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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39261493
71204
10.1038/s41598-024-71204-w
Article
Analysis of forming mechanism and influencing factors of thermoacoustic plate end temperature difference
Wang Jianxin
Liu Xiangbin liu0604103414@163.com

Meng Nan
https://ror.org/044rgx723 grid.462400.4 0000 0001 0144 9297 School of Mechanical Engineering, Inner Mongolia University of Science and Technology, Baotou, 014010 China
11 9 2024
11 9 2024
2024
14 2121914 5 2024
26 8 2024
© The Author(s) 2024
2024
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In order to explore the formation mechanism and influencing factors of the temperature difference between two ends of the plate stack, an expression of the temperature change of the stack with time was established based on the two-dimensional heat conduction equation. Based on this, the finite element model of heat transfer between a single thermoacoustic plate stack and the gas above it is established in Ansys, and the temperature of the plate stack is solved. When the sound field is constant, the variation law of the temperature of the stack with the working time and space is obtained, and the formation mechanism of the temperature difference between the two ends of the plate stack is revealed. From the calculation results, it is found that the net heat transfer between the gas and the plate stack is mainly reflected in the two ends of the plate stack, and the contribution of the air mass in the middle part is mainly the relay heat transfer. In the process of working, part of the sound work is converted into the internal energy of the air mass, which makes the gas temperature on the surface of the plate rise as a whole. The working frequency, stack length and stack thermal conductivity are taken as the influencing factors. When no load is added, the variation of the temperature of the high and low end of the stack with the working time under different working conditions is analyzed. And the theory of series between short plates is put forward to explain the formation mechanism of large temperature difference between the two ends of the plates. In order to further reduce the cooling temperature of thermoacoustic refrigerator, a new research method and exploration direction are proposed.

Keywords

Thermoacoustic refrigerator
Two-dimensional heat conduction
Temperature at both ends of the plate
The formation of large temperature differences
Subject terms

Energy science and technology
Engineering
Physics
The national natural science foundation of China52266005 Wang Jianxin issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

At present, energy utilization and environmental protection are the hot topics of research and thermoacoustic refrigeration system has great development potential in the field of new generation refrigeration due to its advantages of low cost and environmental friendliness. In the initial research work, the driving sound source of the thermoacoustic refrigerator is mostly the loudspeaker. With the development of thermoacoustic technology, thermoacoustic heat engine is gradually used as an incentive in thermoacoustic refrigerator, and its biggest advantage is that the thermoacoustic heat engine can be used as a power of waste heat or solar energy to work1–3. Plate stack is the core component of thermoacoustic refrigerator. The temperature difference between the two ends is very important to the cooling temperature of thermoacoustic refrigerator, which has attracted many scholars to explore it. The research group of Swift4,5 is the first research team to systematically describe the thermal interaction between the plate stack and the working gas. They established a one-dimensional heat conduction model of the plate stack temperature, ignoring the influence of heat conduction from the high-temperature end to the low-temperature end inside the plate stack, and ignoring the change of sound pressure within the length range of the stack. Tijani et al.6–8 based on the theory established by Swift, the relationship between the length of the plate and the position of the plate is analyzed. At the operating frequency of 450 Hz, the stack temperature difference of 75 K is obtained. Ahmed I. Abd El-Rahman of Cairo.

University9 designed a thermoacoustic refrigerator driven by dual sound sources. The working medium was air. When the driving ratio was 7%, a maximum temperature difference of 27° was obtained at both ends of the plate stack.

Through the real-time monitoring of the temperature at the end of the stack, it is found that the temperature at the low end first drops to a certain value and then slowly rises until the system reaches a stable state. Compared with the theoretical calculation results of Swift, the variation trend of plate stack temperature is basically the same, but the numerical difference is large, and the mechanism of the difference needs to be further explored on the basis of the existing theory.

After decades of efforts, thermoacoustic refrigeration technology has been greatly developed, but the thermoacoustic refrigerator still has some problems such as low efficiency and imperfect theory. The heat conduction between gas and plate is a very complicated physical process, and its heat conduction equation is difficult to be solved accurately. With the development of numerical simulation technology, more and more methods such as finite element method and finite volume analysis are applied to the analysis of flow field and temperature field of thermoacoustic refrigerator. Based on the linear theoretical model, X Li et al.10 used the finite element method to analyze the flow field of a traveling wave thermoacoustic engine and obtained the distribution of the flow field in the resonator. The research group of Jan A. De Jong11 adopted the finite element method to analyze the working stability of the standing wave thermoacoustic system, and further verified the correctness of the obtained results by comparing them with the experimental results. T Hofmeister12 used the finite element method to analyze the linear stability of the thermoacoustic system, and explained the phenomenon caused by the disturbance. The research team of Ahmed I. Abd El-Rahman13 established a three-dimensional model of plate stack and gas, solved it with hexahedral grid, and predicted the steady-state response of the system.

Aniruddha S. Worlikar14 used the compressible fluid model with low Mach number to analyze the unsteady two-dimensional thermal stratified flow near idealized thermoacoustic plate stack, and the calculation of plate stack temperature was in good agreement with the experimental data. Later, D. Marx et al.15–17 used numerical simulation to solve the Navier–Stokes equations of unsteady and compressible flows, and studied the changes of the flow field and temperature field near the plate stack. It was found that compared with the results of linear theoretical calculation, the calculated temperature difference still had some differences even at low Mach number. Zoontjens18,19 studied the influence of different plate boundary shape profiles on thermoacoustic effects under different driving ratios. Because of the different shape of the edge of the plate, two kinds of mesh are set up: triangle and quadrilateral. The results show that the expanded and passivated shapes increase the heat transfer rate at low drive ratio, but delay the heat transfer rate at high drive ratio. Li Meng20 used the finite element numerical simulation method to study the influence of laminated materials on the temperature difference between the two ends of the laminated plates, and obtained the result that a small thermal conductivity is conducive to increasing the temperature difference between the laminated plates.

In terms of experiments, many scholars have also carried out exploration, by analyzing the heat transfer process between gas and plate and the influencing factors, to find ways to reduce the refrigeration temperature, in order to meet the requirements of refrigeration temperature under different working conditions. HARIHARAN N. M. and SIVASHANMUGAM P.21 of the Indian Institute of Technology adopted the curved surface response method. Taking the temperature difference as the design goal, the factors such as the position of the stack, the length of the stack and the working frequency of the stack are optimized.

Emanuele Sarpero et al.22, from the University of Lyon, used 3D printing technology to design different plate shapes and conducted experimental research on the temperature of the stack. When the operating frequency was 192 Hz, a temperature difference of 18.42 K was obtained at both ends of the stack. In addition, Gokay et al.23 investigated the temperature difference of the plate and the system performance in response to acoustic waves in five different forms (sine, square, triangle, sawtooth and trapezoid). The results show that when the diameter of the resonant tube is equal to or less than 50 mm, the temperature difference obtained by the sine wave is the largest. When the pipe diameter is 70 mm and 100 mm, the best waveforms are sawtooth wave and trapezoid wave, respectively. Mahmoud A. Alamir24 conducted experimental research on the standing wave thermoacoustic refrigerator, and obtained that the highest and lowest temperatures in the resonator were in the center of the high temperature and low temperature end of the plate stack.

At present, in the study of the temperature distribution law of the plate, the numerical simulation is based on certain assumptions, and does not consider the change of the sound field in the stack region; Although the law of temperature variation at both ends of the plate has been obtained in the experimental study, the formation mechanism has not been thoroughly explained from the theoretical aspect. In a word, there is no unified theoretical model for the study of the temperature distribution in the thermoacoustic refrigerator. This in itself illustrates the complexity of the problem and the incompleteness of existing theories. So the purpose of this paper is twofold. First, a two-dimensional heat transfer equation of lamination temperature, which is more suitable for the actual situation, is established. The gas temperature is taken as the boundary condition to solve the stack temperature, and the distribution law of stack temperature and its correlation with each working parameter are analyzed by Ansys simulation calculation. Second, the theory of series between short plates is put forward. The formation mechanism of large temperature difference between the two ends of the plate is explained systematically from the theoretical aspect. The conclusions obtained in this paper greatly enrich the thermoacoustic theory and provide theoretical basis for the development of engineering prototype.

Theoretical calculation of plate temperature

Taking the parallel plate stack and gas heat transfer process as the research object, the coordinate system as shown in Fig. 1 is established. It is assumed that the direction along the length of the board is x direction, and the direction perpendicular to the board is y direction.Fig. 1 Schematic diagram of plate stack structure of thermoacoustic refrigerator.

During the working process of the thermoacoustic refrigerator, the temperature change at any point on the stack can be expressed as:1 ∂Ts∂t=ks∂2Ts∂ys2+ks∂2Ts∂xs2

In the above formula, ks=Ksρscs , Ks, ρs, cs respectively, the heat transfer coefficient, density and specific heat capacity of the sheet. The first term at the right end of the equal sign represents the influence of heat conduction along y inside the plate, which is mainly determined by the thermoacoustic effect between the gas and the plate. The second term represents the influence of the internal heat conduction along x to the plate, that is, the heat leakage of the plate itself.

The air mass moves reciprocally between the plates, and there is a flow boundary layer dominated by turbulence between the main flow region and the boundary layer. Considering the viscosity of the gas working medium, there is a very thin laminar flow bottom layer between the turbulent boundary layer and the laminated wall. Because of the bedding of the bottom layer of laminar flow, the wall surface can be considered smooth and the contact is tight for the air mass, and the temperature of the plate and the gas contact surface is equal. In reference 25, the expression of sheet temperature change rate is given considering only the thermoacoustic effect25. The effects of heat conduction in both x and y directions are considered in this paper. According to reference 25, formula (1) can be expressed as:2 Ts1=Tc1cosh[(1+j)ys/δs]cosh[(1+j)l/δs]+ks∂2Ts1∂vs2

In Eq. (2), Tc is the gas–solid interface temperature, which is consistent with the change of gas temperature; subscript 1 represents the first-order fluctuation; vs is the result of the time derivative of xs. The temperature of the stack is mainly affected by the gas temperature, thermal conductivity and the stack temperature gradient. It can be seen that the temperature of the gas in contact with the stack can be taken as the boundary condition in the numerical simulation when studying the temperature distribution of the stack.

According to the study of swift25, the temperature change of gas between plates can be expressed as:3 T1=βTmρmcpp1-1ρmω21-σσ-1cosh1+jyδvcosh1+jy0δvdp1dxdTmdx-Tmβρmcpp1-dp1dxdTmdxσ-1ρmω21+εsfvfkcosh1+jy/δK1+εscosh1+jy0/δK

The temperature condition at the gas–solid contact surface is: T1=Tc1

By bringing Eq. (3) into Eq. (2), the temperature at any point on the stack can be obtained as:4 Ts1x,ys=εs1+εsβTmρmcpp1+1ρmω2dp1dxdTmdx1σ-11-fvfkcosh1+jyδvcosh1+jy0δv+ks∂2Ts1∂xs2

where βis the isobaric volume expansion parameter; Tm is the initial temperature of the gas,℃;ωis the angular velocity of gas vibration, rad/s; σis the Prandtl number;ρm is the initial density of the gas, kg/m3;fv, fK and εs depend on the nature and shape of the gas and the plate. According to reference 25, the parameters in the parallel plate can be expressed as follows:5 fv=tanh[(1+j)y0δv](1+j)y0δv

6 fK=tanh[(1+j)y0δK](1+j)y0δK

7 εs=ε0tanh[(1+j)y0δK]tanh[(1+j)lδK,s]

8 ε0=ρmcpδKρscsδK,s

where δv is the viscous penetration depth, δK gas thermal penetration depth, δK、s solid thermal penetration depth, the expressions are as follows:9 δv=2vω

10 δK=2Kρcpω

11 δK,s=2Ksρscsω

In the process of air mass movement, the heat transfer with the plate is continuous. The position of the air mass is different, and the heat transfer changes accordingly, which results in the temperature distribution inside the plate and the temperature of the air mass change from time to time. When calculating formula (4), it is difficult to get its analytical solution directly. In this paper, the finite element method is used to discretize the process of air mass movement, and the finite element model of heat transfer between air mass and plate stack is established in Ansys, and the transient thermal analysis of heat transfer between air mass and plate stack at each position is carried out.

Physical model simplification and finite element model establishment

Physical model simplification. The plates in the resonator are parallel plates, and the gas movement between the plates and the energy exchange behavior of the upper and lower plates are the same. In order to investigate the temperature formation process at both ends of the stack, the heat transfer process was studied by taking a gas microcluster and a single sheet stack in the motion range around it as an example.

Finite element model establishment. The finite element model consists of two parts. The upper air column is based on the shape of the stack gap. In order to ensure that the gas fully acts on the stack during vibration, the length must be less than the stack, and the thickness should be half of the width of the stack gap, which is simplified into a cuboid. The bottom is also simplified into hexahedral plates, and only the length of the stack of plates within the range of the action of this air microcluster is taken, so the two are the same width.

According to the data in Table 1, the finite element model of plate stack and gas is established in Ansys, and the mesh shape is regular hexahedron. See Reference 26 for specific parameter Settings26. Suppose the side length of the hexahedron is h, and the smaller the h, the denser the grid. Figure 2 shows the variation of temperature difference between the two ends of the plate stack with h in one gas working cycle. Table 1 Environmental conditions and physical property parameters of materials.

Property	Value	Units	
Initial temperature	25	°C	
Initial pressure	1 × 105	Pa	
Dimensions of the gas (Length × width × height)	0.016 × 0.002 × 0.001	m	
Gas density	1.2	Kg m−3	
Specific heat capacity of gas	1000.4	J Kg−1K−1	
Gas thermal conductivity	0.0269	W m−1K−1	
Dimensions of the stack (Length × width × height)	0.024 × 0.002 × 0.002	m	
Stack density	2510	Kg m−3	
Laminated specific heat capacity	740.7	J Kg−1K−1	
Thermal conductivity of sheets	2.5	W m−1K−1	

Fig. 2 The variation of plate temperature difference with mesh side length at different operating frequencies.

When the working frequency is 16 and 72 Hz, the mesh side length is reduced from 1 to 0.5 mm, and the temperature difference rate at both ends of the plate stack is 2.83% and 1.92%, respectively, with no significant change trend. Therefore, the mesh side length is determined to be 1 mm, and the finite element model formed after the overall mesh division is completed is shown in Fig. 3. At the initial moment, the temperature of the stack and the air mass is 25 ℃, and the air mass is in the equilibrium position.Fig. 3 Finite element model.

Numerical simulation of the formation process of temperature difference between two ends of the plate stack

In order to study the formation of temperature difference between the two ends of the plate stack, the following assumptions were made in the finite element calculation:The air mass has the same temperature in the plane perpendicular to the direction of movement and is in phase with the pressure;

At the beginning of each working cycle, the air mass is in the equilibrium position, and its temperature is linearly distributed in the direction of movement;

The sound field is stable, and the change law of sound pressure conforms to the harmonic curve, without considering the viscous resistance between the gas and the plate.

According to the above assumptions, formula (4) is simplified, and the governing equation for the temperature of the stack can be obtained as follows:12 Ts1(x,ys)=γ-1γTmpmp1-1ωdTdxu1cosh1+jyδvcosh1+jy0δv+ks∂2Ts1∂vsx2

Ts1 (x,ys) indicates the first-order fluctuation of the stack temperature, that is, the change of the stack temperature over time. The actual temperature of the plate stack can be expressed as:13 Tsx,ys=Ts0x,ys+Ts1x,ys

In Eq. (13), Ts0 (x, ys) represents the initial temperature at a point in the stack. At the same time of the reciprocating movement of the gas on the surface of the plate, the heat conduction between the gas and the plate is carried out. Cause the internal temperature of the plate to change. Therefore, the gas temperature is taken as the boundary condition in numerical simulation, and the plate stack temperature is solved.

Determination of initial temperature

At the beginning of the first working cycle, the stack and the air mass have the same initial temperature. In the process of air mass movement, the temperature fluctuation generated under the action of sound pressure is assigned to the air mass. According to Formula (2), the temperature of the plate stack is determined by the heat conduction between the air mass and the plate stack and the internal heat conduction of the plate stack in the thermoacoustic effect. The air mass moves the heat from one side of the equilibrium position to the other side, and the temperature distribution inside the plate and the air mass changes after each working cycle. Starting from the second working cycle, and for each subsequent working cycle, the initial temperature of the air mass and the stack needs to be re-determined.The initial air mass temperature is determined. When the air mass vibrates around the equilibrium position, the temperature rise or fall is analyzed with the temperature of the equilibrium position as a reference, so the initial temperature of the air mass needs to be determined at the beginning of each working cycle. After the end of each working cycle, the temperature at both ends of the surface of the stack is read out from the temperature cloud map. The leftmost end is the high temperature end, and the temperature is recorded as Tg; the rightmost end is the low temperature end, and the temperature is recorded as Td. According to the outer dimension of the surface of the stack, the temperature is calculated and arranged according to the linear distribution law along the length direction. Assume that the grid number on the upper surface of the stack is x, and the grid number on the lower surface of the gas column is y, as shown in Fig. 4. The dashed line area in the figure is the contact area between the air mass and the stack.

Fig. 4 Temperature linear arrangement diagram.

where14 ΔT=Tg-Tdx

The temperature of the node near the low-temperature end of the i grid is:15 Ti=Tg-iΔTi=0,1.2.3…m

In the temperature calculated by Eq. (13), the temperature in the dotted line region in Fig. 4 is assigned to the air mass as the initial temperature of the air mass in the next working period.(2) The initial temperature of the stack is determined. With each step the air mass moves, it conducts heat with the plate. The temperature change at a certain point on the plate is the result of the heat conduction between the air mass and the plate and the heat leakage inside the plate. At the beginning of each step, the temperature value of the stack in the thermal analysis result of the previous step is read as the initial temperature of the stack in this step.

(3) Load application. The gas vibrates simply near its equilibrium position and exerts a displacement and temperature load on the whole gas. With the equilibrium position as the starting point and the right as the positive direction, the displacement is a sine function of time. The temperature load caused by the sound field pressure can be calculated according to the first term on the right of the equal sign of Eq. (3), which is also a sine function with respect to time. The temperature variation of the air mass is the largest where the displacement on both sides of the equilibrium position is the largest, that is, the temperature of the air mass is the maximum and minimum.

Result analysis

Figure 5 shows the temperature distribution of the air mass at different positions of the stack during the first three working cycles. It can be intuitively seen from the figure that under the influence of variable temperature gas, the temperature difference between the two ends of the plate is formed, with red representing high temperature and blue representing low temperature. When the gas carries out simple harmonic vibration, the influence on the plate at every moment will be superimposed in the plate, so there is a temperature gradient formed along the direction of sound wave propagation and the normal direction on the plate surface. The temperature distribution is not uniform at the beginning of operation. Along the direction of acoustic wave propagation, the temperature at both ends of the stack changes greatly, while the temperature in the middle part of the stack changes little. It can be seen that in the thermoacoustic effect, the net heat transfer between the plate stack and the air mass is mostly reflected in the two ends of the plate stack, which is also the result of the heat being transferred by the air mass relay. By comparing the temperature changes of the plates in different working cycles in Fig. 5, it is found that with the increase of heat transfer times of air mass, the temperature difference between the two ends of the plates gradually increases, and in the plates, the heat gradually diffuses from the high temperature end to the low temperature end. Therefore, the gas pumps the heat from the low temperature end of the stack to the high temperature end, resulting in the temperature difference between the two ends of the stack, and the temperature change in the middle part of the stack is mainly affected by the heat conduction inside the stack.Fig. 5 Plate temperature variation diagram.

Figure 6 shows the temperature changes at both ends of the plate stack in the thermoacoustic effect. When starting to work, the heat at the low temperature end is gradually moved to the high temperature end, so that the temperature at the high temperature end increases and the temperature at the low temperature end decreases. The temperature difference between the two ends of the plate increases rapidly in the initial work, and then increases more and more slowly, and finally is in a stable state. It is worth noting that before the system reaches thermal equilibrium, the temperature at the low temperature end does not continue to decrease, but first decreases to a certain temperature and then slowly increases, but the increase speed is relatively slow compared with the high temperature end. The physical property parameters of sheet and gas in Table 1 are the same as those in reference 9. From the simulation results, it is found that the trend of temperature change at both ends of the plate stack is in good agreement with the experimental results in reference 9, but there is a deviation in the numerical value. This deviation is explained in the study in reference 9, which is due to the system vibration caused by the loosening of parts in the experiment and some nonlinear factors.Fig. 6 Temperature at both ends of the stack: (a) temperature change trend of the high and low end of the stack, (b) temperature difference between the two ends of the stack.

Under the action of the sound field, the micro-air mass vibrates around its equilibrium position, transferring heat from one end of the equilibrium position to the other end. The whole gas column on the surface of the stack can be regarded as composed of multiple gas microclusters. During the vibration of the gas column, the heat transfer process of all air masses and the stack is synchronous, that is, all air masses absorb heat from the stack and transfer heat to the stack simultaneously. Because of the continuity of the gas, the plate is not able to perceive the process of moving the heat of the single air mass, but the temperature of the whole gas column in the surface is changing periodically. In fact, the process of the change of the gas column temperature is also the process of the acoustic work, which is to be converted into the heat of the gas, so the sound field is causing the gas to move the heat and the gas temperature is rising.

During the operation of the system, the sound field causes the gas to vibrate. At the same time, part of the sound power is converted into the internal energy of the gas, so that the heat of the gas is increased.

The amount of heat change at the high temperature end of the stack is:16 ΔQhot=Q-Qs+QW

The amount of heat change at the low temperature end of the stack is:17 ΔQlow=Q-Qs-QW

In Formula (16) and (17) :ΔQhot: The change value of the heat at the high temperature end when the gas works for one cycle;

ΔQlow: The change value of the heat at the low temperature end when the gas works for one cycle;

Q: The heat absorbed by a gas from the cold end. That is, the heat of the air mass moving from the low temperature end to the high temperature end;

Qs: The heat inside the plate is transferred from the high temperature end to the low temperature end;

QW: Heat energy converted from sound energy.18 ΔQhot-ΔQlow=2Qw

According to Eq. (18), the gas transfers heat from the low temperature end of the stack to the high temperature end, while consuming sound power. The heat absorbed by the high temperature end of the stack is greater than the heat released by the low temperature end of the stack. So the increase of the temperature at the hot end of the stack is greater than the decrease of the temperature at the cold end of the stack. Therefore, in the design of thermoacoustic refrigerator, In order to provide a stable cooling temperature, the heat exchange value between the high-temperature end heat exchanger and the outside world is greater than the heat exchange value between the low-temperature end heat exchanger and the outside world.

Analysis of the impact factors of the two sides of the plate

The effect of frequency on the temperature of the plate

The sound source driven by electromagnetism, such as loudspeaker and exciter, is widely used in the experimental research of thermoacoustic refrigerator. In practice, there are insufficient excitation intensity (amplitude) and low radiation efficiency. Increasing the operating frequency can improve the sound power density, but also bring some additional problems: such as excessive heat, sound field dispersion, etc., and most exciter when the frequency and cavity inflation pressure increase, the output displacement will decrease. Therefore, the development of powerful low frequency large amplitude sound source is a difficult problem to be solved in thermoacoustic refrigerator. In the design of thermoacoustic refrigerator, Tijani6 made an analysis from the perspective of vibration, improved the existing loudspeaker, designed a gas spring structure, so that the resonance frequency of the sound source matched the resonance frequency of the gas in the resonator, increased the sound field pressure, and the working frequency was 400 Hz. Ahmed I. Abd El-Rahman9 designed a dual-sound source thermoacoustic refrigerator. In order to increase the sound field pressure, the output displacement amplitude of the sound source was set to 19 mm and the working frequency was 42 Hz.

In fact, there is an optimal operating frequency in thermoacoustic refrigerator. Kriengkrai Assawamartbunlue27 reported the optimal operating frequency and influencing factors in detail. Based on this, the research group proposed to explore ways to reduce the optimal operating frequency of the system and reduce the volume of the system under the premise of meeting the refrigeration requirements by adjusting the structural parameters of the system and the physical property parameters of the working medium and the variation of the stack temperature of the system under low frequency is also discussed. Therefore, combined with the research of Ahmed I. Abd El-Rahman, this paper determined the frequency as 16 Hz-72 Hz.

As shown in Fig. 7, it is the relationship between the stack temperature and working time at different working frequencies when the sound pressure amplitude is constant. In the figure, the horizontal coordinate represents the ratio of working time t to the vibration period Ts of the air mass, and the vertical coordinate represents the temperature at the end of the plate stack. It can be found from the figure that when the sound pressure amplitude and the physical property parameters of the stack are constant, the temperature variation trend of the low temperature end and the high temperature end of the stack is basically the same under different working frequencies, but the temperature value will be different. The lower the frequency, the more significant the trend of the temperature recovery of the plate. When the working frequency is 16 Hz and 32 Hz, the temperature difference between the two sides of the plate is slightly different, but it is not obvious that there will be significant changes when the frequency is 72 Hz. In reference 28, Praitoon Chaiwongsa et al. conducted an experimental study on the performance of a standing wave thermoacoustic refrigerator28. When the input sound power is 30w and the operating frequency is 160 Hz, 163 Hz and 165 Hz respectively, after the system reaches a stable state, the temperature at the low temperature end of the plate is 26℃, 24℃ and 23℃ respectively, while the temperature difference between the two ends of the plate stack does not change significantly, which is consistent with the simulation results in this paper. Such a change trend can be understood as within a certain range of operating frequencies. When the frequency increases, the vibration period of the air mass decreases, and the heat transfer time between the air mass and the plate decreases. The heat absorbed by the air mass from the low temperature end of the plate is reduced. In the stable working state of the system without load, the heat absorbed by the air mass from the low temperature end is equal to the heat transferred from the high temperature end to the low temperature end in the plate stack during a working cycle. That is, the heat removed from the low temperature end by the air mass is equal to the heat transferred to the low temperature end from the heat leakage inside the plate. Therefore, when the frequency increases, the heat transferred by the air mass from the low temperature end decreases, and the heat leakage of the plate stack also decreases, and the heat leakage of the plate stack is positively correlated with the thermal conductivity of the plate stack and the temperature difference between the two ends. Therefore, when the sound pressure amplitude and the thermal conductivity of the plate stack are constant, when the frequency increases, the heat leakage of the plate stack decreases, and the temperature difference between the two ends of the plate stack decreases. Therefore, reducing the operating frequency under the premise of keeping the sound pressure unchanged is conducive to increasing the temperature difference between the two ends of the plate. However, in the design of thermoacoustic refrigerator, to reduce the frequency and keep the sound pressure unchanged, it is necessary to increase the amplitude. Therefore, the development of powerful low frequency large amplitude sound source is an urgent problem to be solved in thermoacoustic refrigerator.Fig. 7 Relationship between the temperature at both ends of the stack and the gas oscillation period: (*) frequency: 16 Hz (o) frequency: 32 Hz (△) frequency: 72 Hz.

Influence of thermal conductivity of the plates on the temperature of both ends of the plates

The thermal conductivity of the plate has a significant influence on the heat conduction between the gas and the plate and the heat leakage of the plate itself. During a working cycle, if the temperature difference between the plate and the gas is determined, the greater the thermal conductivity of the plate, the more conducive to the heat exchange between the gas and the plate, and the more heat conducted in the same time. Therefore, the heat transfer of air mass from the low temperature end to the high temperature end during a working cycle will increase with the increase of the thermal conductivity of the stack. At the same time, the heat flow from the high temperature end to the low temperature end also increases with the increase of the thermal conductivity of the stack. If the effect of the overall temperature rise of the gas is not considered, the heat of the low temperature of the plate is used as the standard for measuring the refrigerating quantity of the heat noise refrigerator.19 ΔQlow=Q-Qs

where ΔQlow: refrigerating capacity;

In the operation of thermoacoustic refrigerator, there are many factors affecting the heat transfer, in addition to the thermal conductivity of gas and plate stack, it is also affected by the sound field parameters, plate stack position and other factors. Therefore, in the design, try to choose a small thermal conductivity of the plate, reduce the internal heat leakage of the plate, and improve the heat transfer of the gas by adjusting other factors. Table 2 lists the selection of thermal conductivity of sheet in some experimental studies. There are many researches on the selection of lamination parameters, which are not listed here. In this paper, The thermal conductivity range of 0.2–15 W/(mK) is determined by referring to a large number of literatures. Table 2 Selected examples of thermal conductivity of sheet in thermoacoustic refrigerator research.

Author	Thermal conductivity of plate	
Tijani et al6	0.16 W/(m K)	
Ahmed I. AbdEl-Rahman et al9	2.5 W/(m K)	
Emanuele Sarpero et al29	0.167 W/(m K)	
Mahmoud A. Alamir et al30	0.45 W/(m K)	
Ramesh Nayak. B et al31	1.5 W/(m K)	

Figure 8 shows the relationship between the temperature at both ends of the stack and the working time of the air mass under different thermal conductivity of the stack. As shown in Fig. 8a, the temperature changes at the low and high temperature ends of the plate are related to the thermal conductivity of the plate. When the thermal conductivity of the stack is 0.2w·m-1·K-1, the temperature at the low temperature end continues to decrease until the system reaches a stable state, and the temperature does not rise significantly during the whole process. Compared with the initial state temperature, when the system works stably, the increase of temperature at the high temperature end is greater than the decrease of temperature at the low temperature end. The same change rule is also obtained in the studies on the temperature difference between plates in references28,32,33. According to Fig. 8b, the temperature difference between the two ends of the stack increases with the decrease of thermal conductivity, but the increase is relatively slow.Fig. 8 Relationship between the temperature at both ends of the stack and the gas working time: (o) Stack thermal conductivity: 15w·m-1·K-1 (*) Stack thermal conductivity: 2.5w·m-1·K-1 (△) Stack thermal conductivity: 0.2w·m-1·K-1  

Influence of stack length on the temperature at both ends of stack.

Figure 9 shows the variation of the temperature at both ends of the stack with the working time of the air mass under different stack lengths .The longer the length of the stack, the greater the temperature difference between the two ends of the stack. Therefore, the temperature difference between the two ends of the stack can be adjusted by changing the length of the stack.Fig. 9 Relationship between temperature at both ends of the stack and working time: (*) Stack length: 0.024 m (o) Stack length: 0.016 m (△) Stack length: 0.012 m.

The theory of short plate in series

In the thermoacoustic effect, the gas in the resonator vibrates around its own equilibrium position, and the temperature change is very weak. However, when the plates are superimposed, a large temperature difference can appear at both ends of the plates through reasonable design. This magical physical phenomenon has attracted the attention of many researchers, and a lot of research has been done on the factors affecting the temperature difference between the two ends of the plate. However, there is no detailed theoretical explanation of how the large temperature difference between the two ends of the plate is generated. Therefore, this paper puts forward the theory of short plate tandem to explain this problem.

When the temperature gradient is formed in the stack, the temperature gradient is also formed in the gas, and the gas is consistent with the temperature gradient on the surface of the stack. Therefore, the gas above the stack has the same temperature distribution as the surface of the stack at the beginning of each working cycle. By discretizing the plate stack and the gas, the entire plate stack can be regarded as a series of multiple short plates, and the gas column above the plate stack can also be regarded as a plurality of micro air masses, as shown in Fig. 10.Fig. 10 Schematic diagram of plate stack and gas column dispersion.

Under the action of the sound field, each air mass synchronously moves the heat from one side of its equilibrium position to the other side to realize the pumping heat. According to reference 34, the amount of heat transferred by the air mass depends on the heat absorbed from the plate during the movement of the air mass from the equilibrium position to the limit position34. Therefore, a stack of one amplitude length can be called a short stack. For the n short plate in Fig. 10, the temperature difference between the two ends can be expressed as:20 ΔTn=T(n,1)-T(n,2)

At the equilibrium position, the air mass has the same temperature distribution as the surface of the short plate. The temperature change of the air mass during the movement is small, and the temperature difference between the two ends of the n short plate is also small. According to the continuity of the temperature change of the stack, the temperature of the low temperature end (right end) of the n short stack is the same as that of the high temperature end (left end) of the n + 1 short plate. By analogy, the temperature boundary conditions at both ends of each short plate stack are:21 T(n-1,2)=T(n,1)

22 T(n,2)=T(n+1,1)

23 Tn+1,2=Tn+2,1

In formula (20), (21), (22) and (23), the first number in the subscript brackets represents the short plate stack, and the second number represents the end of the short plate stack (1 represents the high temperature end and 2 represents the low temperature end). For example, T (n-1,2) ndicates the low temperature end (right end) of the n-1 short stack. From this we can see that the temperature difference between the two ends of the whole stack is equal to the algebraic sum of the temperature difference between the two ends of all the short stacks. Although the temperature difference ∆T at both ends of a short plate stack is small, as the number of short plates in series increases, the temperature difference at both ends of the entire plate stack also increases. Therefore, according to the design requirements, the significant temperature difference between the two ends of the stack can be obtained by increasing the number of short stacks. As shown in Fig. 9 in 4.3, the longer the stack, the greater the temperature difference between the two ends.

In the process of the system from the initial state to the stable state, the temperature difference between the two ends of the short-plate stack increases gradually, and the gas temperature change T1 calculated by Eq. (3) becomes smaller and smaller. When the system reaches a stable working state, T1 no longer changes. With reference to reference 35, Eq. (3) is calculated, and it can be obtained as follows35:24 ηγ-1γTmPmp1=1jωu1dTmdx

η≤1, is the proportional coefficient. Tp1=γ-1γTmPmp1, Represents the amount of fluctuation in air mass temperature caused by sound pressure, Substitute B=1jωu1 into Eq. (18), The temperature gradient of the short sheet surface can be expressed as:25 dTmdx=ηTp1B

When η = 1, the temperature gradient of the plate reaches the maximum, and Eq. (19) can be simplified as:26 dTmdx=Tp1B

According to Eq. (26), when the amplitude of sound pressure and the displacement amplitude of air mass vibration are determined, the temperature gradient of short plate stack at the same position can be determined. According to the continuity of the temperature of the plate, the temperature gradient of the whole plate can be obtained. The temperature difference between the two ends of the plate is:27 ΔT=dTmdx·L

In formula (27), L represents the length of the plate. The temperature difference between the two ends of the plate stack is an important design parameter of thermoacoustic refrigerator. Therefore, under certain conditions according, the idea of short plate tandem can provide a theoretical basis for determining the length of plate.

Conclusions

The two-dimensional heat transfer expression of stack temperature is established, and the temperature graph of stack temperature with working time is obtained by numerical calculation. It can directly reflect the internal temperature variation law of the plate, which is helpful to further understand the thermoacoustic effect. Through numerical simulation, the temperature variation law at both ends of the plate pile was studied:The net heat exchange between the gas and the plate is mainly reflected at both ends of the plate. At the beginning of the work, the temperature changes at both ends of the stack are sharp, and the temperature changes in the middle part of the stack are gentle.

In the process of system work, the sound field in promoting gas vibration, to achieve the pump heat at the same time, part of the sound work into the internal energy of the gas, so that the gas heat increase. The heat absorbed by the high temperature end of the stack is greater than the heat released by the low temperature end of the stack, so the temperature increase of the hot end of the stack is greater than the temperature decrease of the cold end of the stack. The temperature at the low temperature end of the stack decreases rapidly first and then increases slowly, which is related to the thermal conductivity of the stack. The smaller the thermal conductivity, the slower the recovery rate of the low temperature end temperature after falling. In the design, a smaller thermal conductivity of the plate is generally selected to reduce the heat leakage inside the plate, and other parameters are adjusted to increase the temperature difference between the two ends of the plate.

Operating frequency, stack length and thermal conductivity of stack have influence on the temperature variation at both ends of stack. When the sound pressure is constant, the longer the stack length, the more significant the temperature difference between the two ends, indicating that increasing the stack length can effectively increase the temperature difference between the two ends of the stack. However, in the thermoacoustic effect, due to the viscous resistance between the stack and the gas, increasing the stack length will lead to the decrease of the system performance coefficient8,22. Therefore, in the design of thermoacoustic refrigerator, the determination of plate length should take into account the cooling temperature and system performance coefficient.

Operating frequency is the key factor affecting the sound field. When the sound pressure amplitude is constant, when the operating frequency variation range is small, the temperature difference between the two ends of the plate stack is not obvious; when the operating frequency changes greatly, the temperature difference between the two ends of the plate stack is obviously different. In the design of thermoacoustic refrigerator, the amplitude and frequency of sound source can be adjusted, and the frequency can be reduced under the premise of constant sound field pressure, which will be conducive to increasing the temperature difference between the two ends of the plate stack. In a thermoacoustic system, the sound power is positively correlated with the sound pressure, and the influence of plate stack temperature difference on the heat flux between plates is very important. Therefore, the heat flux and sound power of the system can be adjusted by changing the sound field parameters, and then the realization of the system cooling temperature and the improvement of the performance coefficient can be comprehensively considered.

The increasing value of the temperature at the high temperature end of the stack is greater than the decreasing value of the temperature at the low temperature end of the stack. Therefore, in the design of thermoacoustic refrigerator, it is necessary to ensure that the heat transferred between the high-temperature end heat exchanger and the outside world is higher than the heat transferred between the low-temperature end heat exchanger and the outside world, so that the system can provide a stable cooling temperature.

The theory of short plates tandem is proposed. This systematically explains the formation mechanism of the large temperature difference between the two ends of the plate from the theoretical aspect. Under certain design conditions, it can be used as the theoretical basis for selecting the length of the stack.

Acknowledgements

The research was supported by the National Natural Science Foundation of China (Grant Number: 52266005)

Author contributions

Xiangbin Liu: the idea is established, the first draft is written, the formula is written and calculated. Wang: Project formulation, preliminary draft review, fund preparation Nan  Meng：Formula modification, The short board tandem theory is supplemented and expanded.

Data availability

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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