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Numerical analysis on thermoacoustic prime mover
Journal of Sound and Vibration 463 (2019) 114946
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Numerical analysis on thermoacoustic prime mover Dongwei Zhang a, Erhui Jiang a, Chao Shen b, *, Junjie Zhou a, Weiwei Yang c, Yaling He c a
Center on the Technology and Equipments for Energy Saving in Thermal Energy System of MOE, School of Mechanical and Power Engineering, Zhengzhou University, Zhengzhou, Henan, 450001, China b School of Civil Engineering, Zhengzhou University, Zhengzhou, Henan, 450001, China c Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shanxi, 710049, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 March 2019 Received in revised form 3 September 2019 Accepted 9 September 2019 Available online 9 September 2019 Handling Editor: L. Huang
A two-dimensional numerical model based on compressible SIMPLE algorithm was developed to simulate the evolution process of self-excited oscillation in a thermoacoustic engine at two different heating conditions. Moreover, the performance analysis of the thermoacoustic engine was researched. On one hand, the influence of stack parameters and charge pressure on the onset temperature of self-excited thermoacoustic oscillation was exhibited and analyzed. On the other hand, the impact of different gas types on the onset temperature difference was investigated. The results indicated that the self-excited thermoacoustic oscillation happened with the same performance for both two heating conditions. The minimum onset temperature difference can be achieved when the dimensionless stack length and position are Lr ¼ 0.055e0.06 and Xr ¼ 0.14e0.16, respectively. For a detail, the optimal stack plate thickness is 33% of the stack channel width when the computational width is 3 times as large as the thermal penetration depth of the working gas. Furthermore, the onset temperature difference increased with the increasing of the charge pressure in prime mover. For different working gas, the onset temperature difference was the highest for argon, which was followed by helium and nitrogen. Finally, the performance was improved with the increase of the temperature difference between the two exchangers in the prime mover. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Thermoacoustic prime mover Self-excited oscillations Compressible SIMPLE algorithm
1. Introduction The energy-conversion process of thermoacoustic devices can be achieved either to produce the acoustic work inducing by a temperature gradient along the stacks (i.e., a prime mover) or to pump the heat from low-temperature region to hightemperature region driving by the acoustic wave (i.e., a heat pump). This kind of technology avoids the use of moving parts and exotic materials, and does not require close tolerances and critical dimensions, which makes it attractive for specific applications [1]. During the past few decades, a great number of works have been investigated on both thermoacoustic engines and refrigerators [2e5]. As a key factor to evaluate the performance of the engine, the onset temperature of
* Corresponding author. E-mail address: [email protected] (C. Shen). https://doi.org/10.1016/j.jsv.2019.114946 0022-460X/© 2019 Elsevier Ltd. All rights reserved.
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Nomenclatures B c cp cv f h0 h1 h2 k L Lr l Ma Pr Dp p Q Rg S T DT DTonset t u v Xr
Ratio of non-dimensionless stack thickness to stack spacing Sound speed/m$s1 specific heat at constant pressure/J$kg1$K1 specific heat at constant volume/J$kg1$K1 Resonant frequency/Hz Half of the stack spacing/m Half of the stack gap/m Half the thickness of the stack/m Thermal Conductivity/W$m$K1 Resonant cavity or engine length/m Non-dimensional stack length Stack length/m Mach number Prandtl number Pressure difference/Pa Pressure/Pa Heat exchange/J Unit mass gas constant/J$kg1$K1 Source terms Temperarure/K Temperarure difference/K Onset temperature difference/K Time/s Velocity at the x direction/m$s1 Velocity at the y direction/m$s1 Non-dimensional stack position
Greek alphabet g Specific heat ratio m Dynamic viscosity/kg$m1$s1 r Density/kg$m3 f General variables of u, v and T equation u Angular frequency/rad$s1 Superscript 0 Last time value p Pressure T Temperature Correction value * The value of the first iteration level ** The value of the second iteration level *** The value of the third iteration level Subscript crit HX net p s res stk u v x y
Threshold Heat Exchanger Net output Pressure Solid Resonator Stack Velocity at the x direction Velocity at the y direction Horizontal Portrait
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thermoacoustic prime mover decides whether the waste heat can be used to drive the engine. Therefore, it is critical to exhibit the evolution of self-excited process and reveal the influence of stack parameters and charge pressure on the onset temperature. In 2015, Tartibu et al. [6] investigated the influence of the positions of the stack on the performance of the thermoacoustic refrigerators. Their results indicated that the location of the stack closing to the pressure antinode has beneficial to improve the maximum coefficient of performance of the device, while the design for maximum cooling would be increased by moving it away from the pressure antinode. In other aspect, the impact of the geometry structure on the cold end temperature of stack in thermoacoustic refrigeration system was investigated by Wantha et al. [7], in 2018. The outcomes suggested that the pin array stack structure has the best performance than spiral and circular pore stacks on the cold end temperature. Besides, the length of the resonator was also studied, in recent years. A reasonable length of the resonator is associated with the development of the energy-conversion and predicts the design of the thermoacoustic devise [8,9]. In addition, the stability of thermo-acoustics acts a major role in the performance and life of the system [10]. In recent years, N. N. Deshmukh et al. [11e14] of the Institute of Technology Bombay took lots of experimental investigation on the suppression of thermosacoustics instability. Their results indicated that the application of mic-reject to adjust the velocity and mass flow can effectively suppress the instability of thermos-acoustics. As a result, it greatly reduced the heat loss. Some meaningful results have been achieved by experimental methods. However, there still have some problems need to be solved, which is the energy conversion process in experimental tests. It is inevitable because the operation process of thermoacoustic engine along with complex unsteady oscillatory flow. Therefore, the numerical simulation has becoming popularity for the advantages of high nonlinear solution on thermoacoustic. The DeltaE was adopted by Ward et al. [15] to solve one-dimensional wave equation and compute low-amplitude thermoacoustic engines, which based on the linear acoustic approximation, in 1994. This kind of model has the advantage of evaluating the thermoacoustic apparatus performs the designing of novel apparatus to the desired performance. However, due to the limitations of its fundamental base, this kind of model is inadequate to simulate high-amplitude thermoacoustic engines and capture the nonlinear phenomena. In 1999, the Lightffot et al. [16] proposed a theoretical prediction model and this result had great agreement on the temperature difference at which oscillation begin compared with the experimental methods. Furthermore, a sort of optimization model was introduced by Tartibu et al. [17], in 2015. This model based on the discontinuous derivatives was proposed to establish a three-criterion nonlinear programing, which results would achieve the optimization of the thermoacoustic devise. Besides, the investigation of CFD code (i.e., Fluent) on the evolution of self-excited process and the nonlinear phenomena in the engine was studied by some researchers. For example, Hantschk et al. [18,19] developed a two-dimensional model and simulated two different kinds of Rijke tubes with an open-open boundary and a closed-open boundary. The evolution of non-linear selfexcited pressure oscillations was revealed and the higher harmonics during the onset of oscillations was achieved. They also indicated that the oscillations in combustion chambers were similar within Rijke tubes and the method could be employed for both of them. Furthermore, a two-dimensional computational model was proposed by Lycklama et al. [20] to study the traveling-wave thermoacoustic engine. Their results suggested that a linear temperature difference across the regenerator would trigger off the dynamic pressure oscillation. Luo et al. [21,22] investigated the impact of two different thermal boundary conditions (a specified heating load and a known surface temperature) on the self-excited oscillation amplification with time at two different thermoacoustic engines. In addition, they also tried to reveal the nonlinear phenomena of vortex formation at the ends of the stacks. A double-acting thermoacoustic heat engine was also designed to absorb concentrated solar heat [23]. In 2008, Zhou et al. [24] simulated and designed a miniature thermoacoustic Stirling engine. The calculated results indicated that an appropriate resonance tube diameter is beneficial to the improvement of the performance of the engine. A further study on the onset characteristics in different thermoacoustic engines was reported by Qiu et al. [25] and Sun et al. [26,27]. Based on linear theory of thermoacoustics, a similar work was also presented by Boroujerdi and Ziabasharhagh [28] to analyze the influence of the oscillating frequency and onset temperature on the performance of a simple standing wave thermoacoustic heat engine. Most of above studies focused on revealing the evolution of self-excited process in thermoacoustic engines. Further studies are still needed to reveal the influence of the plate and working gas parameters on the onset temperature of thermoacoustic engines. In this work, a two-dimensional numerical model based on the compressible SIMPLE algorithm was developed to simulate the evolution process of self-excited thermoacoustic oscillation. Firstly, the evolution processes of self-excited thermoacoustic oscillation at two different heating conditions were revealed. Then, the impacts of the stack parameters, the charge pressure and working gas on the onset temperature were also studied and analyzed. Finally, some conclusions have drawn.
2. Numerical simulations and theoretical analysis The compressible SIMPLE algorithm was adapted to the model of the low Mach compressible oscillatory flow in thermoacoustic engine. The simulation model including the core components of the engine was developed and the evolution of the self-excited process was analyzed.
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Fig. 1. (a) Model for thermoacoustic prime mover. The dark domain is the computation domain, which is shown magnified in (b).
2.1. Numerical model and algorithm Fig. 1 shows the physical model and simulation domain for the thermoacoustic prime mover with half wavelength, which is composed of the prime mover and a resonant tube. The part of prime mover includes a parallel-plate stack, a hot-end heat exchanger and a cold-end heat exchanger. In this system, the stacks contain many identical parallel plates and it is ratio of the gap width and tube diameter is very small. The engine can be simplified into a two-dimensional channel as marked with dark region. This two-dimensional domain can obtain adequate computational accuracy of oscillatory flow field in the engine with less computing time. Other simplifications of the thermoacoustic prime mover are as follows: (1) The working substance is regarded as ideal gas. (2) The walls of the engine are adiabatic. (3) The gas flow in the engine is considered laminar compressible flow. In thermoacoustic engine, the working fluid experiences compression and expansion process and the oscillatory flow has low Mach number (Ma<0.1). Therefore, the simulation of the self-excited process has much higher requirements for the algorithm. A compressible SIMPLE algorithm was adopted in this work, which is based on the pressure correction method to calculate the flow and the heat transfer problems [29], and the details of calculation process of SIMPLE algorithm was presented in Fig. 2. Furtherly, the inner node method was adopted to establish the staggered grid structure for discrete regions, as shown in Fig. 3. The governing equations are discretized by the finite-volume method and the cell centered scheme is adopted to get the structured grid for the domain discretization, and the details of solving process as following. (1) The new variables (u** and v**) were gained by combing the variables of the previous iteration layer (u*, v* and r*) and the progressive layer (u0, r0 and T0). (2) Saving the pressure correction equation to obtain the value of the pressure correction (p0 ). (3) The new variables (p***, u*** and v***) were got based on above correction values, and then the new density also was calculated by the equation of r** ¼ p**/(RT*). (4) Based on the energy equation to obtain the new temperature (T**).
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Fig. 2. Algorithm diagram.
(5) Take a judgment on the convergence of variables value, and if it has converged to conduct the calculation of next time layer, while return to step (1) until the value reaches convergence. (6) Store the current variables value to u0, r0 and T0, and then judge whether the set calculation time was completed. Exit the procedure when it reached the set calculation time. Otherwise return the step (1) to take the calculation of new time layer. More details of this algorithm can be found in Refs. [28e30].
2.2. Governing equations and boundary conditions In thermoacoustic engine, the flow is compressive and time-dependent. The terms of compressive and viscidity dissipations should be considered. The governing equations of the two-dimensional unsteady fluid flow and heat transfer are unified in a similar expression in Cartesian coordinate system, which can be given by
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Fig. 3. Staggered grid diagram [29].
vðrfÞ vðrufÞ vðrvfÞ v vf v vf þ þ ¼ þ Rf þ Sf Gf Gf vt vx vy vx vx vy vy
(1)
where R4 and S4 are the general source terms. Detailed expressions for G4, R4and S4 are presented in Table 1. Notice that the term S4 and m/3 in the momentum conservation equations and 2m/3 in the energy conservation equation is obtained from the second molecular viscosity coefficient, l ¼ 2m=3 (Stokes hypothesis), which m is constant in here. Totally, the conservation equations for mass, momentum and energy contain five unknown quantities: u, v, p, r and T. To ensure the closure of the governing equations, the state equation for ideal gas is also solved to close the equations:
p ¼ rRg T
(2)
In the domain of parallel-plate stack, the energy conservation equation for solid phase is also needed
rs
vTs ksx v2 Ts ksy v2 Ts ¼ þ vt cs vx2 cs vy2
(3)
where ksx and ksy are the thermal conductivities of solid phase for the horizontal and vertical axes, respectively. In prime mover, there exists temperature gradient between the hot-end exchanger and the cold-end heat exchanger. Two different heating conditions were adopted because of different heat sources in practical applications. For the one halfpffiffiffiffiffiffiffiffiffiffiffi wavelength standing wave resonator, the frequency can be obtained from the formula of f ¼ kRg T =2L. Therefore, the length of the resonator is 2 m which has a frequency of 85.96 Hz for nitrogen with a charge pressure of 1.5 MPa. The time step
Table 1 Content of the varies in the unified governing equations. Governing equations
f
Gf
Rf
Sf
Mass x-Momentum
1 u
0
0
y- Momentum
v
m
0 vp vx vp vy
Energy
T
m
m Pr
m v
. . i 1 hvp þ V,ðp V Þ pðV, V Þ Cp vt
.
ðV, V Þ 3 vy . m v ðV, V Þ þ rg 3 vy 8 > " 2 # > > vu 2 vv > > > 2 þ > vx vy m<
9 > > > > > > > =
> Cp > > > > ! > vv vu 2 2 > > > ðV$ V Þ2 > > > > þ vx þ vy > 3 : ;
þ
r _ $q
Cp
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is set to be 106 s, as a result of 1.18 104 time integration points of each oscillation period. The computational domain is dispersed with structured staggered grid and the 2D mesh grid system is 140 32 which could get a reasonable computational time and simulation results.
2.3. Validation of numerical algorithm The numerical algorithm developed in this work has been applied to different cases of thermoacoustic engine. Firstly, computational codes were developed to solve low Mach number, compressible oscillatory flow. Two numerical examples of lid driven flow and shock tube were shown in Refs. [31,32]. Secondly, the two-dimensional numerical simulations of resonant oscillations in a rectangular chamber and the thermoacoustic refrigerators were exhibited. The results with low amplitude oscillations agree well with that of DeltaE model based on linear theory. The large amplitude results accurately captured the nonlinear phenomena which also agree well with the experimental results reported by Seanger and Hudson [33]. More detailed results can be seen in Refs. [34,35]. Furthermore, a two-dimensional numerical simulation on a half standing-wave thermoacoustic engine was performed. The self-excited oscillations process and the acoustical characteristics of the pressure and velocity wave were presented and analyzed, which was verified the method applicability for thermoacoustic heat engine [36].
2.4. Analysis on the self-excited process of the thermoacoustic prime mover It has known that the standing-wave thermoacoustic engine self-excites and maintains the oscillatory movement, which realizes the conversion from heat energy to acoustic power [37]. The self-excited evolution is as follows. In the beginning, the hot heat exchanger is heated continuously but the cold heat exchanger is kept at a low temperature (e.g. ambient temperature). Thus the temperature gradient along the stacks increases quickly. Once the temperature gradient reaches the starting points, the time-average thermodynamic stream is formed along the stacks. These gas disturbance triggered by the temperature gradient has a short accelerated process in the stack channel. Then, the gas disturbance must overcome the dissipation in the movement to reach the whole resonator. The dissipation is lead by the heat exchangers and resonator tube during its dissemination process. Finally, the gas disturbance forms continuously and eventually is enlarged by the resonator. In the above processes, all the factors that improve the energy of the disturbance gas or reduce the movement dissipation can make the energy conversion from the heat to the acoustic work more efficiently, thus reducing the onset temperature difference. The Q was introduced by Atchley [38e40] to evaluate the self-excited evolution of a thermoacoustic standingwave engine, which is defined as:
Q¼
uEst Enet
(4)
where u is the angular frequency, and Est is the energy stored in the engines which can be calculated by integrating the timeaveraged acoustic energy density throughout the entire resonator. Enet is the net power of the entire engine. The expression is as follows:
Enet ¼ Estk þ EHX þ Eres
(5)
where the subscripts ‘stk’, ‘HX’ and 'res' refer to the stack, the (hot and cold) heat exchangers, and the resonator. In above three energy items, EHX and Eres are the dissipations in the heat exchangers and resonator tube, which has a negative effect on the self-excited process. Estk represents the positive acoustic energy sourced from the temperature gradient along the stacks. There is the critical temperature gradient VTcrit resulted in Estk ¼ 0. Thermoacoustic engine is unstable equilibrium at critical temperature gradient, when VT < VTcrit , Estk<0; and VT > VTcrit , Estk>0. It can be seen that the main active force for the self-excited process is the temperature gradient in the domain of the stacks, and the hinder force is the dissipations in the heat exchangers and resonator tube. The gas disturbance generated by the critical temperature gradient must overcome the self-resistance of the stack. Because the initial disturbance is delivered from the high-temperature region to the low area and experienced decompression exothermic process, the gas disturbance is accelerated during this process and finally moves outside of the stack. When the accelerated gas disturbance could overcome the dissipation of the heat exchangers and resonator tube, the thermoacoustic engine starts the self-excited evolution. Otherwise, when the gas disturbance is too small or the acceleration is insufficient, the self-excited process of the thermoacoustic engine can not happen. For convenient description, some dimensionless parameters are defined in this work. They are the dimensionless stack position, the dimensionless stack length and the dimensionless stack thickness. The first one is the ratio of the distance between the left end of the stack and close end of the resonator to the resonator length: Xr ¼ xp/L. The second one is the ratio of the length of the stack to the resonator length: Lr ¼ l/L. The last one is the ratio of the thickness to the spacing of the stack: B ¼ h2/h1.
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3. Self-excited evolution process and optimized parameters of onset temperature In this part, the evolution processes of self-excited thermoacoustic oscillation at two different heating conditions are exhibited. Then, the impacts of the stack parameters, the charge pressure and working gas on the onset temperature are analyzed and revealed. For the optimization of the stack parameters, the position, the length, the thickness and the spacing of the stack interact with each other. Thus, the influences of the position and the length on the onset temperature are presented with constant thickness and spacing of the stack. Secondly, based on the optimized position and length, the impacts of the thickness and spacing on the onset temperature are analyzed. Finally, the other two factors of the charge pressure and working gas influencing on the onset temperature are given with the optimized parameters of the stack.
3.1. Self-excited evolution process with two different heating conditions In this paper, the impact of different heating conditions on the process of the self-excited evolution was investigated, which with a constant value of the temperature and heat flux. At two different heating conditions, the dynamic pressure evolutions and spectrum analysis in position G in Fig. 1 was presented in Fig. 4 and Fig. 5. The gas in thermoacoustic engine is nitrogen with charge pressure of 1.5 MPa. The constant temperature difference between the two heat exchangers is 600 K, and the constant heat flux in the domain of hot heat exchanger is 5 MW∙m2, which could also generate a temperature difference of 600 K at the two ends of the stacks. It can be seen from Fig. 4 that the charge pressures are all increasing with heating before the self-excited evolution. With the development of the self-excited process, the amplitude of the oscillatory pressure increases but the mean pressure evidently decreases with time. When the pressure reaches stable oscillation, they have the
Fig. 4. Evolution processes of self-excited pressure oscillations in right end of the resonator with two different heating conditions: (a) constant temperature gradient condition; (b) constant heat flux boundary condition.
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Fig. 5. Spectrum diagrams at different heating conditions; (a) constant temperature difference; (b) constant heat flux.
almost the same pressure amplitudes at the two heating boundaries. It can be indicated that the same temperature gradient with different heat modes has the same performance of the pressure oscillation. It can also be seen from Fig. 4 that the boundary of constant temperature difference could generate the temperature gradient along the stack more quickly. So the self-excited process begins near 2 s. For the boundary of constant heating flux, it need some preheating time, therefore, the thermoacoustic engine starts near 4 s once the calculation began. Because of the longer heating time, the boundary of constant heating flux has a larger start-up pressure than that of constant temperature difference. But for the evolution of the thermoacoustic engine self-excited process, the boundary of constant temperature difference could establish the temperature gradient more quickly and it could easily get the onset temperature with different parameters. Therefore, in this paper, the heating boundaries of constant temperature difference much better than that with constant heat flux. In addition, Fig. 5 exhibits that the frequency oscillation has the same trend at two different heating boundaries. As above discussed, the performance of the pressure oscillation of resonator shows much well at any heating boundaries. However, the value of fundamental frequency of resonator much closer to the theoretical value of 89.96 Hz, at the heating boundary of constant temperature difference. Thus, it can also consider that the numerical simulation of resonator adopted the heating boundary of constant temperature different shows excellent performance. As previous introduced in introduction section, the D. M. Sun et al. [22,23] has conducted a success simulation on the onset characteristics in thermoacoustic engines. Compare on this method or algorithm, our algorithm has the advantage of simulating pressure wave nonlinear oscillation. Therefore, in this work, the numerical model based on compressible SIMPLE algorithm with the constant temperature difference heating boundary was adopted in following work.
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3.2. Influence of the stack parameters on the onset temperature The stack is the most important component in the thermoacoustic core (the hot and cold heat exchangers are also included), which has much influence on the onset temperature of the engine. In this part, different parameters of the parallel plate length, position, thickness and spacing are studied and analyzed. The onset temperature change with these parameters is revealed and presented. 3.2.1. The length of the stack It is known that large temperature gradient along the stack benefits for the thermoacoustic self-excited evolution. It has two ways to obtain large temperature gradient, one is increasing the temperature difference, and the other is decreasing the length of the stack. However, it should be also noticed that the stack need a certain length providing for the sufficient acceleration length of the gas disturbance, and the small stack length will lead to direct contraction of oscillatory gas between the hot and cold heat exchanger, thus degrades the energy conversion efficiency in the engine. Fig. 6 shows the onset temperature difference change with the dimensionless stack length of three charge pressures. The charge gas is nitrogen with the dimensionless stack position of Xr ¼ 0.18 and the dimensionless stack thickness of B ¼ 0.33. It can be seen that the onset temperature difference reaches the lowest value with a dimensionless stack length of 0.055e0.06. When the dimensionless stack length is larger than 0.18, the thermoacoustic engine is difficult to reach self-excited process. The onset temperature difference of the dimensionless stack length shows a slight increase from 0.055 to 0.065 with the increasing of the charge pressure from 1.0 MPa to 2.0 MPa. As mentioned before, when the dimensionless stack length of Lr is small, the sufficient temperature gradient can be obtained with smaller temperature difference between the two heat exchangers. Thus the gas disturbance can be generated easily near the plate. However, a relatively small stack length leads to a short acceleration process of the gas disturbance in the stack domain. The gas disturbance can not overcome the dissipation to reach self-excited state. As a result, the larger temperature gradient is needed in the domain of the stack to generate the gas disturbance with higher energy. So it need higher temperature difference when the stack length is too short. As the stack length reaches Lr ¼ 0.055, the length required for acceleration of the gas disturbance is enough and the temperature gradient also reaches the critical value. Thus the stack length has the optimal value to generate and accelerate the gas disturbance, which exactly enable overcome the heat exchangers and resonator resistance, and then reach self-excited state. Although the acceleration process is longer with increase of the length, the temperature gradient generating the gas disturbance is constant in the stack domain. Therefore, the temprature difference between the two heat exchangers increases with the increase of the stack length. With the charge pressure increasing, a large temperature gradient is needed to overcome the resistance of the stack and the flow resistance in the resonator. As a result, the dimensionless stack length which is corresponding the lowest onset temperature increases with the increasing of the charge pressure. 3.2.2. The position of the stack The charge pressure is 1.5 MPa and the dimensionless stack thickness is B ¼ 0.33. Fig. 7 presents the onset temperature difference changes with the stack position at four different stack lengths. It can be seen that the onset temperature difference firstly descends and then increases as the stack moving from the close end to the middle of the resonator. The position with the lowest onset temperature increase from Xr ¼ 0.14 to Xr ¼ 0.16 with the increase of the stack length. The onset temperature difference increases quickly with the increase of the stack length and the engine almost can not reach self-excited process when the stack right end is close to the middle of the resonator.
Fig. 6. The onset temperature difference changes with the stack length in different charge pressures.
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When the stack is near the close end of the resonator, the gas disturbance is generated at the velocity node and the pressure antinode. Although the gas disturbance adequately experiences compression and expansion process, the velocity amplitude is too small which leads to short acceleration process, thus the initial disturbance can not be accelerated to reach to the critical point of the self-excited evolution. As a result, the large temperature gradient is needed to increase the initial gas disturbance near the close end of the resonator. With the stack moving to the middle of the resonator, the compression and expansion process of the gas disturbance is shortened but the acceleration process is evidently increased. So the onset temperature difference is reduced and reached the minimum at Xr ¼ 0.14e0.16, as shown in Fig. 7. When the stack moves to near the middle of the resonator at which is close to the velocity antinode and the pressure node, the gas disturbance has much longer acceleration process but inadequate compression and expansion process. Therefore, the gas disturbance energy has slightly upgraded in the compression and expansion process. It needs large temperature gradient to obtain much violent initial gas disturbance and the onset temperature evidently increases. When the stack is in the middle of the resonator (e.g. Xr þ Lr > 0.48), the gas disturbance has no acceleration process here and the thermoacoustic engine can not reach the selfexcited evolution. With the increase of the stack length, the initial gas disturbance has longer acceleration process during the stack moving to the middle of the resonator. The position with the lowest onset temperature increase from Xr ¼ 0.14 to Xr ¼ 0.16. However, because initial gas disturbance need constant critical temperature gradient, the onset temperature increase with the increase of the stack length. 3.2.3. The thickness and the spacing of the parallel plate The thickness and the spacing of the parallel plate determine the thermal contraction characteristic between the working gas and the plate wall. The Rayleigh criterion indicated that the proper parameters of the stack should lead to inadequate thermal contact between the gas parcels and the solid wall. Thus the several times of the thermal penetration depths are recommended in the actual thermoacoustic engine. In this part, the whole computational width (h0) is three thermal penetration depths and the dimensionless stack thickness can be adjusted according to the occupied grids of the thickness and the spacing, respectively. Fig. 8 reveals the onset temperature difference changes with the dimensionless thickness of the stack at three different charge pressures. The charge gas is nitrogen with the stack length of Lr ¼ 0.055. It can be seen from the figure that the onset temperature difference firstly decrease and then increase with the plate thickness increase. The onset temperature difference reaches the lowest in the dimensionless stack thickness of B ¼ 0.33. When B is larger than 0.9, the thermoacoustic engine hardly reaches the selfexcited evolution. Because of the constant computational width, the thickness and the spacing has a reciprocal change with each other. Therefore, the small thickness has correspondingly large spacing of the stack. There is small resistance during the process of generating gas disturbance near the stack, which corresponds to small temperature gradient. But the small thickness also has limited ability of storing and releasing heat, and this leads to the underpowered acceleration process. The onset temperature difference is large by combining the above two factors with small thickness of the stack. With the increase of the plate thickness, the critical temperature gradient to generate the initial gas disturbance has slightly increase but the plate ability of storing and releasing heat is increase evidently. So the overall effect is decreasing the onset temperature of the engine. When the plate ability of storing and releasing heat is matching with the acceleration process in the channel, the onset temperature reaches the lowest, as show in Fig. 8 for B ¼ 0.33. With further increase of the plate thickness, the acceleration function is no longer increasing but the growing thickness occupies the channel between the two plates. As a result, the amount of the initial gas disturbance is decreased and the engine needs larger temperature gradient to reach self-excited state.
Fig. 7. The onset temperature difference changes with the stack position at different stack lengths.
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Fig. 8. The onset temperature difference changes with the dimensionless stack thickness at different charge pressures.
3.3. Influence of the charge pressure and working gas on the onset temperature The charge pressure influences on the onset temperature by the following aspects, namely the energy of the initial gas disturbance and the resistance in the movement. When the charge pressure is low, the gas disturbance is generated easily near the stack and has low resistance in the movement. Commonly, the lower critical onset temperature gradient and the smaller gas density refer to the more easily to gain the gas disturbance. As a result, it accelerates the disturbance of gas during the movement process of the stack. The higher critical temperature is needed to generate the initial gas disturbance at the higher charge pressure. Besides, the resistance would is increased during the movement in the resonator with the growing of charge pressure. Although the acceleration function increases in the stack domain, the onset temperature of the thermoacoustic engine also increases with the increase of the charge pressure. Fig. 9 exhibits the onset temperature difference with charge pressure with three different working gases. The stack adopts the optimized parameters in previous section. It can be seen from Fig. 9 that the onset temperature difference has slight decreases and then increases with the increase of the charge pressure. The onset temperature reaches the lowest at the charge pressure of 0.5 MPa. The properties of the working gas have much influence on the onset temperature difference. Arnott et al. [41] indicated pffiffiffiffiffiffiffiffiffiffiffiffi that the onset temperature gradient is proportional to the g 1 excluding the acoustic dissipation outside of the stack. Therefore, the engine has the lower onset temperature, when the specific heat ratio of g is closer to 1. It is known from the engineering thermodynamics that the specific heat ratio of different kinds of the gas, which is 1.67 for the monatomic gases, 1.4 for the diatomic gases and 1.286 for the polyatomic gases. So it can conclude that the onset temperature decreases with the increasing of the number of atoms, namely, the energy-conversion has the best performance with polyatomic gases, followed by diatomic and monatomic gases. Besides, the onset temperature increases with the addition of the molar mass at same number of atoms.
Fig. 9. The onset temperature difference changes with charge pressure at different working gases.
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Fig. 9 also shows the onset temperature difference changes with three different working gases. It can be seen that the onset temperature difference was the highest for argon, followed by helium and nitrogen. These phenomena can be attributed to the specific heat ratio and molar mass. As described previously, the gas with higher number of atoms and smaller molar mass has better energy-conversion performance. Therefore, the thermoacoustic engine has the best conversion capability with the nitrogen, while the argon is worst. 3.4. Comparison with experimental results and discuss In order to evaluate validity of the numerical simulation, the comparison with experimental results is further exhibited. The experimental structure and parameter are shown in Fig. 10 and Fig. 11. Furthermore, in order to fit the experimental setup the length of the resonator is adjusted in the numerical simulation process, and the charge gas is nitrogen. The resonator length is 4.5 m with the fundamental frequency of 38.24 Hz in calculation, which has the error of 2.1% compared with the experimental results of 39.06 Hz. A more detailed presentation can be seen in Refs. [42,43]. Fig. 12 gives the comparison of onset temperature differences between the simulated and experimental results with different charge pressures. It can be seen that the calculated result of onset temperature difference at different charge pressure near the experimental values, and the error is within 1%. Therefore, it can be considered the calculated result is in good agreement with the experimental values and the simulation result is reasonable. It is well known that the gas disturbance plays an important role during the thermoacoustic self-excited process. In the development of the gas disturbance, the charge pressure has the impact on the generation of the gas disturbance. Besides, Fig. 12 shows that the onset temperature difference increased with the increasing of the charge pressure. Thus, the performance of the thermoacoustic reduces with the addition of the charge pressure. The possible reasons to explain this phenomenon can be attributed the following two aspects. On one hand, the charge pressure could minimize the critical temperature gradient and increase the acceleration process of the initial disturbance in the domain of the stack. On the other hand, for the application of the thermoacoustic engine, the large charge pressure is chosen to gain enough energy density of the oscillatory flow. Therefore, the engine with small charge pressure has low onset temperature difference, which has beneficial to the advancement of the energy-conversion.
Fig. 10. Thermoacoustic engine physical picture.
Fig. 11. Thermoacoustic engine structure diagram.
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Fig. 12. Comparison of onset temperature between the calculated and experimental results changes with charge pressure. The charge gas is nitrogen.
4. Conclusions In this work, a two-dimensional numerical model was introduced to simulate the evolution process of self-excited thermoacoustic based on the compressible SIMPLE algorithm. The evolution processes of self-excited thermoacoustic oscillation at two different heating conditions were revealed. The impacts of the stack parameters, the charge pressure and working gas on the onset temperature were also analyzed. The main findings are as follows: (1) The numerical simulation of thermoacoustic prime mover adopted the heating boundary of constant temperature gradient shows much better than with the constant heat flux heating boundary. (2) The influence of the stack parameters on the onset temperature were revealed and analyzed. The results indicate that he minimum onset temperature difference can be achieved when the dimensionless stack length and position are Lr ¼ 0.055e0.06 and Xr ¼ 0.14e0.16, respectively. In addition, the optimal stack plate thickness was obtained with value of 33% of the stack channel width, when the computational width is 3 times the thermal penetration depth of the working gas. (3) The onset temperature difference between the two exchangers was increased with the growing of the charge pressure. (4) For three different working gases, the onset temperature difference was the highest for argon, followed by helium and nitrogen.
Acknowledgements The study was supported by National Natural Science Foundation of China (Grants No. 51706208) and Henan provincial key science and technology research projects (Grants No. 192102310244). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jsv.2019.114946. References [1] G.W. Swift, Thermoacoustic engines, J. Aoust. Soc. Am. 84 (4) (1988) 1145e1180. [2] Y. Wang, Y.L. He, G.H. Tang, W.Q. Tao, Simulation of two dimensional oscillating flow using the Lattice Boltzmann method, Int. J. Mod. Phys. C 17 (5) (2006) 615e630. [3] Y. Wang, Y.L. He, Q. Li, G.H. Tang, Numerical simulations of gas resonant oscillations in a closed tube using Lattice Boltzmann method, Int. J. Heat Mass Transf. 51 (51) (2008) 3082e3090. [4] Y. Wang, Y.L. He, J. Huang, Q. Li, Impliciteexplicit finite-difference Lattice Boltzmann method with viscid compressible model for gas oscillating patterns in a resonator, Int. J. Numer. Methods Fluids 59 (2009) 853e872. [5] G.H. Tang, X.J. Gu, R.W. Barber, D.R. Emerson, Y.H. Zhang, Lattice Boltzmann simulation of nonequilibrium effects in oscillatory gas flows, Phys. Rev. E 78 (2) (2008), 026706(8). [6] L.K. Tartibu, Maximum cooling and maximum efficiency of thermoacoustic refrigerator, Heat Mass Transf. 52 (2016) 95e102. [7] C. Wantha, The impact of stack geometry and mean pressure on cold end temperature of stack in thermoacoustic refrigeration systems, Heat Mass Transf. 54 (2018) 2153e2161. [8] C. Wantha, K. Assawmartbunlue, Experimental investigation of the effect of drive housing and resonance tube on the temperature difference across a thermoacoustic stack, Heat Mass Transf. 49 (2013) 887e896.
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[9] S.H. Tasnim, S. Mahmud, R.A. Fraser, Compressible pulsating convection through regular and random porous media: the thermoacoustic case, Heat Mass Transf. 48 (2012) 329e342. [10] G. Hu, J.L. Wang, Z. Su, G.P. Li, K.C.S. Kwok, Performance evaluation of twin piezoelectric wind energy harvesters under mutual interference, Appl. Phys. Lett. 115 (7) (2019), 073901. [11] N.N. Deshmukh, S.D. Sharma, Suppression of thermo-acoustic instability using air injection in horizontal Rijke tube, J. Energy Inst. 90 (2017) 485e495, https://doi.org/10.1016/j.joei.2016.03.001. [12] N.N. Deshmukh, S.D. Sharma, Effect of pointed and diffused air injection on premixed flame confined in a Rijke tube, Cogent. Eng. 3 (2016). https://doi. org/10.1080/23311916.2016.1156280. [13] N.N. Deshmukh, S. Sharma, Experiments on heat content inside a Rijke tube with suppression of thermo-acoustics instability, Int. J. Spray Combust. Dyn. 9 (2017) 85e101, https://doi.org/10.1177/1756827716655007. [14] A. Ibrahim, A. Elbeltagy, M. Emam, M. Abdou, Suppressing harmonics in standing-wave thermoacoustic engines, Res. Gate. Net. (2013) 1e12. [15] W.C. Ward, G.W. Swift, Design environment for low-amplitude thermoacoustic engines, J. Acoust. Soc. Am. 94 (6) (1994) 3671e3672. [16] J.A. Lightfoot, W.P. Arnott, H.E. Bass, R. Raspet, Experimental study of a radial mode thermoacoustic prime mover, J. Acoust. Soc. Am. 105 (1999) 2652e2662. https://doi.org/10.1121/1.426881. [17] L.K. Tartibu, B. Sun, M.A. Kaunda, Lexicographic multi-objective optimization of thermoacoustic refrigerator's stack, Heat Mass Transf. 51 (2015) 649e660. [18] C.-C. Hantschk, D. Vortmeyeer, Numerical simulation of self-excited thermoscoustics instabilities in a Rijke tube, J. Sound Vib. 277 (1999) 511e522. [19] C.-C. Hantschk, D. Vortmeyer, Numerical simulation of unsteady interactions between flow, heat conduction and acoustics within a Rijke tube, Chem. Eng. Technol. 9 (2000) 758e763. [20] J.A. Lycklama, M.E.H. Tijani, S. Spoelstra, Simulation of a traveling-wave thermoacoustic engine, Acoust. Soc. Am. 118 (2005) 2265e2270. [21] G.Y. Yu, E.C. Luo, Study of nonlinear processes of a large experimental thermoacoustic-stirling heat engine by using computational fluid dynamics, J. Appl. Phys. 102 (2007) 4901e4907. [22] G.Y. Yu, W. Dai, E.C. Luo, CFD simulation of a 300 Hz thermoacoustic standing wave engine, Cryogenics 50 (9) (2010) 615e622. [23] S. Zhang, Z.H. Wu, R.D. Zhao, G.Y. Yu, W. Dai, E.C. Luo, S. Zhang, Z.H. Wu, R.D. Zhao, et al., Study on a basic unit of a double-acting thermoacoustic heat engine used for dish solar power, Energy Convers. Manag. 85 (9) (2014) 718e726. [24] G. Zhou, Q. Li, Z.Y. Li, Q. Li, A miniature thermoacoustic stirling engine, Energy Convers. Manag. 49 (2008) 1785e1792. [25] L.M. Qiu, B.H. Lai, Y.F. Li, D.M. Sun, Numerical simulation of the onset characteristics in a standing wave thermoacoustic engine based on thermodynamic analysis, Int. J. Heat Mass Transf. 55 (2012) 2200e2203. [26] D.M. Sun, K. Wang, L.M. Qiu, B.H. Lai, Y.F. Li, X.B. Zhang, Theoretical and experimental investigation of onset characteristics of standing-wave thermoacoustic engines based on thermodynamic analysis, Appl. Acoust. 81 (3) (2014) 50e57. [27] K. Wang, D.M. Sun, J. Zhang, J. Zou, K. Wu, L.M. Qiu, Z.Y. Huang, Numerical simulation on onset characteristics of traveling-wave thermoacoustic engines based on a time-domain network model, Int. J. Therm. Sci. 94 (2015) 61e71. [28] A.A. Boroujerdi, M. Ziabasharhagh, Analytical determination of oscillating frequencies and onset temperatures of standing wave thermoacoustic heat engines, Int. J. Heat Mass Transf. 98 (2016) 401e414. [29] W.Q. Tao, Numerical Heat Transfer [M], in: Xi'an, second ed., Xi'an Jiaotong University Press, 2006. [30] J. Huang, R. Li, Y.L. He, Z.G. Qu, Solutions for variable density low Mach number flows with a compressible pressure-based algorithm, Appl. Therm. Eng. 27 (11e12) (2007) 2104e2112. [31] J. Huang, Numerical Simulation and Experimental Study on Theomoacoustic Heat Engine, Xi'an Jiaotong University, 2007. [32] R. Li, Numerical Simulation of Thermoacoustic Refrigerators and Experimental Study of Mixed Medium Thermoacoustic Primer Movers, Xi'an Jiaotong University, 2007. [33] R.A. Saenger, G.E. Hudson, Periodic shock waves in resonating gas columns, J. Acoust. Soc. Am. 32 (1960) 961e970. [34] H.B. Ke, Y.W. Liu, Y.L. He, Y. Wang, J. Huang, Numerical simulation and parameter optimization of thermo-acoustic refrigerator driven at large amplitude, Cryogenics 50 (2010) 28e35. [35] H.B. Ke, Numerical and Experimental Investigations on the Feat and Mass Transfer and its Mechanisms in Thermoacoustic, Xi'an Jiaotong University, 2012. [36] D.W. Zhang, Y.L. He, Y. Wang, J. Huang, Simulation of a standing-wave thermoacoustic engine using compressible SIMPLE algorithm, CP1207, in: 6th International Symposium on Multiphase Flow, Heat Mass Transfer and Energy Conversion, Xi’an, China, July 11-15 2009, America Institute of Physics, 2010, pp. 939e944. [37] G.B. Chen, Advanced Cryogenic Refrigeration Techniques, second ed., China Machine Press, Beijing, 2003. [38] A.A. Atchley, Standing wave analysis of a thermoacoustic prime mover below onset of self-oscillation, J. Acoust. Soc. Am. 92 (5) (1992) 2907e2914. [39] A.A. Atchley, Analysis of the initial buildup of oscillations in a thermoacoustic prime mover, J. Acoust. Soc. Am. 95 (3) (1994) 1661e1664. [40] A.A. Atchley, F.M. Kuo, Stability curves for a thermoacoustic prime mover, J. Acoust. Soc. Am. 95 (3) (1994) 1401e1404. [41] W.P. Arnott, J.R. Belcher, R. Raspet, H.E. Bass, Stability analysis of a helium-filled thermoacoustic engine, J. Acoust. Soc. Am. 96 (1) (1994) 370e375. [42] C. Shen, Y.L. He, Y.G. Li, H.B. Ke, D.W. Zhang, Y.W. Liu, Performance of solar powered thermoacoustic engine at different tilted angles, Appl. Therm. Eng. 29 (2009) 2745e2756. [43] C. Shen, H.X. Li, D.W. Zhang, P. Yu, N. Pan, S.F. Wang, Study on the heat transfer characteristic of solar powered thermoacoustic prime mover at different tilted angles, Appl. Therm. Eng. 103 (2016) 1126e1134.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Numerical analysis on thermoacoustic prime mover Dongwei Zhang a, Erhui Jiang a, Chao Shen b, *, Junjie Zhou a, Weiwei Yang c, Yaling He c a
Center on the Technology and Equipments for Energy Saving in Thermal Energy System of MOE, School of Mechanical and Power Engineering, Zhengzhou University, Zhengzhou, Henan, 450001, China b School of Civil Engineering, Zhengzhou University, Zhengzhou, Henan, 450001, China c Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shanxi, 710049, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 March 2019 Received in revised form 3 September 2019 Accepted 9 September 2019 Available online 9 September 2019 Handling Editor: L. Huang
A two-dimensional numerical model based on compressible SIMPLE algorithm was developed to simulate the evolution process of self-excited oscillation in a thermoacoustic engine at two different heating conditions. Moreover, the performance analysis of the thermoacoustic engine was researched. On one hand, the influence of stack parameters and charge pressure on the onset temperature of self-excited thermoacoustic oscillation was exhibited and analyzed. On the other hand, the impact of different gas types on the onset temperature difference was investigated. The results indicated that the self-excited thermoacoustic oscillation happened with the same performance for both two heating conditions. The minimum onset temperature difference can be achieved when the dimensionless stack length and position are Lr ¼ 0.055e0.06 and Xr ¼ 0.14e0.16, respectively. For a detail, the optimal stack plate thickness is 33% of the stack channel width when the computational width is 3 times as large as the thermal penetration depth of the working gas. Furthermore, the onset temperature difference increased with the increasing of the charge pressure in prime mover. For different working gas, the onset temperature difference was the highest for argon, which was followed by helium and nitrogen. Finally, the performance was improved with the increase of the temperature difference between the two exchangers in the prime mover. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Thermoacoustic prime mover Self-excited oscillations Compressible SIMPLE algorithm
1. Introduction The energy-conversion process of thermoacoustic devices can be achieved either to produce the acoustic work inducing by a temperature gradient along the stacks (i.e., a prime mover) or to pump the heat from low-temperature region to hightemperature region driving by the acoustic wave (i.e., a heat pump). This kind of technology avoids the use of moving parts and exotic materials, and does not require close tolerances and critical dimensions, which makes it attractive for specific applications [1]. During the past few decades, a great number of works have been investigated on both thermoacoustic engines and refrigerators [2e5]. As a key factor to evaluate the performance of the engine, the onset temperature of
* Corresponding author. E-mail address: [email protected] (C. Shen). https://doi.org/10.1016/j.jsv.2019.114946 0022-460X/© 2019 Elsevier Ltd. All rights reserved.
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Nomenclatures B c cp cv f h0 h1 h2 k L Lr l Ma Pr Dp p Q Rg S T DT DTonset t u v Xr
Ratio of non-dimensionless stack thickness to stack spacing Sound speed/m$s1 specific heat at constant pressure/J$kg1$K1 specific heat at constant volume/J$kg1$K1 Resonant frequency/Hz Half of the stack spacing/m Half of the stack gap/m Half the thickness of the stack/m Thermal Conductivity/W$m$K1 Resonant cavity or engine length/m Non-dimensional stack length Stack length/m Mach number Prandtl number Pressure difference/Pa Pressure/Pa Heat exchange/J Unit mass gas constant/J$kg1$K1 Source terms Temperarure/K Temperarure difference/K Onset temperature difference/K Time/s Velocity at the x direction/m$s1 Velocity at the y direction/m$s1 Non-dimensional stack position
Greek alphabet g Specific heat ratio m Dynamic viscosity/kg$m1$s1 r Density/kg$m3 f General variables of u, v and T equation u Angular frequency/rad$s1 Superscript 0 Last time value p Pressure T Temperature Correction value * The value of the first iteration level ** The value of the second iteration level *** The value of the third iteration level Subscript crit HX net p s res stk u v x y
Threshold Heat Exchanger Net output Pressure Solid Resonator Stack Velocity at the x direction Velocity at the y direction Horizontal Portrait
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thermoacoustic prime mover decides whether the waste heat can be used to drive the engine. Therefore, it is critical to exhibit the evolution of self-excited process and reveal the influence of stack parameters and charge pressure on the onset temperature. In 2015, Tartibu et al. [6] investigated the influence of the positions of the stack on the performance of the thermoacoustic refrigerators. Their results indicated that the location of the stack closing to the pressure antinode has beneficial to improve the maximum coefficient of performance of the device, while the design for maximum cooling would be increased by moving it away from the pressure antinode. In other aspect, the impact of the geometry structure on the cold end temperature of stack in thermoacoustic refrigeration system was investigated by Wantha et al. [7], in 2018. The outcomes suggested that the pin array stack structure has the best performance than spiral and circular pore stacks on the cold end temperature. Besides, the length of the resonator was also studied, in recent years. A reasonable length of the resonator is associated with the development of the energy-conversion and predicts the design of the thermoacoustic devise [8,9]. In addition, the stability of thermo-acoustics acts a major role in the performance and life of the system [10]. In recent years, N. N. Deshmukh et al. [11e14] of the Institute of Technology Bombay took lots of experimental investigation on the suppression of thermosacoustics instability. Their results indicated that the application of mic-reject to adjust the velocity and mass flow can effectively suppress the instability of thermos-acoustics. As a result, it greatly reduced the heat loss. Some meaningful results have been achieved by experimental methods. However, there still have some problems need to be solved, which is the energy conversion process in experimental tests. It is inevitable because the operation process of thermoacoustic engine along with complex unsteady oscillatory flow. Therefore, the numerical simulation has becoming popularity for the advantages of high nonlinear solution on thermoacoustic. The DeltaE was adopted by Ward et al. [15] to solve one-dimensional wave equation and compute low-amplitude thermoacoustic engines, which based on the linear acoustic approximation, in 1994. This kind of model has the advantage of evaluating the thermoacoustic apparatus performs the designing of novel apparatus to the desired performance. However, due to the limitations of its fundamental base, this kind of model is inadequate to simulate high-amplitude thermoacoustic engines and capture the nonlinear phenomena. In 1999, the Lightffot et al. [16] proposed a theoretical prediction model and this result had great agreement on the temperature difference at which oscillation begin compared with the experimental methods. Furthermore, a sort of optimization model was introduced by Tartibu et al. [17], in 2015. This model based on the discontinuous derivatives was proposed to establish a three-criterion nonlinear programing, which results would achieve the optimization of the thermoacoustic devise. Besides, the investigation of CFD code (i.e., Fluent) on the evolution of self-excited process and the nonlinear phenomena in the engine was studied by some researchers. For example, Hantschk et al. [18,19] developed a two-dimensional model and simulated two different kinds of Rijke tubes with an open-open boundary and a closed-open boundary. The evolution of non-linear selfexcited pressure oscillations was revealed and the higher harmonics during the onset of oscillations was achieved. They also indicated that the oscillations in combustion chambers were similar within Rijke tubes and the method could be employed for both of them. Furthermore, a two-dimensional computational model was proposed by Lycklama et al. [20] to study the traveling-wave thermoacoustic engine. Their results suggested that a linear temperature difference across the regenerator would trigger off the dynamic pressure oscillation. Luo et al. [21,22] investigated the impact of two different thermal boundary conditions (a specified heating load and a known surface temperature) on the self-excited oscillation amplification with time at two different thermoacoustic engines. In addition, they also tried to reveal the nonlinear phenomena of vortex formation at the ends of the stacks. A double-acting thermoacoustic heat engine was also designed to absorb concentrated solar heat [23]. In 2008, Zhou et al. [24] simulated and designed a miniature thermoacoustic Stirling engine. The calculated results indicated that an appropriate resonance tube diameter is beneficial to the improvement of the performance of the engine. A further study on the onset characteristics in different thermoacoustic engines was reported by Qiu et al. [25] and Sun et al. [26,27]. Based on linear theory of thermoacoustics, a similar work was also presented by Boroujerdi and Ziabasharhagh [28] to analyze the influence of the oscillating frequency and onset temperature on the performance of a simple standing wave thermoacoustic heat engine. Most of above studies focused on revealing the evolution of self-excited process in thermoacoustic engines. Further studies are still needed to reveal the influence of the plate and working gas parameters on the onset temperature of thermoacoustic engines. In this work, a two-dimensional numerical model based on the compressible SIMPLE algorithm was developed to simulate the evolution process of self-excited thermoacoustic oscillation. Firstly, the evolution processes of self-excited thermoacoustic oscillation at two different heating conditions were revealed. Then, the impacts of the stack parameters, the charge pressure and working gas on the onset temperature were also studied and analyzed. Finally, some conclusions have drawn.
2. Numerical simulations and theoretical analysis The compressible SIMPLE algorithm was adapted to the model of the low Mach compressible oscillatory flow in thermoacoustic engine. The simulation model including the core components of the engine was developed and the evolution of the self-excited process was analyzed.
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Fig. 1. (a) Model for thermoacoustic prime mover. The dark domain is the computation domain, which is shown magnified in (b).
2.1. Numerical model and algorithm Fig. 1 shows the physical model and simulation domain for the thermoacoustic prime mover with half wavelength, which is composed of the prime mover and a resonant tube. The part of prime mover includes a parallel-plate stack, a hot-end heat exchanger and a cold-end heat exchanger. In this system, the stacks contain many identical parallel plates and it is ratio of the gap width and tube diameter is very small. The engine can be simplified into a two-dimensional channel as marked with dark region. This two-dimensional domain can obtain adequate computational accuracy of oscillatory flow field in the engine with less computing time. Other simplifications of the thermoacoustic prime mover are as follows: (1) The working substance is regarded as ideal gas. (2) The walls of the engine are adiabatic. (3) The gas flow in the engine is considered laminar compressible flow. In thermoacoustic engine, the working fluid experiences compression and expansion process and the oscillatory flow has low Mach number (Ma<0.1). Therefore, the simulation of the self-excited process has much higher requirements for the algorithm. A compressible SIMPLE algorithm was adopted in this work, which is based on the pressure correction method to calculate the flow and the heat transfer problems [29], and the details of calculation process of SIMPLE algorithm was presented in Fig. 2. Furtherly, the inner node method was adopted to establish the staggered grid structure for discrete regions, as shown in Fig. 3. The governing equations are discretized by the finite-volume method and the cell centered scheme is adopted to get the structured grid for the domain discretization, and the details of solving process as following. (1) The new variables (u** and v**) were gained by combing the variables of the previous iteration layer (u*, v* and r*) and the progressive layer (u0, r0 and T0). (2) Saving the pressure correction equation to obtain the value of the pressure correction (p0 ). (3) The new variables (p***, u*** and v***) were got based on above correction values, and then the new density also was calculated by the equation of r** ¼ p**/(RT*). (4) Based on the energy equation to obtain the new temperature (T**).
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Fig. 2. Algorithm diagram.
(5) Take a judgment on the convergence of variables value, and if it has converged to conduct the calculation of next time layer, while return to step (1) until the value reaches convergence. (6) Store the current variables value to u0, r0 and T0, and then judge whether the set calculation time was completed. Exit the procedure when it reached the set calculation time. Otherwise return the step (1) to take the calculation of new time layer. More details of this algorithm can be found in Refs. [28e30].
2.2. Governing equations and boundary conditions In thermoacoustic engine, the flow is compressive and time-dependent. The terms of compressive and viscidity dissipations should be considered. The governing equations of the two-dimensional unsteady fluid flow and heat transfer are unified in a similar expression in Cartesian coordinate system, which can be given by
6
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Fig. 3. Staggered grid diagram [29].
vðrfÞ vðrufÞ vðrvfÞ v vf v vf þ þ ¼ þ Rf þ Sf Gf Gf vt vx vy vx vx vy vy
(1)
where R4 and S4 are the general source terms. Detailed expressions for G4, R4and S4 are presented in Table 1. Notice that the term S4 and m/3 in the momentum conservation equations and 2m/3 in the energy conservation equation is obtained from the second molecular viscosity coefficient, l ¼ 2m=3 (Stokes hypothesis), which m is constant in here. Totally, the conservation equations for mass, momentum and energy contain five unknown quantities: u, v, p, r and T. To ensure the closure of the governing equations, the state equation for ideal gas is also solved to close the equations:
p ¼ rRg T
(2)
In the domain of parallel-plate stack, the energy conservation equation for solid phase is also needed
rs
vTs ksx v2 Ts ksy v2 Ts ¼ þ vt cs vx2 cs vy2
(3)
where ksx and ksy are the thermal conductivities of solid phase for the horizontal and vertical axes, respectively. In prime mover, there exists temperature gradient between the hot-end exchanger and the cold-end heat exchanger. Two different heating conditions were adopted because of different heat sources in practical applications. For the one halfpffiffiffiffiffiffiffiffiffiffiffi wavelength standing wave resonator, the frequency can be obtained from the formula of f ¼ kRg T =2L. Therefore, the length of the resonator is 2 m which has a frequency of 85.96 Hz for nitrogen with a charge pressure of 1.5 MPa. The time step
Table 1 Content of the varies in the unified governing equations. Governing equations
f
Gf
Rf
Sf
Mass x-Momentum
1 u
0
0
y- Momentum
v
m
0 vp vx vp vy
Energy
T
m
m Pr
m v
. . i 1 hvp þ V,ðp V Þ pðV, V Þ Cp vt
.
ðV, V Þ 3 vy . m v ðV, V Þ þ rg 3 vy 8 > " 2 # > > vu 2 vv > > > 2 þ > vx vy m<
9 > > > > > > > =
> Cp > > > > ! > vv vu 2 2 > > > ðV$ V Þ2 > > > > þ vx þ vy > 3 : ;
þ
r _ $q
Cp
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is set to be 106 s, as a result of 1.18 104 time integration points of each oscillation period. The computational domain is dispersed with structured staggered grid and the 2D mesh grid system is 140 32 which could get a reasonable computational time and simulation results.
2.3. Validation of numerical algorithm The numerical algorithm developed in this work has been applied to different cases of thermoacoustic engine. Firstly, computational codes were developed to solve low Mach number, compressible oscillatory flow. Two numerical examples of lid driven flow and shock tube were shown in Refs. [31,32]. Secondly, the two-dimensional numerical simulations of resonant oscillations in a rectangular chamber and the thermoacoustic refrigerators were exhibited. The results with low amplitude oscillations agree well with that of DeltaE model based on linear theory. The large amplitude results accurately captured the nonlinear phenomena which also agree well with the experimental results reported by Seanger and Hudson [33]. More detailed results can be seen in Refs. [34,35]. Furthermore, a two-dimensional numerical simulation on a half standing-wave thermoacoustic engine was performed. The self-excited oscillations process and the acoustical characteristics of the pressure and velocity wave were presented and analyzed, which was verified the method applicability for thermoacoustic heat engine [36].
2.4. Analysis on the self-excited process of the thermoacoustic prime mover It has known that the standing-wave thermoacoustic engine self-excites and maintains the oscillatory movement, which realizes the conversion from heat energy to acoustic power [37]. The self-excited evolution is as follows. In the beginning, the hot heat exchanger is heated continuously but the cold heat exchanger is kept at a low temperature (e.g. ambient temperature). Thus the temperature gradient along the stacks increases quickly. Once the temperature gradient reaches the starting points, the time-average thermodynamic stream is formed along the stacks. These gas disturbance triggered by the temperature gradient has a short accelerated process in the stack channel. Then, the gas disturbance must overcome the dissipation in the movement to reach the whole resonator. The dissipation is lead by the heat exchangers and resonator tube during its dissemination process. Finally, the gas disturbance forms continuously and eventually is enlarged by the resonator. In the above processes, all the factors that improve the energy of the disturbance gas or reduce the movement dissipation can make the energy conversion from the heat to the acoustic work more efficiently, thus reducing the onset temperature difference. The Q was introduced by Atchley [38e40] to evaluate the self-excited evolution of a thermoacoustic standingwave engine, which is defined as:
Q¼
uEst Enet
(4)
where u is the angular frequency, and Est is the energy stored in the engines which can be calculated by integrating the timeaveraged acoustic energy density throughout the entire resonator. Enet is the net power of the entire engine. The expression is as follows:
Enet ¼ Estk þ EHX þ Eres
(5)
where the subscripts ‘stk’, ‘HX’ and 'res' refer to the stack, the (hot and cold) heat exchangers, and the resonator. In above three energy items, EHX and Eres are the dissipations in the heat exchangers and resonator tube, which has a negative effect on the self-excited process. Estk represents the positive acoustic energy sourced from the temperature gradient along the stacks. There is the critical temperature gradient VTcrit resulted in Estk ¼ 0. Thermoacoustic engine is unstable equilibrium at critical temperature gradient, when VT < VTcrit , Estk<0; and VT > VTcrit , Estk>0. It can be seen that the main active force for the self-excited process is the temperature gradient in the domain of the stacks, and the hinder force is the dissipations in the heat exchangers and resonator tube. The gas disturbance generated by the critical temperature gradient must overcome the self-resistance of the stack. Because the initial disturbance is delivered from the high-temperature region to the low area and experienced decompression exothermic process, the gas disturbance is accelerated during this process and finally moves outside of the stack. When the accelerated gas disturbance could overcome the dissipation of the heat exchangers and resonator tube, the thermoacoustic engine starts the self-excited evolution. Otherwise, when the gas disturbance is too small or the acceleration is insufficient, the self-excited process of the thermoacoustic engine can not happen. For convenient description, some dimensionless parameters are defined in this work. They are the dimensionless stack position, the dimensionless stack length and the dimensionless stack thickness. The first one is the ratio of the distance between the left end of the stack and close end of the resonator to the resonator length: Xr ¼ xp/L. The second one is the ratio of the length of the stack to the resonator length: Lr ¼ l/L. The last one is the ratio of the thickness to the spacing of the stack: B ¼ h2/h1.
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3. Self-excited evolution process and optimized parameters of onset temperature In this part, the evolution processes of self-excited thermoacoustic oscillation at two different heating conditions are exhibited. Then, the impacts of the stack parameters, the charge pressure and working gas on the onset temperature are analyzed and revealed. For the optimization of the stack parameters, the position, the length, the thickness and the spacing of the stack interact with each other. Thus, the influences of the position and the length on the onset temperature are presented with constant thickness and spacing of the stack. Secondly, based on the optimized position and length, the impacts of the thickness and spacing on the onset temperature are analyzed. Finally, the other two factors of the charge pressure and working gas influencing on the onset temperature are given with the optimized parameters of the stack.
3.1. Self-excited evolution process with two different heating conditions In this paper, the impact of different heating conditions on the process of the self-excited evolution was investigated, which with a constant value of the temperature and heat flux. At two different heating conditions, the dynamic pressure evolutions and spectrum analysis in position G in Fig. 1 was presented in Fig. 4 and Fig. 5. The gas in thermoacoustic engine is nitrogen with charge pressure of 1.5 MPa. The constant temperature difference between the two heat exchangers is 600 K, and the constant heat flux in the domain of hot heat exchanger is 5 MW∙m2, which could also generate a temperature difference of 600 K at the two ends of the stacks. It can be seen from Fig. 4 that the charge pressures are all increasing with heating before the self-excited evolution. With the development of the self-excited process, the amplitude of the oscillatory pressure increases but the mean pressure evidently decreases with time. When the pressure reaches stable oscillation, they have the
Fig. 4. Evolution processes of self-excited pressure oscillations in right end of the resonator with two different heating conditions: (a) constant temperature gradient condition; (b) constant heat flux boundary condition.
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Fig. 5. Spectrum diagrams at different heating conditions; (a) constant temperature difference; (b) constant heat flux.
almost the same pressure amplitudes at the two heating boundaries. It can be indicated that the same temperature gradient with different heat modes has the same performance of the pressure oscillation. It can also be seen from Fig. 4 that the boundary of constant temperature difference could generate the temperature gradient along the stack more quickly. So the self-excited process begins near 2 s. For the boundary of constant heating flux, it need some preheating time, therefore, the thermoacoustic engine starts near 4 s once the calculation began. Because of the longer heating time, the boundary of constant heating flux has a larger start-up pressure than that of constant temperature difference. But for the evolution of the thermoacoustic engine self-excited process, the boundary of constant temperature difference could establish the temperature gradient more quickly and it could easily get the onset temperature with different parameters. Therefore, in this paper, the heating boundaries of constant temperature difference much better than that with constant heat flux. In addition, Fig. 5 exhibits that the frequency oscillation has the same trend at two different heating boundaries. As above discussed, the performance of the pressure oscillation of resonator shows much well at any heating boundaries. However, the value of fundamental frequency of resonator much closer to the theoretical value of 89.96 Hz, at the heating boundary of constant temperature difference. Thus, it can also consider that the numerical simulation of resonator adopted the heating boundary of constant temperature different shows excellent performance. As previous introduced in introduction section, the D. M. Sun et al. [22,23] has conducted a success simulation on the onset characteristics in thermoacoustic engines. Compare on this method or algorithm, our algorithm has the advantage of simulating pressure wave nonlinear oscillation. Therefore, in this work, the numerical model based on compressible SIMPLE algorithm with the constant temperature difference heating boundary was adopted in following work.
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3.2. Influence of the stack parameters on the onset temperature The stack is the most important component in the thermoacoustic core (the hot and cold heat exchangers are also included), which has much influence on the onset temperature of the engine. In this part, different parameters of the parallel plate length, position, thickness and spacing are studied and analyzed. The onset temperature change with these parameters is revealed and presented. 3.2.1. The length of the stack It is known that large temperature gradient along the stack benefits for the thermoacoustic self-excited evolution. It has two ways to obtain large temperature gradient, one is increasing the temperature difference, and the other is decreasing the length of the stack. However, it should be also noticed that the stack need a certain length providing for the sufficient acceleration length of the gas disturbance, and the small stack length will lead to direct contraction of oscillatory gas between the hot and cold heat exchanger, thus degrades the energy conversion efficiency in the engine. Fig. 6 shows the onset temperature difference change with the dimensionless stack length of three charge pressures. The charge gas is nitrogen with the dimensionless stack position of Xr ¼ 0.18 and the dimensionless stack thickness of B ¼ 0.33. It can be seen that the onset temperature difference reaches the lowest value with a dimensionless stack length of 0.055e0.06. When the dimensionless stack length is larger than 0.18, the thermoacoustic engine is difficult to reach self-excited process. The onset temperature difference of the dimensionless stack length shows a slight increase from 0.055 to 0.065 with the increasing of the charge pressure from 1.0 MPa to 2.0 MPa. As mentioned before, when the dimensionless stack length of Lr is small, the sufficient temperature gradient can be obtained with smaller temperature difference between the two heat exchangers. Thus the gas disturbance can be generated easily near the plate. However, a relatively small stack length leads to a short acceleration process of the gas disturbance in the stack domain. The gas disturbance can not overcome the dissipation to reach self-excited state. As a result, the larger temperature gradient is needed in the domain of the stack to generate the gas disturbance with higher energy. So it need higher temperature difference when the stack length is too short. As the stack length reaches Lr ¼ 0.055, the length required for acceleration of the gas disturbance is enough and the temperature gradient also reaches the critical value. Thus the stack length has the optimal value to generate and accelerate the gas disturbance, which exactly enable overcome the heat exchangers and resonator resistance, and then reach self-excited state. Although the acceleration process is longer with increase of the length, the temperature gradient generating the gas disturbance is constant in the stack domain. Therefore, the temprature difference between the two heat exchangers increases with the increase of the stack length. With the charge pressure increasing, a large temperature gradient is needed to overcome the resistance of the stack and the flow resistance in the resonator. As a result, the dimensionless stack length which is corresponding the lowest onset temperature increases with the increasing of the charge pressure. 3.2.2. The position of the stack The charge pressure is 1.5 MPa and the dimensionless stack thickness is B ¼ 0.33. Fig. 7 presents the onset temperature difference changes with the stack position at four different stack lengths. It can be seen that the onset temperature difference firstly descends and then increases as the stack moving from the close end to the middle of the resonator. The position with the lowest onset temperature increase from Xr ¼ 0.14 to Xr ¼ 0.16 with the increase of the stack length. The onset temperature difference increases quickly with the increase of the stack length and the engine almost can not reach self-excited process when the stack right end is close to the middle of the resonator.
Fig. 6. The onset temperature difference changes with the stack length in different charge pressures.
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When the stack is near the close end of the resonator, the gas disturbance is generated at the velocity node and the pressure antinode. Although the gas disturbance adequately experiences compression and expansion process, the velocity amplitude is too small which leads to short acceleration process, thus the initial disturbance can not be accelerated to reach to the critical point of the self-excited evolution. As a result, the large temperature gradient is needed to increase the initial gas disturbance near the close end of the resonator. With the stack moving to the middle of the resonator, the compression and expansion process of the gas disturbance is shortened but the acceleration process is evidently increased. So the onset temperature difference is reduced and reached the minimum at Xr ¼ 0.14e0.16, as shown in Fig. 7. When the stack moves to near the middle of the resonator at which is close to the velocity antinode and the pressure node, the gas disturbance has much longer acceleration process but inadequate compression and expansion process. Therefore, the gas disturbance energy has slightly upgraded in the compression and expansion process. It needs large temperature gradient to obtain much violent initial gas disturbance and the onset temperature evidently increases. When the stack is in the middle of the resonator (e.g. Xr þ Lr > 0.48), the gas disturbance has no acceleration process here and the thermoacoustic engine can not reach the selfexcited evolution. With the increase of the stack length, the initial gas disturbance has longer acceleration process during the stack moving to the middle of the resonator. The position with the lowest onset temperature increase from Xr ¼ 0.14 to Xr ¼ 0.16. However, because initial gas disturbance need constant critical temperature gradient, the onset temperature increase with the increase of the stack length. 3.2.3. The thickness and the spacing of the parallel plate The thickness and the spacing of the parallel plate determine the thermal contraction characteristic between the working gas and the plate wall. The Rayleigh criterion indicated that the proper parameters of the stack should lead to inadequate thermal contact between the gas parcels and the solid wall. Thus the several times of the thermal penetration depths are recommended in the actual thermoacoustic engine. In this part, the whole computational width (h0) is three thermal penetration depths and the dimensionless stack thickness can be adjusted according to the occupied grids of the thickness and the spacing, respectively. Fig. 8 reveals the onset temperature difference changes with the dimensionless thickness of the stack at three different charge pressures. The charge gas is nitrogen with the stack length of Lr ¼ 0.055. It can be seen from the figure that the onset temperature difference firstly decrease and then increase with the plate thickness increase. The onset temperature difference reaches the lowest in the dimensionless stack thickness of B ¼ 0.33. When B is larger than 0.9, the thermoacoustic engine hardly reaches the selfexcited evolution. Because of the constant computational width, the thickness and the spacing has a reciprocal change with each other. Therefore, the small thickness has correspondingly large spacing of the stack. There is small resistance during the process of generating gas disturbance near the stack, which corresponds to small temperature gradient. But the small thickness also has limited ability of storing and releasing heat, and this leads to the underpowered acceleration process. The onset temperature difference is large by combining the above two factors with small thickness of the stack. With the increase of the plate thickness, the critical temperature gradient to generate the initial gas disturbance has slightly increase but the plate ability of storing and releasing heat is increase evidently. So the overall effect is decreasing the onset temperature of the engine. When the plate ability of storing and releasing heat is matching with the acceleration process in the channel, the onset temperature reaches the lowest, as show in Fig. 8 for B ¼ 0.33. With further increase of the plate thickness, the acceleration function is no longer increasing but the growing thickness occupies the channel between the two plates. As a result, the amount of the initial gas disturbance is decreased and the engine needs larger temperature gradient to reach self-excited state.
Fig. 7. The onset temperature difference changes with the stack position at different stack lengths.
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Fig. 8. The onset temperature difference changes with the dimensionless stack thickness at different charge pressures.
3.3. Influence of the charge pressure and working gas on the onset temperature The charge pressure influences on the onset temperature by the following aspects, namely the energy of the initial gas disturbance and the resistance in the movement. When the charge pressure is low, the gas disturbance is generated easily near the stack and has low resistance in the movement. Commonly, the lower critical onset temperature gradient and the smaller gas density refer to the more easily to gain the gas disturbance. As a result, it accelerates the disturbance of gas during the movement process of the stack. The higher critical temperature is needed to generate the initial gas disturbance at the higher charge pressure. Besides, the resistance would is increased during the movement in the resonator with the growing of charge pressure. Although the acceleration function increases in the stack domain, the onset temperature of the thermoacoustic engine also increases with the increase of the charge pressure. Fig. 9 exhibits the onset temperature difference with charge pressure with three different working gases. The stack adopts the optimized parameters in previous section. It can be seen from Fig. 9 that the onset temperature difference has slight decreases and then increases with the increase of the charge pressure. The onset temperature reaches the lowest at the charge pressure of 0.5 MPa. The properties of the working gas have much influence on the onset temperature difference. Arnott et al. [41] indicated pffiffiffiffiffiffiffiffiffiffiffiffi that the onset temperature gradient is proportional to the g 1 excluding the acoustic dissipation outside of the stack. Therefore, the engine has the lower onset temperature, when the specific heat ratio of g is closer to 1. It is known from the engineering thermodynamics that the specific heat ratio of different kinds of the gas, which is 1.67 for the monatomic gases, 1.4 for the diatomic gases and 1.286 for the polyatomic gases. So it can conclude that the onset temperature decreases with the increasing of the number of atoms, namely, the energy-conversion has the best performance with polyatomic gases, followed by diatomic and monatomic gases. Besides, the onset temperature increases with the addition of the molar mass at same number of atoms.
Fig. 9. The onset temperature difference changes with charge pressure at different working gases.
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Fig. 9 also shows the onset temperature difference changes with three different working gases. It can be seen that the onset temperature difference was the highest for argon, followed by helium and nitrogen. These phenomena can be attributed to the specific heat ratio and molar mass. As described previously, the gas with higher number of atoms and smaller molar mass has better energy-conversion performance. Therefore, the thermoacoustic engine has the best conversion capability with the nitrogen, while the argon is worst. 3.4. Comparison with experimental results and discuss In order to evaluate validity of the numerical simulation, the comparison with experimental results is further exhibited. The experimental structure and parameter are shown in Fig. 10 and Fig. 11. Furthermore, in order to fit the experimental setup the length of the resonator is adjusted in the numerical simulation process, and the charge gas is nitrogen. The resonator length is 4.5 m with the fundamental frequency of 38.24 Hz in calculation, which has the error of 2.1% compared with the experimental results of 39.06 Hz. A more detailed presentation can be seen in Refs. [42,43]. Fig. 12 gives the comparison of onset temperature differences between the simulated and experimental results with different charge pressures. It can be seen that the calculated result of onset temperature difference at different charge pressure near the experimental values, and the error is within 1%. Therefore, it can be considered the calculated result is in good agreement with the experimental values and the simulation result is reasonable. It is well known that the gas disturbance plays an important role during the thermoacoustic self-excited process. In the development of the gas disturbance, the charge pressure has the impact on the generation of the gas disturbance. Besides, Fig. 12 shows that the onset temperature difference increased with the increasing of the charge pressure. Thus, the performance of the thermoacoustic reduces with the addition of the charge pressure. The possible reasons to explain this phenomenon can be attributed the following two aspects. On one hand, the charge pressure could minimize the critical temperature gradient and increase the acceleration process of the initial disturbance in the domain of the stack. On the other hand, for the application of the thermoacoustic engine, the large charge pressure is chosen to gain enough energy density of the oscillatory flow. Therefore, the engine with small charge pressure has low onset temperature difference, which has beneficial to the advancement of the energy-conversion.
Fig. 10. Thermoacoustic engine physical picture.
Fig. 11. Thermoacoustic engine structure diagram.
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Fig. 12. Comparison of onset temperature between the calculated and experimental results changes with charge pressure. The charge gas is nitrogen.
4. Conclusions In this work, a two-dimensional numerical model was introduced to simulate the evolution process of self-excited thermoacoustic based on the compressible SIMPLE algorithm. The evolution processes of self-excited thermoacoustic oscillation at two different heating conditions were revealed. The impacts of the stack parameters, the charge pressure and working gas on the onset temperature were also analyzed. The main findings are as follows: (1) The numerical simulation of thermoacoustic prime mover adopted the heating boundary of constant temperature gradient shows much better than with the constant heat flux heating boundary. (2) The influence of the stack parameters on the onset temperature were revealed and analyzed. The results indicate that he minimum onset temperature difference can be achieved when the dimensionless stack length and position are Lr ¼ 0.055e0.06 and Xr ¼ 0.14e0.16, respectively. In addition, the optimal stack plate thickness was obtained with value of 33% of the stack channel width, when the computational width is 3 times the thermal penetration depth of the working gas. (3) The onset temperature difference between the two exchangers was increased with the growing of the charge pressure. (4) For three different working gases, the onset temperature difference was the highest for argon, followed by helium and nitrogen.
Acknowledgements The study was supported by National Natural Science Foundation of China (Grants No. 51706208) and Henan provincial key science and technology research projects (Grants No. 192102310244). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jsv.2019.114946. References [1] G.W. Swift, Thermoacoustic engines, J. Aoust. Soc. Am. 84 (4) (1988) 1145e1180. [2] Y. Wang, Y.L. He, G.H. Tang, W.Q. Tao, Simulation of two dimensional oscillating flow using the Lattice Boltzmann method, Int. J. Mod. Phys. C 17 (5) (2006) 615e630. [3] Y. Wang, Y.L. He, Q. Li, G.H. Tang, Numerical simulations of gas resonant oscillations in a closed tube using Lattice Boltzmann method, Int. J. Heat Mass Transf. 51 (51) (2008) 3082e3090. [4] Y. Wang, Y.L. He, J. Huang, Q. Li, Impliciteexplicit finite-difference Lattice Boltzmann method with viscid compressible model for gas oscillating patterns in a resonator, Int. J. Numer. Methods Fluids 59 (2009) 853e872. [5] G.H. Tang, X.J. Gu, R.W. Barber, D.R. Emerson, Y.H. Zhang, Lattice Boltzmann simulation of nonequilibrium effects in oscillatory gas flows, Phys. Rev. E 78 (2) (2008), 026706(8). [6] L.K. Tartibu, Maximum cooling and maximum efficiency of thermoacoustic refrigerator, Heat Mass Transf. 52 (2016) 95e102. [7] C. Wantha, The impact of stack geometry and mean pressure on cold end temperature of stack in thermoacoustic refrigeration systems, Heat Mass Transf. 54 (2018) 2153e2161. [8] C. Wantha, K. Assawmartbunlue, Experimental investigation of the effect of drive housing and resonance tube on the temperature difference across a thermoacoustic stack, Heat Mass Transf. 49 (2013) 887e896.
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