, 5599 W for the second engine and 10440 W for the third engine. The calculated value of εRT, εeng, heng, εcooler and hcooler here verifies the above deduced relationships. Furthermore, each engine stage has different acoustic impedance. The phase difference varied from 46.5 to 16.9 for the 1st engine stage, 43.7 to 0.65 for the 2nd engine stage, and 29.5 to 1.22 for the 3rd engine stage. With a 19.09 kW overall input power, a cooling power of 1.17 kW was obtained in the liquefaction temperature range near 110 K, corresponding to the overall relative Carnot efficiency of 15%. The TAHE and the PTC efficiencies were 45.8% and 32.6%, respectively. This result demonstrates that this LTWTDC has great potential for application in NG liquefaction. 4.1.2. Key parameters distributions The axis distributions of the acoustic power and the phase difference between the pressure wave and the volume flow rate are presented in Fig. 6. The X-axis begins at the MAHX of the first
engine and moves clockwise around the looped design shown in Fig. 1. As shown in Fig. 6, the X-axis is divided into eight parts: 1st engine, 2nd RT, 2nd engine, 3rd RT, 3rd engine, 4th RT, cooler and 1st RT. Since there is minimal power dissipation in the RT and the engine components, acoustic power is amplified by three times in the REGs, from 1009 W to 2565 W, 1984 We4601 W, and 3797 We8514 W, with power amplification ratios of 2.56, 2.42 and 2.24, respectively. Approximately 6890 W of acoustic power was input into the cooler and 4913 W was consumed by the REG to produce cooling power at the cooling temperature. After some power dissipation in the first RT, 1001 W acoustic power was recovered into the first engine to repeat the cycle. To ensure high cooling efficiency, it was necessary to keep the pressure wave and the volume flow rate in phase at the REGs. Due to the distribution of phase difference between the pressure wave and the volume flow rate, an in-phase relationship was obtained in the cooler REG, which means an efficient energy conversion could be achieved in the cooler. In the engine, the phase difference varied from 44 to 22 for the 1st REG, 42 to 27 for the 2nd REG, and 27 to 16 for the 3rd REG. Although the acoustic fields deviated from the ideal conditions, they were dominated by the traveling-wave field, and thus a high efficiency could be obtained. 4.1.3. Comparisons with the existing systems Fig. 7 shows two current LTWTDC configurations. Fig. 7 (a) shows a by-pass type configuration which symmetrically includes multiple engine-stages in a loop with each branch-connected cooler [14,15]. In this configuration, acoustic power is all
Table 2 Thermodynamic performance of the system. Where P0 is mean pressure, f is working frequency, Pr is pressure ratio at inlet of the energy-conversion stage, Pr¼(P1þP0)/(P0eP1), Za is acoustic impedance with amplitude and phase, the subscripts of eng and cooler denote engine stage and cooler stage, the subscripts of in and out denotes inlet of the stage and outlet of the stage. Working Conditions
Calculated Performance
P0
7 MPa
Th
923 K
T0
293 K
Tc
110 K
f Pr-eng
55 Hz #1: 1.11 #2: 1.13 #3: 1.16 #1: 2.49 #2: 2.38 #3: 2.17 0.26 #1: 4.56 Eþ07 Pa s/m3:-46.5 #2: 3.58 Eþ07 Pa s/m3:-43.7 #3: 2.99 Eþ07 Pa s/m3:-29.5 1.84 Eþ07 Pa s/m3:-38.8 1171 W 45.8%
Pr-cooler Qh
1.18 #1: 3051 W #2: 5599 W #3: 10440 W #1: 49.2% #2: 47.1% #3:45.3% 23% #1: 2.3 Eþ07 Pa s/m3:16.9 #2: 1.74 Eþ07 Pa s/m3:-0.65 #3: 1.09 Eþ07 Pa s/m3:1.22 3.25 Eþ05 Pa s/m3:40.6 15% 32.7%
εeng
εcooler Zeng-in
Zcooler-in Qc
hTAHE
heng hcooler Zeng-out
Zcooler-out
hoverall hPTC
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Fig. 6. Axis distributions of the acoustic power and phase difference in the overall system.
Fig. 7. Schematic diagram of the existing three-stage LMSTDCs.
dissipated as waste heat at the warm end of the pulse tube, and the three separated coolers make it difficult to obtain concentrated cooling power. Fig. 7 (b) shows an asymmetric configuration that consists of three engine stages and a cooler stage with several identical resonance tubes [10]. In this configuration, it is impossible to match the different acoustic impedances of each engine stage with the identical-diameter RTs. The proposed configuration in this paper can tackle the above-
mentioned problems in current LTWTDC configurations. When compared with the by-pass LMSTDC, the dissipated acoustic power at the warm end of the cooler can be recovered in a loop, contributing to a higher cooling efficiency; furthermore, it contains no phase shifters and gas reservoirs, which makes its structure less complex. When compared with the cascade-type with identical RTs, the different-diameter RTs are helpful to achieve better matching of acoustic impendence. Here, the system performance of
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three configurations is numerically compared by the Sage program. The simulation results in Table 3 show that the proposed configuration can achieve higher efficiency or larger cooling power over two existing configurations. 4.2. Impact of the area-ratio of the resonance tubes The resonance tube plays a crucial role in both phase shifting and power transmission. In terms of the asymmetric system, different-diameter RTs are used to match different acoustic impedance of each engine stage, thus providing the desired acoustic fields. In addition, owing to the increasing volume flow rate with increased-stage RTs, the diameters of the RTs are supposed to be changed to reduce power transmission loss. The arearatio between the two adjacent RTs is a crucial parameter in determining overall performance. Fig. 8 shows an engine stage and some crucial parameters. Assuming that there is no viscous dissipation in the engine stage, the ideal volume-flow rate increases by the temperature ratio of two ends of the REG, i.e., jU1dj/jU1uj ¼ Th/Tc. The primary cause of power loss in the RT is the viscous resistance associated with mean-flow velocity u given by u ¼ jU1j/A. To ensure the same oscillation velocity at each RTs, the RT flow area should be increased by a factor equal to the temperature ratio between two ends of the engine REG, i.e., Ad/Au]Th/Tc, if there is no branch at the RT. From the perspective of practical design, the impact of the arearatio of the RT does not equal the ideal value because of viscous dissipation and additions of the TBT, SAHX and RT. However, to simplify the calculation process, it was assumed that the area-ratio between the two adjacent RTs, RRT, was the same at all the engine stages. RRT is defined by:
RRT ¼
A#2RT A#3RT A#4RT ¼ ¼ A#1RT A#2RT A#3RT
(27)
To clarify the effect of RRT on system performance, calculations were performed and are presented in the following sections. In the calculations, component dimensions were the same as those in Table 1 except for the diameters of the 2nd, 3rd RT and 4th RT. That is to say, we fixed the diameter of the 1st RT at 15 mm and changed the diameter of all the other RTs to satisfy Equation (27). In this way, all the RT diameters could be represented by RRT. The working conditions were the same as those outlined in Table 2. The effects of RRT on system performance, phase relation, normalized impedance amplitude and exergy loss are discussed in the following sections. 4.2.1. Effect of RT area-ratio on system performance Fig. 9 shows the influence of RRT on the cooling capacity and the overall relative Carnot efficiency. It shows that RRT is crucial for achieving good system performance. The cooling power and the
Fig. 8. Schematic diagram of an engine stages and some crucial parameters. jU1j is the amplitude of the volume flow rate, A is the flow area of RT. Subscripts u and d stand for the upstream and the downstream, respectively.
Fig. 9. Dependence of system performance on the area-ratio of resonance tubes.
efficiency show the same trend with the increasing RRT: an initial increase followed by a decrease after reaching a peak. When the RRT is approximately 1.5, the best system performance of 1171 W cooling power and 15% overall relative Carnot efficiency are achieved. This optimal RRT is significantly different from the temperature ratio between the two ends of the REG, which is approximately 3. If RRT is too large or too small it negatively impacts cooling performance; thus it should be carefully chosen. When RRT is 1, the proposed configuration becomes similar to De Blok's configuration shown in Fig. 5 (b). In this case, only 541 W cooling power and 11.1% efficiency are achieved, which are lower than the values obtained using the optimal area-ratio. For the optimal RRT, the diameters of 1st to 4th RTs are 15 mm, 18 mm, 22 mm and 27 mm, respectively. 4.2.2. Effect of RT area-ratio on phase relation In general, the regenerator of the thermoacoustic engine or the
Table 3 Simulation results of system performance in three configurations. For three configurations, all the engine components as well as the main cooler components are set to be same as Table 1. The length of RTs are changed to achieve similar frequency. The diameters of the cooler as well as RTs are optimized for the highest efficiency. Same working conditions are used as Table 2. Configuration
f/Hz
Qh/W
Qc/W
hTAHE ]
hPTC
hoverall
Proposed type
55
1171
45.8%
32.7%
15%
By-pass type
55
30.5%
13.2%
49
#1: 390 #2: 390 #3: 390 541
43.3%
Cascade-type with identical RTs
#1: #2: #3: #1: #2: #3: #1: #2: #3:
34%
32.6%
11.1%
3051 5599 10440 7310 7310 7310 2075 4075 5732
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jZnorm j ¼
AjP1 j
(28)
rCjU1 j
where r, C, and A are the gas density, speed of sound and flow crosssectional area, respectively. For the REG, the recommended impedance value is within the range of 15e30 [24]. In a system with multiple energy-conversion stages, high acoustic impedance should be maintained for all of the RGs and RTs to reduce the viscous loss. Figs. 12 and 13 show the influence of the RRT on the normalized acoustic impedance at the inlet and outlet of the regenerator and resonance tube, respectively. As shown in Fig. 12, RRT has a significant influence on the 3rd REG
θ/o
20
#1 RT-Inlet #1 RT-Outlet #2 RT-Inlet #2 RT-Outlet #3 RT-Inlet #3 RT-Outlet #4 RT-Inlet #4 RT-Outlet
0 -20 -40 -60 -80
0.8 1.0 1.2 1.4 1.6 1.8 2.0 RRT
Fig. 11. Dependence of phase difference of resonance tubes on the area-ratio of resonance tubes.
30 25 #1 Engine REG-Inlet
20
#1 Engine REG-Outlet #2 Engine REG-Inlet #2 Engine REG-Outlet
15
#3 Engine REG-Inlet #3 Engine REG-Outlet Cooler REG-Inlet Cooler REG-Outlet
10 5 0
0.8 1.0 1.2 1.4 1.6 1.8 2.0 RRT
Fig. 12. Dependence of impedance amplitude of regenerators on the area-ratio of resonance tubes.
1.0 0.8
#1 RT-Inlet #1 RT-Outlet #2 RT-Inlet #2 RT-Outlet #3 RT-Inlet #3 RT-Outlet #4 RT-Inlet #4 RT-Outlet
0.6 0.4
20 0 #1 Engine REG-Inlet #1 Engine REG-Outlet
θ/o
40
|Znorm|
4.2.3. Effect of RT area-ratio on normalized impedance amplitude The normalized impedance amplitude is a crucial parameter for system performance. Low normalized acoustic impedance causes large viscous losses owing to high acoustic velocities, thereby significantly reducing performance. Normalized impedance amplitude jZnormj is expressed as the ratio between the specific acoustic impedance and the gas characteristic impedance, which is defined by:
60
|Znorm|
pulse tube cryocooler achieves high energy-conversion efficiency when an in-phase relationship is obtained inside. In addition, in the resonance tubes, such an in-phase relationship also reduces the power loss because it lowers the volume flow rate for a given acoustic power [23]. Figs. 10 and 11 show the influence of the arearatio on the phase difference between the pressure wave and the volume flow rate at the inlet and outlet of the regenerator and resonance tube, respectively. Similar alteration trends in the phase difference are observed in the inlet and the outlet in two figures. Fig. 10 shows that RRT has the greatest effect on the 3rd engine, owing to the largest change of phase from 71 to 29 . When RRT is smaller than 1, the phase difference of the 3rd engine REG is smaller than 45 , which significantly deviates from the in-phase relationship and becomes one of the causes of low overall efficiency. Increasing RRT to the optimal value of 1.5 brings the phase difference close to 0 . When RRT is larger than 1.8, the phase difference of the 1st engine REG and the 3rd engine REG are smaller than 30 , which becomes one of the main reasons for low efficiency. In Fig. 11, when RRT is small, the 3rd RT and the 4th RT are dominated by the standing-wave acoustic fields, which have negative effects on the overall cooling performance. However, the degraded acoustic fields can be improved by increasing RRT. In conclusion, these results demonstrate that the optimal value of RRT to be selected for balance the phase fields is 1.5.
-20
#2 Engine REG-Inlet
-40
Cooler REG-Inlet Cooler REG-Outlet
#2 Engine REG-Outlet #3 Engine REG-Inlet #3 Engine REG-Outlet
-60 0.8 1.0 1.2 1.4 1.6 1.8 2.0 RRT Fig. 10. Dependence of phase difference of the regenerators on the area-ratio of resonance tubes.
951
0.2
0.8 1.0 1.2 1.4 1.6 1.8 2.0 RRT
Fig. 13. Dependence of impedance amplitude of resonance tubes on area-ratio of resonance tubes.
impedance and minimal influence on the other RGs. For most RGs, the normalized acoustic impedance is greater than 10, which is within the recommended value range. However, low impedance (below 5) is present in the 3rd engine REG when RRT is as large as 2. This low impedance induces a large acoustic loss in the REG, which could be improved by decreasing the value of RRT. As shown in
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4.2.4. Effect of RT area-ratio on exergy loss Exergy represents the ability to do useful work when a thermal source reservoir at ambient temperature is freely accessible [25], thus determining the maximum performance of the system. To evaluate the component performance, the exergy -loss ratio is employed. The exergy-loss ratio is defined as a ratio of component P exergy loss to overall input heat exergy, i.e. (1-T0/Th) Qh. Fig. 14 shows the dependence of the gross exergy-loss ratio for the RTs and the REGs. When RRT is small, the RTs and the REGs contribute large and equal fractions of exergy loss. Taking RRT of 0.8 as an example, the gross exergy-loss ratio for the RTs reaches the highest value of 44.7%, and the exergy loss in the REGs also accounts for a large portion (52.3%) of the overall input exergy. As RRT increases, the gross exergy of the RTs decreases, while the gross REG exergy loss decreases to a minimum before increasing. When RRT is large enough, the REGs dominate the exergy loss while the RTs make a minimal contribution. When RRT is 2, the gross exergyloss ratios for the REGs and RTs are 56.2% and 17.3%, respectively. Figs. 15 and 16 show the individual exergy-loss ratios for each of RTs and REGs. As shown in Fig. 13, with an increase in RRT, the 3rd engine REG makes the largest contribution to the exergy loss. When RRT increases to 2, the exergy-loss ratio of the 3rd REG exceeds 28%. Based on the analysis of REG impedance, here the normalized impedance amplitude is very small, thus inducing large viscosity loss. As shown in Fig. 14, with the small RRT, 3rd RT and the 4th RT dominating the exergy loss, this leads to low efficiency in the system. Taking RRT of 1 as an example, the exergy loss ratios of the 3rd RT and the 4th RT are 12% and 20%. Similarly, the large loss is induced by the small normalized impedance amplitude. In conclusion, to achieve efficient cooling, the exergy loss in the RTs and REGs should be simultaneously balanced at low values by optimizing the RRT. 4.3. Impact of the area-ratio of the engine regenerators
60%
Exergy loss ratio
40% 30% All RTs All REGs
0.8
1.0
1.2
1.4 1.6 RRT
1.8
20%
10%
0%
0.8
1.0
1.2
1.4 1.6 RRT
1.8
2.0
Fig. 15. Dependence of the exergy-loss ratio of each regenerator on area-ratio of resonance tubes.
25%
#1 RT #2 RT #3 RT #4 RT
20% 15% 10% 5% 0%
0.8 1.0
1.2 1.4 RRT
1.6 1.8 2.0
engine regenerator should be equally enlarged for power matching. For a better understanding of the efficient working principles of the system, the effects of the area-ratio of the engine regenerators is therefore studied here. RREG is assumed to be the same for each engine stage, which can be expressed by:
RREG ¼
50%
20%
#1 Engine REG #2 Engine REG #3 Engine REG Cooler REG
Fig. 16. Dependence of exergy-loss ratio of each resonance tube on area-ratio of resonance tubes.
Besides the area-ratio of the resonance tubes, the area-ratio of the two adjacent engine stages RREG also effects the overall system performance. From a simple point of view, as the power capacity increases with the increased engine stage, the diameter of the
10%
30%
Exergy loss ratio
Fig. 13, the 3rd RT and the 4th RT have low acoustic impedance when RRT is small. For these reasons, selecting the optimal RRT enables high impedances for all the RGs and RTs, which improves overall system performance.
Exergy loss ratio
952
2.0
Fig. 14. Dependence of the gross exergy-loss ratio of resonance tubes and regenerators on area-ratio of resonance tubes.
A#2REG A#3REG ¼ A#1REG A#2REG
(29)
In the calculations, we maintain all the parameters outlined in Table 1, expect for the diameters of the 2nd and 3rd engine stages. The calculation results are shown in Fig. 17. Clearly, RREG that is too large or too small has a negative impact on system performance. According to the calculation, the optimal area-ratio range is from 1 to 1.4. When taking the area-ratio as 1.2, an overall relative Carnot efficiency of 15.2% and the highest cooling power of 1.3 kW can be achieved. However, this result is close to the efficiency of 15% and a cooling power of 1.17 kW when RREG is 1. Considering the practical aspects of engine machining, RREG is better if is it 1. 5. Conclusions A looped multi-stage thermoacoustically-driven cryocooler is
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from China Postdoctoral Council, and National Natural Science Foundation of China (Grant No. 51627809, 51576204, 51876213).
References
Fig. 17. Dependence of system performance on the area-ratio of the regenerators. RRT ¼ 1.5.
introduced in this paper. In this configuration, multiple engine stages and a cooler stage are connected in a loop by differentdiameter resonance tubes, which are characterized by high efficiency, large capacity and simple structure. The different-diameter resonance tubes are used to match the different acoustic impedance of each engine stage, thus improving system efficiency. The investigation shows that: The engine-stage number, heating temperature and cooling temperature should be carefully chosen based on the power matching of the system. The heating temperature increases with fewer engine stages or lower cooling temperatures. When the system is well-matched, the system efficiency decreases with more engine stages. To avoid low efficiency or over-high heating temperatures, engine-stage numbers should not be too small or too large. The area-ratio between two adjacent engine resonance tubes is found to be a crucial factor in achieving efficient thermoacoustic conversion and power transmission. An area-ratio that is either too small or too large has a negative effect on system performance because a large exergy loss is induced by improper phase relations and low acoustic impedance. The optimal area-ratio of resonance tube is 1.5. Furthermore, the effect of the engine-regenerator arearatio displays a similar trend, and the optimal values range from 1 to 1.4. The three-stage system obtains a cooling power of 1.17 kW and an overall efficiency of 15%, with the optimal resonance tube arearatio of 1.5 and identical engine regenerators. The engine and the cooler efficiencies are 45.8% and 32.6%, respectively. This performance is superior to the performance of two existing configurations, and demonstrates that this thermoacoustically-driven cryocooler is promising for application in natural gas liquefaction, which will be tested in the next phase of experimental work. Acknowledgements This work was supported by Beijing Natural Science Foundation (Contract No. 3194061), National Key Research and Development Program of China (Contract No. 2016YFB0901403), International Postdoctoral Exchange Fellowship Program (Grant No. 20180012)
[1] Arman B, Wollan JJ, Swift GW, Backhaus S. Thermoacoustic natural gas liquefiers and recent developments. In: Cryogenic and refrigeration proceedings of ICCR. Hang Zhou. International Academic Publisher and World Publishing Corporation; 2003. p. 123e7. [2] Backhaus S, Swift GW. A thermoacoustic Stirling heat engine. Nature 1999;399:335e8. [3] Arman B, Wollan JJ, Kotsubo V, Backhaus S, Swift G. Operation of thermoacoustic Stirling heat engine driven large multiple pulse tube refrigerator. In: Ross Jr RG, editor. Cryocooler 13. New York: Springer Science and Business Media Inc.; 2004. p. 181e7. [4] Kees DB. Novel 4-stage traveling wave thermoacoustic power generator. In: International conference on nanochannels, and minichannels; 2010. [5] Li DH, Chen YY, Luo EC, Wu ZH. Study of a liquid-piston traveling-wave thermoacoustic heat engine with different working gases. Energy 2014;74: 158e63. [6] Mariko S, Shinya H. Four-stage loop-type cascade traveling-wave thermoacoustic engine. Appl Therm Eng 2016;104:258e62. [7] Jin T, Yang R, Liu YL, Tang K. Thermodynamic characteristics during the onset and damping processes in a looped thermoacoustic prime mover. Appl Therm Eng 2016;100:1169e72. [8] Jin T, Yang R, Wang Y, et al. Low temperature difference thermoacoustic prime mover with asymmetric multi-stage loop configuration. Sci Rep 2017;7:7665. [9] Tang K, Feng Y, Jin T, et al. Effect of Gedeon streaming on thermal efficiency of a travelling-wave thermoacoustic engine. Appl Therm Eng 2017;115: 1089e100. [10] Kees DB. Multi-stage travelling wave thermoacoustic in practice. In: The 19th international congress of sound and vibration. Vilnius, Lithuania: International Institute of Acoustics and Vibration and Vilnius University; 2010. p. 1e8. [11] Zhang XQ, Chang JZ, Cai SS, Hu J’A. A multi-stage travelling wave thermoacoustic engine driven refrigerator and operation features for utilizing low grade energy. Energy Convers Manag 2016;114:224e33. [12] Jin T, Yang R, Wang Y, Feng Y, Tang K. Acoustic field characteristics and performance analysis of a looped traveling-wave thermoacoustic refrigerator. Energy Convers Manag 2016;123:243e51. [13] Wang HZ, Zhang LM, Yu GY, Hu JY, Luo EC, Ma Ying, Jiang Chao, Liu Xianxian. A looped heat-driven thermoacoustic refrigeration system with directcoupling configuration for room temperature cooling. Sci Bull 2019;64:8e10. [14] Xu JY, Zhang LM, Hu JY, Wu ZH, Bi TJ, Dai W, et al. An efficient looped multiple-stage thermoacoustically-driven cryocooler for liquefaction and recondensation of natural gas. Energy 2016;101:427e33. [15] Zhang LM, Hu JY, Wu ZH, Luo EC, Xu JY, Bi TJ. A 1 kW-class multi-stage heatdriven thermoacoustic cryocooler system operating at liquefied natural gas temperature range. Appl Phys Lett 2015;107:033905. [16] Gedeon D. DC gas flow in Stirling and pulse-tube cryocoolers. Cryocooler; 9: 385e392. [17] Gedeon D. SAGE: object-oriented software for cryocooler design. In: Proceedings of 8th international cryocooler conference; 1995. p. 281e92. [18] Hu JY, Luo EC, Zhang LM, Chen YY, Wu ZH, Gao B. Analysis of a displacercoupled multi-stage thermoacoustic-Stirling engine. Energy 2018;145: 507e14. [19] Xu JY, Hu JY, Zhang LM, Luo EC. A looped three-stage cascade traveling-wave thermoacoustically-driven cryocooler. Energy 2016;112:804e9. [20] Luo KQ, Hu JY, Luo EC, Zhang LM, Sun YL, Zhou Y. Theoretical investigation on the optimal PU phase relationships of regenerative cooling systems with highest efficiency. Cryogenics 2019;98:5e11. [21] Dietrich M, Yang LW, Thummes G. High-power Stirling-type pulse tube cryocooler: observation and reduction of regenerator temperature-inhomogeneities. Cryogenics 2007;47:306e14. [22] Chen LB, Wu XL, Liu XM, Pan CZ, Zhou Y. Numerical and experimental study on the characteristics of 4 K gas-coupled Stirling-type pulse tube cryocooler. Int J Refrig 2018;88:204e10. [23] Radebaugh R, Lewis M, Luo EC. Inertance tube optimization for pulse tube refrigerators. Adv. Cryogenics Eng. 2006;51(A):59e70. [24] Backhaus S, Swift GW. A thermoacoustic-Stirling heat engine: detailed study. J Am Chem Soc 2000;107(6):3148e66. [25] Swift GW. Thermoacoustics: a unifying perspective for some engines and refrigerators. Acoustical Society of America; 2002.
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A cascade-looped thermoacoustic driven cryocooler with differentdiameter resonance tubes. Part I: Theoretical analysis of thermodynamic performance and characteristics Jingyuan Xu a, b, Jianying Hu a, *, Ercang Luo a, **, Limin Zhang a, Wei Dai a a b
Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing, 100190, China Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 January 2019 Received in revised form 30 April 2019 Accepted 2 June 2019 Available online 3 June 2019
A looped-cascade thermoacoustic driven cryocooler is proposed and theoretically studied in this paper. This system is capable of achieving ideal acoustic fields by employing different-diameter resonance tubes. It overcomes the limitations of current configurations and possesses the advantages of high efficiency, large capacity and compact size. First, power matching between the engine-stage number and the working temperatures is investigated. Theoretical results show that either too few or too many engine stages induce negative effects: the former results in low efficiency and the latter results in overhigh heating temperatures. Then, thermodynamic characteristics of the three-stage system are presented. Simulation results show that a cooling power of 1.17 kW and an overall relative Carnot efficiency of 15% can be achieved at 110 K, which is superior to the performance of the existing looped configurations. The distributions of key parameters are also presented for a better understanding of the energy conversion process. Finally, the effects of the crucial parametersdresonance-tube area-ratio and engineregenerator area-ratiodare presented. Simulation results show that having an either too low or too high area-ratio has a significant negative effect on system performance due to improper phase relations and low acoustic impedance. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Thermoacoustic cooling Thermoacoustic Cryocooler Cascade looped
1. Introduction Thermoacoustic heat engines (TAHE) use thermal energy to generate acoustic power (mechanical power) and have therefore become a popular topic in research relating to new sustainable energy conversion technology. Using the pressure wave generated in the TAHE to drive the pulse tube cryocooler (PTC), the thermoacoustically-driven pulse tube cryocooler (TDPTC) is capable of converting externally added heat into cooling power and provides heat source flexibility, structural simplicity and high reliability without any moving parts. TDPTC is considered to be a promising candidate to meet the demands of small-scale liquefaction natural gas (LNG) plants in remote areas: part of the natural gas (NG) is burned as a heat source of the TAHE and the remaining NG is liquefied by the PTC, which therefore does not require electrical
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (J. Hu), [email protected] (E. Luo). https://doi.org/10.1016/j.energy.2019.06.009 0360-5442/© 2019 Elsevier Ltd. All rights reserved.
power. In the past twenty years, several attempts have been made to use TDPTC for NG liquefaction, and great progress has been achieved. In 1997, the Los Alamos National Laboratory, the Cryoenco company and NIST jointly developed a TDPTC for NG liquefaction for the first time. This system employed a standing-wave TAHE to drive an orifice PTC by using NG energy as a heat source. The system provided a cooling power of 2.1 kW at 125 K (i.e. a liquefaction capacity of 140 gallons per day), which was capable of liquefying 40% of the NG while burning 60% LNG [1]. Since then, by employing three parallel PTCs and a Swift's standingeandetraveling TAHE [2], a TDPTC with the aim of achieving more than three times liquefaction capacity has been developed. The highest cooling power reported to date has been 3.8 kW at a cooling temperature of 150 K [3]. Despite the achievements of these TDPTCs, two key defects still exist in these systems which limits their further applications: a large-volume standing-wave resonance tube is required in the systems, resulting in uncompact size and low power density, and the acoustic fields of resonance tube are dominated by a standing wave, leading to low efficiency in power transmission.
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Inspired by the multi-cylinder Stirling engine, the looped multistage TAHE can provide a feasible solution for these problems. The configuration incorporates multiple engine stages in a loop using slim resonance tubes, which are characterized by compact size, large power density and high potential efficiency. De Blok developed a four-stage looped multi-stage TAHE for the first time [4]. Experimental results showed that the low onset temperature of 299 K was achieved by using argon as the working gas, which promoted the development of thermoacoustic technology in the utilization of low-grade thermal energy. Since then, many research groups have been devoted to developments of a looped multi-stage TAHE [5e9], using it as the PTC drive. In 2012, a looped multi-stage thermoacoustically-driven cryocooler (LMSTDC) was developed by De Blok. It has an asymmetric configuration that consisted of three engine stages and a cooler stage with several identical resonance tubes. With 1300 W input power at a heating temperature of 484 K, it obtained a cooling power of 95.1 W at 230 K [10]. Based on the system, Zhang et al. then performed a detailed numerical investigation of onset characteristics and performance analysis [11]. Later, Jin et al. investigated a one-stage LMSTDC. Powered by heat at 483 K-523 K, the numerical results show that the overall energy efficiency exceeded 13% with a cooling temperature of 270 K [12]. Recently, Luo et al. developed a LMSTDC with a direct-coupling configuration for room temperature cooling. The cooling capacity of 4.5 kW and a COP of 0.19 were obtained with the heating temperature of 613 K and the cooling temperature of 283 K [13]. In contrast to the afore-mentioned systems aimed at roomtemperature refrigeration, Luo et al. proposed a bypass-type
LMSTDC for NG liquefaction. The configuration included multiple symmetrical engine-stages in a loop with each branch-connected PTC. Experimental results show that a cooling power of 1.2 kW and an overall relative Carnot efficiency of 8% were experimentally achieved in the three-stage system at 130 K [14,15]. This paper introduces a cascade-looped LMSTDC with differentdiameter resonance tubes, operating in natural gas liquefaction temperature ranges. The proposed configuration avoids structural complexity, can cascade amplified acoustic power, and has wellcoupled acoustic impedance. Section 2 shows the system configuration and matching principle. Section 3 introduces the simulation model. Section 4 presents the results and discussion, including thermodynamic characteristics analysis, the effects of the resonance tube area-ratio, and the effects of the regenerator area-ratio. Section 5 presents conclusions the conclusions. 2. Configuration and matching principle 2.1. Configuration Fig. 1 shows the schematic diagram of the proposed cascadelooped LMSTDC. In this configuration, multiple engine stages and a cooler stage are connected in a loop by different-diameter resonance tubes. There is no theoretical limit on the number of engine stages, which refers to the stage number of the system. Each engine consists of a regenerator (REG) sandwiched between a main ambient heat exchanger (MAHX) and a high-temperature heat exchanger (HHX), a thermal buffer tube (TBT) and a secondary
Fig. 1. Schematic diagram of the looped multi-stage thermoacoustically-driven cryocooler. The green arrows indicate the acoustic power flow. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
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ambient heat exchanger (SAHX). The cooler consists of a REG sandwiched between a MAHX and a cold-temperature heat exchanger (CHX), a pulse tube (PT) and a secondary ambient heat exchanger (SAHX). The diameter of the RTs increases with the increasing engine stage. An elastic membrane at the inlet of the 1st engine suppresses the DC flow [16]. In the system, the TAHE is capable of converting thermal energy into acoustic power, and the PTC is used to convert acoustic power into cooling capacity. The working gas within the RT acts like a gastype expansion piston for the upstream energy-conversion stage as well as a gas-type compression piston for the downstream stage. RTs therefore not only play a crucial role in transmitting acoustic power, but also in matching the acoustic fields of each energyconversion stage. As for the proposed asymmetric configuration, the same-diameter RTs are not conducive to providing desired acoustic fields, owing to the inconsistent acoustic impedance at each stage. For this reason, different-diameter RTs are employed to match the acoustic impedance of the system. The working principle of the cooling system is as follows: each HHX is heated by a hightemperature heating source and the MAHX is cooled by chilled water, which results in a temperature gradient across the engine REG. When the axial temperature gradient exceeds a critical value, a self-excited thermoacoustic oscillation begins. The acoustic power flow is shown as green arrows in Fig. 1. The acoustic power is successively amplified multiple times before it is consumed by the cooler. The remaining acoustic power is recovered by the 1st engine and then the circulation is repeated.
transmitted by the RTs, and consumed in the PTC. The power amplification capability of the TAHE is mainly determined by the engine-stage number and the heating temperature; the power consumption capability of the PTC is related to the cooling temperature. The power amplification capability should be well matched with the consumption capability in the system. If power amplification capability is far less than power consumption capability, the system can even be unable to obtain cooling power at NG temperature ranges. Therefore, power matching of the system is quite crucial to the overall system performance. Based on the power matching principle, the relationships between the engine-stage number N, the heating temperature Th and the cooling temperature Tc are investigated. As shown in Fig. 2, the looped configuration is spread to a straight-line configuration, i.e., the left-most end actually coincides with the right-most end. We define an acoustic power ratio ε as a ratio of acoustic power at the outlet to that at the inlet. In the calculation, N is typically chosen as 1 to 6. Th is set to be the same for each engine stage; thus the acoustic power ratio of each engine εeng is the same. Furthermore, owing to the similar power transmission characteristics, the acoustic power ratio of each resonance tube εRT is set to be same. The acoustic power at the most-right end Wc0 is:
2.2. Power matching principle
Nþ1 εN eng εRT εcooler ¼ 1
In a LMSTDC system, acoustic power is amplified in the TAHE,
0
0
Nþ1 Wc ¼ εN eng εRT εcooler W0
(1)
εcooler is the acoustic power ratio of the cooler unit. When Wc0 ¼ W00 , the system is considered to be power-matched, which can be expressed as:
(2)
The relationships between acoustic power ratio and the working
Fig. 2. Acoustic power at each energy-conversion stage. T0 is 293 K. WN-10 and WN is the acoustic power at the inlet and outlet of #N engine stage, WN0 and Wc is the acoustic power at the inlet and outlet of the cooler. The left-most end coincides with the right-most end in realistic system.
Fig. 3. Numerical models for engine and cooler unit, and calculations results of relationship between working temperature and acoustic power ratio. Solid points are numerical simulation data, the lines and equations are fitted, T0 ¼ 293 K.
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temperatures for an engine- or a cooler-unit are separately calculated. The numerical model and calculation results are shown in Fig. 3. Each model consists of an energy-conversion unit sandwiched by two pistons. For different dimensions of the energyconversion unit, the input power is directly proportional to the diameter, while the acoustic power ratio almost remains the same. The input power is therefore kept constant under the optimal dimension of the unit. The piston parameters are optimized to offer optimal acoustic impendence for the energy conversion process. The acoustic power ratio is taken when the maximum output acoustic power or the maximum cooling power is achieved for the engine- or cooler-unit. According to calculation results in Fig. 3, εeng and εcooler are:
εeng ¼ 0:002378 jTh j þ 0:03
(3)
εcooler ¼ 0:002156 jTc j þ 0:0272
(4)
where jThj and jTcj are numerical values of heating temperature and cooling temperature. For example, with heating temperatures of 900 K, 750 K and 600 K, εeng are 2.17, 1.81 and 1.45; With cooling temperatures of 110 K, 170 K and 230 K, εcooler are 0.26, 0.39 and 0.52. Incorporating Equations (3) and (4) into Equation (2), the relationships between the engine-stage number and working temperatures at the power matching condition can be found as presented in Fig. 4. Here, εRT is set to be 0.8 for all the RTs. For a fixed engine-stage number, the heating temperature decreases when the cooling temperature increases. For example, in a three-stage system, the heating temperatures are 1001 K, 830 K and 690 K when the cooling temperatures are 70 K, 130 K and 230 K. For a fixed cooling temperature, the heating temperature decreases with more engine stages. Taking a closer look at the cooling temperature of 130 K, the heating temperatures are about 1800 K, 1060 K, 830 K, 740 K, 686 K and 657 K when the engine-stage number increases from 1 to 6. This indicates that the engine-stage number can not be small in the case of an over-high heating temperature. The power matching principles provide crucial guidance for the design of the system and choice of working temperatures. In power-matched conditions, the influence of the engine-stage number on cooling efficiency is investigated at 130 K. For the Nstage engine, the input acoustic power WN-10, the out acoustic power WN, and input heating power Qh-N can be expressed by:
Fig. 4. Relationship between engine-stage number and working temperatures at the power matching condition. The contour lines are Tc. εRT ¼ 0.8.
0
N1 W 0N1 ¼ εN1 eng εRT W 0
(5)
0
N1 WN ¼ εN eng εRT W 0
(6)
. 0 N1 QhN ¼ WN W N1 heng ¼ εN1 εeng 1 W 00 heng eng εRT (7) where heng is the engine efficiency. The overall input heating power Qh is:
Qh ¼ Qh1 þ Qh2 þ … þ QhN # " N1 0 ð1εRT Þεeng ð1ðεeng εRT Þ N N1 þ εeng εRT 1 W 0 1εeng εRT ¼
(8)
heng
the cooling power Qc is
0 0 N Qc ¼ WN Wc hcooler ¼ W0 εN eng εRT ð1 εcooler Þhcooler
(9)
where hcooler is the cooler efficiency. The overall relative Carnot efficiency hoverall is defined by:
hoverall ¼
Qc
T0 Tc
1
Qh 1 TT0h
εeng N εRT Nð1 εcooler Þhcooler heng 1 εeng εRT ¼ N 1 εeng εRT εeng 1 T0 1 Tc 1 TT0h
(10)
We calculate heng and hcooler based on the numerical models in Fig. 3 hcooler is calculated as 0.4 when Tc is 130 K. The relationship between heng and Th is:
heng ¼ 0:00178 Th 8:19 107 Th j2 0:414
(11)
Taking Th of 900 K, 650 K and 500 K as examples, heng is 0.52, 0.39 and 0.22. According to the results in Fig. 3, when N increases from 2 to 6 at Tc of 130 K, the corresponding optimal Th is 1060 K, 830 K, 740 K, 686 K and 657 K. Incorporating the above results into Equation (10), the overall Carnot efficiency hoverall of different engine-stage number N in power-matched conditions is shown in Fig. 5. The efficiency decreases with more engine stages. As for εRT ¼ 0.7, the system efficiency varies from 19% to 10.5% when N increases from 2 to 6. In addition, the results show that the resonance-tube efficiency is crucial to the overall efficiency. Based on above analysis, in the NG liquefaction temperature range, the engine-stage number should be carefully chosen in terms of both input heating power and system efficiency. An engine-stage number that is too large or small will have a negative impact: the former induces high heating temperature; the latter causes low system efficiency. Based on a compromise, a three-stage system was chosen as a typical example in the following study.
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where Nu, k, Sx and Tw eT are the Nusselt number, gas conductivity, wetted perimeter and the temperature difference between the negative z surface and section-average respectively. Term q is formulated in terms of effective gas conductivity ke,
q ¼ ke
vT A vx
(17)
In a screen-type regenerator, f and Nu are given by
Fig. 5. Dependence of overall Carnot efficiency on the engine-stage number at the power-matched condition. Tc ¼ 130 K, T0 ¼ 293 K.
f ¼ 129=Re þ 2:91R0:103 e
(18)
Nu ¼ 1 þ 0:99P 0:66 b1:79 e
(19)
where Re is Reynolds number, Pe is Peclet number, and b is porosity. In a plated-fin type heat exchanger in the turbulent case, f and Nu are given by
f ¼ 0:11ðε=dh þ 68=Re Þ0:25
(20)
3. Simulation model
P 0:33 Nu ¼ 0:035R0:75 e r
(21)
Numerical simulations were implemented in the Sage program [17]. This program is a graphical interface that supports simulation and optimization of an underlying class of engineering models, such as the Stirling-cycle, pulse-tube and low-T cooler. The model class contains many model instances. Each of them is a particular collection of component building blocks. To form a complete system model, all the model instances need to be incorporated, connected and assembled in the conservation of mass, momentum and energy, with particular input of the parameter values. Then the well-established model can be solved, mapped or optimized. Sage program is widely used for thermoacoustic machines [13e15,18,19] and pulse tube cryocoolers [20e22]. In the gas domain, the continuity, momentum and energy equations are [17],
where ε is average height of surface irregularities, Pr is Prandtl number. The acoustic power is defined as
W¼
1 jP jjU jcos q 2 1 1
(22)
where U1 is volume flow rate, q is the phase difference between the pressure wave and the volume flow rate. The acoustic impedance Za is given by
Za ¼
P1 P P ¼ Re 1 þ iIm 1 U1 U1 U1
(23)
vrA vruA þ ¼0 vt vx
(12)
The relative Carnot efficiency of the TAHE, hTAHE, the relative Carnot efficiency of the PTC, hPTC, and the relative Carnot efficiency of the overall system, hoverall, are defined by:
vruA vuruA vP1 þ þ A FA ¼ 0 vt vx vx
(13)
hTAHE ¼
Win Wout ; P Qh 1 TT0h
(24)
3
vreA vA v þ P1 þ ðureA þ uP1 A þ qÞ Qw ¼ 0 vt vt vx
where P1, u, A and r are the pressure wave, mean-flow velocity, cross section area and gas density respectively. e is mass-specific total gas energy with e ¼ kþu2/2 (k is mass-specific internal gas energy). F amounts to the viscous pressure gradient, Qw is the heat flow per unit length through the negative z surface due to film heat transfer, and q is the gas axial-conductive heat flux. Terms F, Qw and q are calculated as empirical functions. The purpose of these gasdynamic equations is to determine three implicit solution variables of r, ruA and re. For a heat exchanger or regenerator with hydraulic diameter dh and length L, term F is formulated in terms of the Darcy friction factor f and total local loss coefficient K,
F ¼ ðf =dh þ K=LÞrujuj=2
(15)
For a matrix gas domains, term Qw is formulated as,
Qw ¼ hSx ðTw TÞ ¼ Nu ðk=dh ÞSx ðTw TÞ
(14)
(16)
hPTC ¼
T0 Tc
Qc
1 (25)
Win Wout
hoveral ¼
Qc
T0 Tc
1
; P Qh 1 TT0h
(26)
3
respectively, where Qc is the cooling power, Qh is the input heat power for each engine stage, Win is the acoustic power input to the cooler, and Wout is the acoustic power output at the first RT. In the calculation, the working gas was 7 MPa helium, the heating temperature was 923 K for the three HHXs, the cooling temperature was 110 K, and the ambient temperature was 293 K. The optimization target was to achieve the highest overall efficiency with 1 kW in the NG liquefaction temperature range. The optimized dimensions of the main components are listed in Table 1. Each engine stage is identical here. It is noted that the RT diameters are 15 mm, 18 mm, 22 mm, 27 mm for the first, second, third and
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Table 1 Optimized dimensions of the three-stage LTWTDC system. All unitless dimensions are in millimeters (mm).
Engine
Cooler
RT
Component
Diameter
Length
Other dimensions
MAHX REG HHX TBT SAHX MAHX REG CHX PT SAHX 1st 2nd 3rd 4th
80
60 70 80 150 40 64 70 30 150 9 2400
Shell-tube type, 20% in porosity, 2 in hydraulic diameter 80% in porosity, 150-mesh, 52 mm in wire diameter Plated-fin type, 37% in porosity, 1 in channel width 5 in wall thickness Shell-tube type, 43% in porosity, 3 in hydraulic diameter Plated-fin type, 26% in porosity, 0.4 in channel width 70% in porosity, 300-mesh, 31 mm in wire diameter Plated-fin type, 18% in porosity, 0.25 in channel width 2.2 in wall thickness 40-mech cooper mesh 1.5 in Wall thickness
110
55 15 18 22 27
forth RT, respectively. 4. Results and discussion 4.1. Thermodynamic characteristics analysis 4.1.1. System performance analysis Table 2lists the thermodynamic performance of the system. From the calculation, the frequency of 55 Hz was found to be the fundamental feature of the self-excited oscillation. The input power increased as the engine stage increased: the power was 3051 W for the first engine, 5599 W for the second engine and 10440 W for the third engine. The calculated value of εRT, εeng, heng, εcooler and hcooler here verifies the above deduced relationships. Furthermore, each engine stage has different acoustic impedance. The phase difference varied from 46.5 to 16.9 for the 1st engine stage, 43.7 to 0.65 for the 2nd engine stage, and 29.5 to 1.22 for the 3rd engine stage. With a 19.09 kW overall input power, a cooling power of 1.17 kW was obtained in the liquefaction temperature range near 110 K, corresponding to the overall relative Carnot efficiency of 15%. The TAHE and the PTC efficiencies were 45.8% and 32.6%, respectively. This result demonstrates that this LTWTDC has great potential for application in NG liquefaction. 4.1.2. Key parameters distributions The axis distributions of the acoustic power and the phase difference between the pressure wave and the volume flow rate are presented in Fig. 6. The X-axis begins at the MAHX of the first
engine and moves clockwise around the looped design shown in Fig. 1. As shown in Fig. 6, the X-axis is divided into eight parts: 1st engine, 2nd RT, 2nd engine, 3rd RT, 3rd engine, 4th RT, cooler and 1st RT. Since there is minimal power dissipation in the RT and the engine components, acoustic power is amplified by three times in the REGs, from 1009 W to 2565 W, 1984 We4601 W, and 3797 We8514 W, with power amplification ratios of 2.56, 2.42 and 2.24, respectively. Approximately 6890 W of acoustic power was input into the cooler and 4913 W was consumed by the REG to produce cooling power at the cooling temperature. After some power dissipation in the first RT, 1001 W acoustic power was recovered into the first engine to repeat the cycle. To ensure high cooling efficiency, it was necessary to keep the pressure wave and the volume flow rate in phase at the REGs. Due to the distribution of phase difference between the pressure wave and the volume flow rate, an in-phase relationship was obtained in the cooler REG, which means an efficient energy conversion could be achieved in the cooler. In the engine, the phase difference varied from 44 to 22 for the 1st REG, 42 to 27 for the 2nd REG, and 27 to 16 for the 3rd REG. Although the acoustic fields deviated from the ideal conditions, they were dominated by the traveling-wave field, and thus a high efficiency could be obtained. 4.1.3. Comparisons with the existing systems Fig. 7 shows two current LTWTDC configurations. Fig. 7 (a) shows a by-pass type configuration which symmetrically includes multiple engine-stages in a loop with each branch-connected cooler [14,15]. In this configuration, acoustic power is all
Table 2 Thermodynamic performance of the system. Where P0 is mean pressure, f is working frequency, Pr is pressure ratio at inlet of the energy-conversion stage, Pr¼(P1þP0)/(P0eP1), Za is acoustic impedance with amplitude and phase, the subscripts of eng and cooler denote engine stage and cooler stage, the subscripts of in and out denotes inlet of the stage and outlet of the stage. Working Conditions
Calculated Performance
P0
7 MPa
Th
923 K
T0
293 K
Tc
110 K
f Pr-eng
55 Hz #1: 1.11 #2: 1.13 #3: 1.16 #1: 2.49 #2: 2.38 #3: 2.17 0.26 #1: 4.56 Eþ07 Pa s/m3:-46.5 #2: 3.58 Eþ07 Pa s/m3:-43.7 #3: 2.99 Eþ07 Pa s/m3:-29.5 1.84 Eþ07 Pa s/m3:-38.8 1171 W 45.8%
Pr-cooler Qh
1.18 #1: 3051 W #2: 5599 W #3: 10440 W #1: 49.2% #2: 47.1% #3:45.3% 23% #1: 2.3 Eþ07 Pa s/m3:16.9 #2: 1.74 Eþ07 Pa s/m3:-0.65 #3: 1.09 Eþ07 Pa s/m3:1.22 3.25 Eþ05 Pa s/m3:40.6 15% 32.7%
εeng
εcooler Zeng-in
Zcooler-in Qc
hTAHE
heng hcooler Zeng-out
Zcooler-out
hoverall hPTC
J. Xu et al. / Energy 181 (2019) 943e953
949
Fig. 6. Axis distributions of the acoustic power and phase difference in the overall system.
Fig. 7. Schematic diagram of the existing three-stage LMSTDCs.
dissipated as waste heat at the warm end of the pulse tube, and the three separated coolers make it difficult to obtain concentrated cooling power. Fig. 7 (b) shows an asymmetric configuration that consists of three engine stages and a cooler stage with several identical resonance tubes [10]. In this configuration, it is impossible to match the different acoustic impedances of each engine stage with the identical-diameter RTs. The proposed configuration in this paper can tackle the above-
mentioned problems in current LTWTDC configurations. When compared with the by-pass LMSTDC, the dissipated acoustic power at the warm end of the cooler can be recovered in a loop, contributing to a higher cooling efficiency; furthermore, it contains no phase shifters and gas reservoirs, which makes its structure less complex. When compared with the cascade-type with identical RTs, the different-diameter RTs are helpful to achieve better matching of acoustic impendence. Here, the system performance of
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J. Xu et al. / Energy 181 (2019) 943e953
three configurations is numerically compared by the Sage program. The simulation results in Table 3 show that the proposed configuration can achieve higher efficiency or larger cooling power over two existing configurations. 4.2. Impact of the area-ratio of the resonance tubes The resonance tube plays a crucial role in both phase shifting and power transmission. In terms of the asymmetric system, different-diameter RTs are used to match different acoustic impedance of each engine stage, thus providing the desired acoustic fields. In addition, owing to the increasing volume flow rate with increased-stage RTs, the diameters of the RTs are supposed to be changed to reduce power transmission loss. The arearatio between the two adjacent RTs is a crucial parameter in determining overall performance. Fig. 8 shows an engine stage and some crucial parameters. Assuming that there is no viscous dissipation in the engine stage, the ideal volume-flow rate increases by the temperature ratio of two ends of the REG, i.e., jU1dj/jU1uj ¼ Th/Tc. The primary cause of power loss in the RT is the viscous resistance associated with mean-flow velocity u given by u ¼ jU1j/A. To ensure the same oscillation velocity at each RTs, the RT flow area should be increased by a factor equal to the temperature ratio between two ends of the engine REG, i.e., Ad/Au]Th/Tc, if there is no branch at the RT. From the perspective of practical design, the impact of the arearatio of the RT does not equal the ideal value because of viscous dissipation and additions of the TBT, SAHX and RT. However, to simplify the calculation process, it was assumed that the area-ratio between the two adjacent RTs, RRT, was the same at all the engine stages. RRT is defined by:
RRT ¼
A#2RT A#3RT A#4RT ¼ ¼ A#1RT A#2RT A#3RT
(27)
To clarify the effect of RRT on system performance, calculations were performed and are presented in the following sections. In the calculations, component dimensions were the same as those in Table 1 except for the diameters of the 2nd, 3rd RT and 4th RT. That is to say, we fixed the diameter of the 1st RT at 15 mm and changed the diameter of all the other RTs to satisfy Equation (27). In this way, all the RT diameters could be represented by RRT. The working conditions were the same as those outlined in Table 2. The effects of RRT on system performance, phase relation, normalized impedance amplitude and exergy loss are discussed in the following sections. 4.2.1. Effect of RT area-ratio on system performance Fig. 9 shows the influence of RRT on the cooling capacity and the overall relative Carnot efficiency. It shows that RRT is crucial for achieving good system performance. The cooling power and the
Fig. 8. Schematic diagram of an engine stages and some crucial parameters. jU1j is the amplitude of the volume flow rate, A is the flow area of RT. Subscripts u and d stand for the upstream and the downstream, respectively.
Fig. 9. Dependence of system performance on the area-ratio of resonance tubes.
efficiency show the same trend with the increasing RRT: an initial increase followed by a decrease after reaching a peak. When the RRT is approximately 1.5, the best system performance of 1171 W cooling power and 15% overall relative Carnot efficiency are achieved. This optimal RRT is significantly different from the temperature ratio between the two ends of the REG, which is approximately 3. If RRT is too large or too small it negatively impacts cooling performance; thus it should be carefully chosen. When RRT is 1, the proposed configuration becomes similar to De Blok's configuration shown in Fig. 5 (b). In this case, only 541 W cooling power and 11.1% efficiency are achieved, which are lower than the values obtained using the optimal area-ratio. For the optimal RRT, the diameters of 1st to 4th RTs are 15 mm, 18 mm, 22 mm and 27 mm, respectively. 4.2.2. Effect of RT area-ratio on phase relation In general, the regenerator of the thermoacoustic engine or the
Table 3 Simulation results of system performance in three configurations. For three configurations, all the engine components as well as the main cooler components are set to be same as Table 1. The length of RTs are changed to achieve similar frequency. The diameters of the cooler as well as RTs are optimized for the highest efficiency. Same working conditions are used as Table 2. Configuration
f/Hz
Qh/W
Qc/W
hTAHE ]
hPTC
hoverall
Proposed type
55
1171
45.8%
32.7%
15%
By-pass type
55
30.5%
13.2%
49
#1: 390 #2: 390 #3: 390 541
43.3%
Cascade-type with identical RTs
#1: #2: #3: #1: #2: #3: #1: #2: #3:
34%
32.6%
11.1%
3051 5599 10440 7310 7310 7310 2075 4075 5732
J. Xu et al. / Energy 181 (2019) 943e953
jZnorm j ¼
AjP1 j
(28)
rCjU1 j
where r, C, and A are the gas density, speed of sound and flow crosssectional area, respectively. For the REG, the recommended impedance value is within the range of 15e30 [24]. In a system with multiple energy-conversion stages, high acoustic impedance should be maintained for all of the RGs and RTs to reduce the viscous loss. Figs. 12 and 13 show the influence of the RRT on the normalized acoustic impedance at the inlet and outlet of the regenerator and resonance tube, respectively. As shown in Fig. 12, RRT has a significant influence on the 3rd REG
θ/o
20
#1 RT-Inlet #1 RT-Outlet #2 RT-Inlet #2 RT-Outlet #3 RT-Inlet #3 RT-Outlet #4 RT-Inlet #4 RT-Outlet
0 -20 -40 -60 -80
0.8 1.0 1.2 1.4 1.6 1.8 2.0 RRT
Fig. 11. Dependence of phase difference of resonance tubes on the area-ratio of resonance tubes.
30 25 #1 Engine REG-Inlet
20
#1 Engine REG-Outlet #2 Engine REG-Inlet #2 Engine REG-Outlet
15
#3 Engine REG-Inlet #3 Engine REG-Outlet Cooler REG-Inlet Cooler REG-Outlet
10 5 0
0.8 1.0 1.2 1.4 1.6 1.8 2.0 RRT
Fig. 12. Dependence of impedance amplitude of regenerators on the area-ratio of resonance tubes.
1.0 0.8
#1 RT-Inlet #1 RT-Outlet #2 RT-Inlet #2 RT-Outlet #3 RT-Inlet #3 RT-Outlet #4 RT-Inlet #4 RT-Outlet
0.6 0.4
20 0 #1 Engine REG-Inlet #1 Engine REG-Outlet
θ/o
40
|Znorm|
4.2.3. Effect of RT area-ratio on normalized impedance amplitude The normalized impedance amplitude is a crucial parameter for system performance. Low normalized acoustic impedance causes large viscous losses owing to high acoustic velocities, thereby significantly reducing performance. Normalized impedance amplitude jZnormj is expressed as the ratio between the specific acoustic impedance and the gas characteristic impedance, which is defined by:
60
|Znorm|
pulse tube cryocooler achieves high energy-conversion efficiency when an in-phase relationship is obtained inside. In addition, in the resonance tubes, such an in-phase relationship also reduces the power loss because it lowers the volume flow rate for a given acoustic power [23]. Figs. 10 and 11 show the influence of the arearatio on the phase difference between the pressure wave and the volume flow rate at the inlet and outlet of the regenerator and resonance tube, respectively. Similar alteration trends in the phase difference are observed in the inlet and the outlet in two figures. Fig. 10 shows that RRT has the greatest effect on the 3rd engine, owing to the largest change of phase from 71 to 29 . When RRT is smaller than 1, the phase difference of the 3rd engine REG is smaller than 45 , which significantly deviates from the in-phase relationship and becomes one of the causes of low overall efficiency. Increasing RRT to the optimal value of 1.5 brings the phase difference close to 0 . When RRT is larger than 1.8, the phase difference of the 1st engine REG and the 3rd engine REG are smaller than 30 , which becomes one of the main reasons for low efficiency. In Fig. 11, when RRT is small, the 3rd RT and the 4th RT are dominated by the standing-wave acoustic fields, which have negative effects on the overall cooling performance. However, the degraded acoustic fields can be improved by increasing RRT. In conclusion, these results demonstrate that the optimal value of RRT to be selected for balance the phase fields is 1.5.
-20
#2 Engine REG-Inlet
-40
Cooler REG-Inlet Cooler REG-Outlet
#2 Engine REG-Outlet #3 Engine REG-Inlet #3 Engine REG-Outlet
-60 0.8 1.0 1.2 1.4 1.6 1.8 2.0 RRT Fig. 10. Dependence of phase difference of the regenerators on the area-ratio of resonance tubes.
951
0.2
0.8 1.0 1.2 1.4 1.6 1.8 2.0 RRT
Fig. 13. Dependence of impedance amplitude of resonance tubes on area-ratio of resonance tubes.
impedance and minimal influence on the other RGs. For most RGs, the normalized acoustic impedance is greater than 10, which is within the recommended value range. However, low impedance (below 5) is present in the 3rd engine REG when RRT is as large as 2. This low impedance induces a large acoustic loss in the REG, which could be improved by decreasing the value of RRT. As shown in
J. Xu et al. / Energy 181 (2019) 943e953
4.2.4. Effect of RT area-ratio on exergy loss Exergy represents the ability to do useful work when a thermal source reservoir at ambient temperature is freely accessible [25], thus determining the maximum performance of the system. To evaluate the component performance, the exergy -loss ratio is employed. The exergy-loss ratio is defined as a ratio of component P exergy loss to overall input heat exergy, i.e. (1-T0/Th) Qh. Fig. 14 shows the dependence of the gross exergy-loss ratio for the RTs and the REGs. When RRT is small, the RTs and the REGs contribute large and equal fractions of exergy loss. Taking RRT of 0.8 as an example, the gross exergy-loss ratio for the RTs reaches the highest value of 44.7%, and the exergy loss in the REGs also accounts for a large portion (52.3%) of the overall input exergy. As RRT increases, the gross exergy of the RTs decreases, while the gross REG exergy loss decreases to a minimum before increasing. When RRT is large enough, the REGs dominate the exergy loss while the RTs make a minimal contribution. When RRT is 2, the gross exergyloss ratios for the REGs and RTs are 56.2% and 17.3%, respectively. Figs. 15 and 16 show the individual exergy-loss ratios for each of RTs and REGs. As shown in Fig. 13, with an increase in RRT, the 3rd engine REG makes the largest contribution to the exergy loss. When RRT increases to 2, the exergy-loss ratio of the 3rd REG exceeds 28%. Based on the analysis of REG impedance, here the normalized impedance amplitude is very small, thus inducing large viscosity loss. As shown in Fig. 14, with the small RRT, 3rd RT and the 4th RT dominating the exergy loss, this leads to low efficiency in the system. Taking RRT of 1 as an example, the exergy loss ratios of the 3rd RT and the 4th RT are 12% and 20%. Similarly, the large loss is induced by the small normalized impedance amplitude. In conclusion, to achieve efficient cooling, the exergy loss in the RTs and REGs should be simultaneously balanced at low values by optimizing the RRT. 4.3. Impact of the area-ratio of the engine regenerators
60%
Exergy loss ratio
40% 30% All RTs All REGs
0.8
1.0
1.2
1.4 1.6 RRT
1.8
20%
10%
0%
0.8
1.0
1.2
1.4 1.6 RRT
1.8
2.0
Fig. 15. Dependence of the exergy-loss ratio of each regenerator on area-ratio of resonance tubes.
25%
#1 RT #2 RT #3 RT #4 RT
20% 15% 10% 5% 0%
0.8 1.0
1.2 1.4 RRT
1.6 1.8 2.0
engine regenerator should be equally enlarged for power matching. For a better understanding of the efficient working principles of the system, the effects of the area-ratio of the engine regenerators is therefore studied here. RREG is assumed to be the same for each engine stage, which can be expressed by:
RREG ¼
50%
20%
#1 Engine REG #2 Engine REG #3 Engine REG Cooler REG
Fig. 16. Dependence of exergy-loss ratio of each resonance tube on area-ratio of resonance tubes.
Besides the area-ratio of the resonance tubes, the area-ratio of the two adjacent engine stages RREG also effects the overall system performance. From a simple point of view, as the power capacity increases with the increased engine stage, the diameter of the
10%
30%
Exergy loss ratio
Fig. 13, the 3rd RT and the 4th RT have low acoustic impedance when RRT is small. For these reasons, selecting the optimal RRT enables high impedances for all the RGs and RTs, which improves overall system performance.
Exergy loss ratio
952
2.0
Fig. 14. Dependence of the gross exergy-loss ratio of resonance tubes and regenerators on area-ratio of resonance tubes.
A#2REG A#3REG ¼ A#1REG A#2REG
(29)
In the calculations, we maintain all the parameters outlined in Table 1, expect for the diameters of the 2nd and 3rd engine stages. The calculation results are shown in Fig. 17. Clearly, RREG that is too large or too small has a negative impact on system performance. According to the calculation, the optimal area-ratio range is from 1 to 1.4. When taking the area-ratio as 1.2, an overall relative Carnot efficiency of 15.2% and the highest cooling power of 1.3 kW can be achieved. However, this result is close to the efficiency of 15% and a cooling power of 1.17 kW when RREG is 1. Considering the practical aspects of engine machining, RREG is better if is it 1. 5. Conclusions A looped multi-stage thermoacoustically-driven cryocooler is
J. Xu et al. / Energy 181 (2019) 943e953
953
from China Postdoctoral Council, and National Natural Science Foundation of China (Grant No. 51627809, 51576204, 51876213).
References
Fig. 17. Dependence of system performance on the area-ratio of the regenerators. RRT ¼ 1.5.
introduced in this paper. In this configuration, multiple engine stages and a cooler stage are connected in a loop by differentdiameter resonance tubes, which are characterized by high efficiency, large capacity and simple structure. The different-diameter resonance tubes are used to match the different acoustic impedance of each engine stage, thus improving system efficiency. The investigation shows that: The engine-stage number, heating temperature and cooling temperature should be carefully chosen based on the power matching of the system. The heating temperature increases with fewer engine stages or lower cooling temperatures. When the system is well-matched, the system efficiency decreases with more engine stages. To avoid low efficiency or over-high heating temperatures, engine-stage numbers should not be too small or too large. The area-ratio between two adjacent engine resonance tubes is found to be a crucial factor in achieving efficient thermoacoustic conversion and power transmission. An area-ratio that is either too small or too large has a negative effect on system performance because a large exergy loss is induced by improper phase relations and low acoustic impedance. The optimal area-ratio of resonance tube is 1.5. Furthermore, the effect of the engine-regenerator arearatio displays a similar trend, and the optimal values range from 1 to 1.4. The three-stage system obtains a cooling power of 1.17 kW and an overall efficiency of 15%, with the optimal resonance tube arearatio of 1.5 and identical engine regenerators. The engine and the cooler efficiencies are 45.8% and 32.6%, respectively. This performance is superior to the performance of two existing configurations, and demonstrates that this thermoacoustically-driven cryocooler is promising for application in natural gas liquefaction, which will be tested in the next phase of experimental work. Acknowledgements This work was supported by Beijing Natural Science Foundation (Contract No. 3194061), National Key Research and Development Program of China (Contract No. 2016YFB0901403), International Postdoctoral Exchange Fellowship Program (Grant No. 20180012)
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