Design optimization and CFD analysis of the dynamic behavior of a standing wave thermoacoustic engine with various geometry parameters and boundary conditions

Thermoacoustic devices are converters of thermal energy into acoustic energy and vice versa. Although these machines contain simple components, the design of these machines is very challenging. In order to predict the behavior and optimize the efficiency of a standing-wave thermoacoustic engine designed to drive a thermally driven thermoacoustic refrigerator, considering changes in geometrical parameters and operating conditions, two analogies have been presented in this paper. The first analogy is based on a CFD simulation carried out to investigate the influence of stack parameters, working gas and boundary conditions on the thermoacoustic process. The second analogy is performed by the use of an optimization algorithm based on the simplified linear thermoacoustic theory to design and optimize the parameters investigated by the CFD study. Stack of parallel plates of normalized stack center positions of 0.007 to 0.26, normalized stack lengths of 0.018 to 0.11, and several gaps and thicknesses of plates and working gases are used. The results from the algorithm give the ability to design any thermoacoustic engine with high efficiency by picking the appropriate parameters. Simulation results show that decreasing thickness and position of the plates gives a significant efficiency. However, there are optimum values for length of the stack and the gap between two plates. The material chosen for the construction of plates should have a low thermal conductivity and gases with higher ratios of specific heats and lower Prandtl numbers are well suitable for thermoacoustic systems.


Introduction
Thermoacoustics is the field of researches that concerns the interaction between acoustic and thermal phenomena. At the end of the eighteenth century, Byron Higgins who succeeded in generating an acoustic wave in a duct by means of a flame heating a wall studied the thermoacoustic phenomenon experimentally [1]. In 1850, Sondhaus who built a thermoacoustic engine called the Sondhaus tube continued this work [2]. In 1945, Rayleigh provided a qualitative description of the operation of the Sondhaus tube by analyzing the relation between pressure and temperature oscillations [3]. However, it is only in the seventies that Rott has provided a linear theoretical analysis of thermoacoustic phenomena [4][5][6]. Later, in the 1980s, prototypes of refrigerators were made and studies, have led to a linear description of the thermoacoustic process in steady state (Wheatley [7], Swift [8]). These studies led to the realization of standing-wave thermoacoustic emitters using quarter-wave and halfwave resonators by Swift [9,10], and travelling-wave thermoacoustic engines using annular resonators, Yazaki et al. [11], Backhaus et al. [12], and Job et al. [13]). Over the last 20 years, several studies have allowed a better understanding of the complex phenomena that can take place in thermoacoustic systems. Actually, it is the transactions between the sound waves and heat transfer considerations that are not yet fully understood. Through analytical, mathematical studies and optimization analysis, these multifaceted interactions can be better assumed. Several existing efforts are carried out to better understand thermoacoustic behavior. In 1997, Minner et al. conducted an optimization approach of thermoacoustic systems through a parametric study [14]. The optimization approach of the design exposes that the thermoacoustic heat pumps performance is delicate to the mean pressure, gas combination, stack length and position, and less affected by the stack plates spacing. In 2001, Besnoin analyzed the flow behavior and the performance of thermoacoustic refrigerator by examining the effect of several parameters include, the stack plates thickness, heat exchanger position and length and the drive ratio, on the performance of thermoacoustic refrigerator [15].
The findings of previous studies show that the cooling charge peaks at an effectively defined combination of length and width of heat exchanger with defined combination of thickness and the gap between plates and the heat exchanger. In 2002, Tijani et al. presented a systematically investigation of the influence of the stack plates spacing on the behavior of thermoacoustic devices by identifying and given an optimal gap between plate for thermoacoustic devices [16]. In 2009, based on an optimization-based method in combination with a finite element approach, Zink et al. give an optimum design of the stack regarding thermal losses in thermoacoustic devices [17]. The results show that for minimizing thermal losses, the regenerative unit should be considered as short as possible. Based on a numerical simulation and experiment, Tang et al. studied the effect of an acoustic pressure amplifier sizes on thermoacoustic machines records [18]. In this study several operating parameters, as well as acoustic power, pressure ratio, hot side temperature of the stack and other parameters, depending on the size of the acoustic pressure amplifier were investigated. Results show that at a critical length, lesser than a quarter of the wavelength, for the incidence of velocity antinode and pressure node of the pressure amplifier the effect of the acoustic power and pressure ratio is more significant. In 2012, Hariharan et al. conducted theoretical and experimental investigations to evaluate the efficiency of a thermoacoustic engine examined in terms of pressure amplitude, onset temperature gradient and frequency by varying resonator and stack parameters [19]. The results show that resonator and stack parameters have a significant role regarding to the design of thermoacoustic driver. The acoustic pressure amplitude and the onset temperature gradient across plates increase with increasing in resonator length, stack length and plates spacing with minimum width of plates, while the frequency rises with lessening in resonator length, stack length, and the gap between plates with higher plate thickness. In 2014, to advance the efficiency of the thermoacoustic engine, Liu et al. introduced a phase adjuster by decreasing the cross-sectional part of the wave-guide [20]. The effects of the inner diameter dimension and position of the adjuster section on the acoustic power produced, pressure amplitude, frequency, and system efficiency were investigated. Results display that compared to those without phase adjuster, the efficiency, acoustic power and pressure amplitude with optimal phase adjuster could be improved by 10%, 4.5%, and 3% respectively. In 2015, Tartibu et al. conducted an experimental study to investigate the maximum efficiency and cooling of thermoacoustic refrigerators with different stack locations and lengths [21]. The results showed that next to the pressure antinode, the position of the regenerative unit has beneficial to increase the optimum performance of the acoustically driven thermoacoustic refrigerator, whereas the position for an optimal cooling power would be enhanced by shifting the stack away from the pressure antinode. In 2015, in travelingwave thermoacoustic engine composed of looped-tube, Tourkov et al. studied the influence of regenerator position on the thermoacoustic process [22]. The results show a notable rise on system efficiency when the stack is shifted away from the extremity of the ambient side. In 2018, based on a parametric investigation, Chen et al. studied the dynamic performance of a thermoacoustic engine with standing waves including a resonator opened at one extremity while the other end closed with a deformable object [23]. The effect of the boundary conditions at both limits of the wave-guide have been studied. Results show that the acoustic radiation effects the onset temperature gradient along the plates as well as the acoustic field along the wave-guide with introducing the deformable plate at the closed side. Rahpeima et al. conducted a study to examine the behavior of cooling power and performance of a standing-wave thermoacoustic refrigerator against thermophysical properties and geometric parameters of the stack. It is found that decreasing thermal conductivity and width of the stack produce lesser cooling temperatures gradient and advanced constants of performance [24]. In 2019, through a thermodynamic analysis of a standing-wave thermoacoustic engine with converging stack of parallel plates, Liu et al. studied the impact of ratio of gap between pales to thermal penetration depth on both acoustic field and device performance [25].
Results indicate that the relevant choice of value of the ratio of plates spacing to thermal penetration depth in converging and parallel stack can enhance both the produced acoustic power and engine productivity. In addition, the cost of consumptive materials can be reduced by replacing the stack of parallel plates by the converging stack with the same ratio. Zhang et al. studied the impact of stack geometry parameters and pressure charge on the temperature gradient along the stack of thermoacoustic prime mover with various working gas [26]. The results shows that when the computational thickness is three times longer as the gas thermal penetration depth the optimal plate thickness is 33% of the plates width and lowest charge pressure in thermoacoustic engine leads to the increase of onset temperature along the stack. In addition, it is found that the temperature gradient is sensitive to the thermophysical proprieties of working gas; this temperature was the lower for nitrogen followed by helium and argon. other more recent studies have been developed to study the efficiency of the stack either by optimizing its geometry or by using a different working fluid [27][28][29][30].
Most of above researches are carried out with a view to optimize the performance of thermoacoustic devices. In fact, the majority of the numerical studies conducted to understand the behavior of thermoacoustic systems are about simple simulations that presenting the effect of some parameters on thermoacoustic energy conversion. However, none of these studies shows the time-dependent evolution from the self-exited process until reaching the steady-state. In this work, a 2D model, based on CFD simulation carried out using the software Comsol Multiphysics, is considered to simulate and investigate the influence of various stack parameters and boundary conditions on thermoacoustic energy conversion of a standing-wave thermoacoustic engine designed to drive a thermally driven thermoacoustic refrigerator [31,32]. The parameters studied are examined by solving Navier-Stokes and energy equations using the finite element method. The parameters considered are stack length, position, plate thickness, gap between two plates and temperature gradient along the stack. The thermo-physical properties of the stack studied are thermal conductivity and heat capacity, considered by variant the regenerative unit material. The time-dependent of the self-excited-process from the initial disturbance to the steady state of the acoustic wave generated by the engine is showed at each case. In addition, an optimization approach is performed by the use of an optimization algorithm based on the simplified linear thermoacoustic theory, carried out using MATLAB code, to design and optimize the parameters investigated by the CFD analysis.

Standing wave thermoacoustic engine
Thermoacoustic machines are composed of a resonator filled with a working fluid and excited by an acoustic source, heat exchangers and the regenerative unit (stack), which is the seat of thermoacoustic effect. Figure 1 shows a simple design standing wave thermoacoustic engine opened at one extremity.
In thermoacoustic devices, the distance between the fluid particle and the stack walls is a key parameter concerning the type of heat exchange process. Distant from the stack walls, the heat exchange is adiabatic, the acoustic pressure and the temperature oscillates steadily in phase. In the immediate vicinity of the plates, heat exchange is isothermal; the acoustic temperature is imposed at all times by the imposed temperature gradient of the stack. For a fluid particle situated at an interval of the order of a thickness of thermal boundary layer δ k , the temperature contact between stack wall and fluid particle is adequately enough for that the heat exchange process takes place, but sufficiently poor so that this heat exchange results in a consecutive variation in fluid temperature due to the phase shift between heat exchange and acoustic vibration which do not obey the same time constants. To come to the point, only the particles placed at a distance identical to a thickness of thermal boundary layer provide to the conversion of thermoacoustic energy. It is then easy to understand why the use of a stack whose pores are correctly dimensioned relative to the distance δ κ makes it possible to optimize the thermoacoustic conversion process. δ κ is defined as [16]: where k is the thermal conductivity, cp the pressure specific heat constant of the working fluid, ρ the mean density, and ω is the angular frequency of oscillations. The conduction heat transfer process is provided by a wide boundary layer. However, the thermoacoustic effects is discourages by the viscous boundary layer δ v that arises next to the plate where losses due to viscous effect occur. δ v is defined as [16] where µ is the dynamic viscosity of the gas. For having effective thermoacoustic effects, the viscous boundary layer is required to be smaller than the thermal boundary layer. The Prandtl number defined as the ratio between the viscous boundary layer and the thermal boundary layer, defined as: Once the steady state has been achieved in self-exited thermoacoustic engines a loud noise released, caused by the pressure amplification disturbances in the working gas, which caused by the temperature gradient (∇T) through the stack. ∇T is required to be upper than (∇Tcrit), which is the critical temperature gradient defined as the temperature gradient involved by gas particles if it were under the effect of oscillations in adiabatic conditions. Started with the equation of heat transfer and under the short stack hypothesis, Swift [8] obtained a critical temperature gradient beyond which the gas oscillations are amplified (engine) and below which the gas performs heat pumping throughout the stack (refrigerator). ∇Tcrit is given as ∇Tcrit depends on the working fluid pressure p, velocity u, specific heat cp, density ρ, thermal expansion coefficient β as well as the operating frequency ω. For thermoacoustic engines, the initial temperature gradient is required to be higher than the critical temperature gradient (∇T/∇Tcrit > 1). While, for systems that function as thermoacoustic heat pumps, /∇Tcrit upper limits its performance (∇T/∇Tcrit < 1).

CFD simulation
The CFD consists in studying fluid flows, or their effects, by the numerical resolution of the equations governing the fluid. CFD has grown from a mathematical interest to become a needed tool in practically all branches of fluid dynamics. In the field of research, this approach is the object of a significant effort, because it allows access to all instantaneous information (velocity, pressure, concentration) for each point of the calculation domain, for a global cost generally low compared to the corresponding experiences. Currently, however, the precise prediction of complex phenomena is still a challenging problematic. Thus, the development and further improvement of simulation software is still ongoing.
Beside the commercial CFD software, Comsol Multiphysics is increasing in popularity. This is not only because it is a compact tool, but also due to the possibility to implement new elements, which is interesting especially for research applications.

Materials and methods
The geometry of the modeled system, boundary conditions and initial parameters used for the CFD simulation are mentioned in this section. The governing equations describing thermoacoustic phenomena, grid generation, and the validation of the simulation are also presented.

Geometrical modeling and boundary conditions
The modeled system is a simple standing-wave thermoacoustic engine with a quarter-wavelength wave guide (resonator), closed at the left side and open at the right side, with 150 mm long and 12 mm wide, inside which a stack of parallel plates is placed. Parallel plates of 0.5 mm wide represent the stack of length 10 mm, spaced 0.5 mm apart and located 30 mm from the vicinity of the closed end. Figure 2 displays the simulation model of the studied thermoacoustic engine. In the present simulation, the borders of resonator are used as adiabatic walls expect at the opened end, where imposed a temperature of 300 K. A temperature gradient is set to be 900 K at the left side and decreases along the extent of the stack down to 300 K at the right side of stack and imposed to the horizontal walls of plates. The heat transfer coefficient between plates and air witch used as working gas is defined to be 50 W/ (m 2 K).

Governing equations
In the parallel plates, the flow of working gas is compressible, viscous and conductive of heat. To establish a simplified model, we start with the Navier-Stokes equations for a compressible flow in the fluid domain (mass conservation, momentum and energy equations), a state law of the working gas, as well as the equation of conduction of the unsteady heat in the solid field. These equations can be expressed as the following: -Mass conservation: -Momentum conservation:

-Energy conservation
Where W p is the pressure work and Q vh is the involvement of viscous heating process given by To link the physical parameters of the fluid and solve the system of equations defined above, it is assumed that the working gas is an ideal gas, the law of state is thus written Finally, the unsteady heat conduction equation for the solid is written where α is the thermal diffusivity of the stack walls defined as the ratio of the coefficient of thermal conductivity to the density and the specific heat at constant pressure (α = k/ρcp).

Grid dependency
The grid is made using triangular cells, approximately 30,000 cells are used. As described above the thermoacoustic energy conversion takes place just next to the boundary layer of the walls, a fine mesh density is required toward and out of the plates region. Figure 3 presents the comparison of a refined and a relaxed mesh of the numerical simulation. The stack is considered as a vacant domain. To properly capture the thermal and flow fields near and in the region of the plates, a higher density mesh with a refined boundary layer mesh around the plates is used.

Validation of the simulation
A comparative study is carried out to determine the accuracy of the numerical model adopted in this study with the findings of Zink et al. [33]. The same accurate characteristics and properties were applied again for the model presented in this study. After completing the model simulation from the transition phase of initial pressure disturbance to the pressure oscillations at steady state and by applying the fast Fourier transform (FFT) analysis to the acoustic pressure oscillations (Fig. 4), results show that the main operating frequency of the simulated model is about 582 Hz. This frequency corresponds with the theoretical operating frequency of a thermoacoustic engine with a quarter-wave resonator with an entire length of 0.15 m. The comparison of the frequency with the findings reported by [33] shows that (11) ∂T ∂t = α∇ 2 T the frequency is less than 1.35% difference, which is an acceptable difference.

CFD results
The acoustic pressure amplitude and the operating frequency are considered as key parameters for estimating the strength of thermoacoustic oscillations. These parameters are of excessive significance to a thermoacoustic engine to drive any thermoacoustic system. This part is shared into seven sections: the simulation results achieved for the model taken as the reference case (model described above, Sect. 3.1.1) are presented. In addition, the comparison of these results with the simulations obtained for different geometrical parameters and boundary conditions are investigated. The time-dependent evolution of the acoustic oscillations achieved from the initial disturbance to the steady state and the frequency as key parameters are showed and compared by graphs.

Results obtained for the reference case
The process of the pressure oscillations at the closed end, the FFT analysis, and the distribution of acoustic pressure at the steady-state depending on the length of thermoacoustic engine are presented in Figs. 5 and 6, respectively. The self-excited process of the acoustic wave produced by imposing a temperature gradient along the stack of parallel plates, from the initial disturbance to the steady state, for the reference sample is showed in Fig. 5a. The time step of 1e − 5 is used for carrying out the simulations including 100,000 time steps (1 s). From the presenting graphs, it can be noticed that the progress of the pressure generated can be splitted into three parts: first phase, which an instability of the system takes place in the first 8 ms. This phase results when the temperature gradient at the ends of the stack reaches the gradient imposed at the ends of the plates. The next phase, the system is approaching equilibrium between the various energy losses from viscous and thermal losses to the surrounding environment; the amplitude of the acoustic pressure continues to increase gradually. Last phase, the progress of the acoustic wave produced by conversion thermal energy into acoustic energy reaches the steady state after heating the stack for around 0.4 s. The pressure stabilizes after approx. 43,000 iterations at the amplitude of 4.6 kPa. Figure 6b displays the distribution of acoustic pressure as function of whole device length for separate segment of the period Tp of oscillation. From this figure, it is observed that the closed end of resonator presents the pressure antinode, where the pressure amplitude is set to be 4.6 kPa, which is the optimal amplitude reached for this case. It is acknowledged that the standing-wave self-excited thermoacoustic machines sustains the oscillatory motions, which converts thermal energy supplied by the hot heat exchanger into acoustic power generated in the stack. The self-excited process can be described along these lines. First, the temperature of the cold heat exchanger is set to a low temperature (ambient temperature of 300 k) while the hot heat exchanger is continuously heated with a high temperature of 900 k. Thus, the temperature gradient between the ends of plates increases rapidly. Once this temperature difference reaches the critical temperature gradient described above, the time-averaged thermodynamic stream is produced beside the stack. The disturbances generated by the heat gradient imposed on the ends of the plates accelerate gradually. These disturbances are required to overpower the dissipation instigated by the heat exchangers in the movement to reach the whole wave-guide during its propagation process. Lastly, disturbances occur continuously and ends up being amplified by the resonator. The FFT analysis of the pressure oscillation is presented in Fig. 5b and shows that the main operating frequency is about 585 Hz.

Effect of stack length
The time-dependent of the self-excited graph of the acoustic wave oscillations, the acoustic pressure amplitude and the operating frequency for various lengths of plates are presented in Figs. 7 and 8. In this part, a modification of stack length is applied (step between 2.5 mm to 15 by 2.5). To maintain the same resonance settings the boundary and initial parameters are set to be constant. Consequently, the disturbance of the thermodynamic parameters of working gas is effected by stack length changes. The time-dependent of the self-excited evolution of the acoustic oscillations, the pressure amplitude reached at the steady state at the closed end and the operating frequency for various lengths of plates are presented in Figs. 7 and 8. The results show that, as the length of stack plates varies from 2.5 to 15 mm, the acoustic pressure oscillations increase to such optimal value then decrease. The pressure amplitude obtained increases from 170 Pa to 4.7 kPa then lessen to 92 Pa by increasing the stack length from 2.5 to 15 mm. Whereas, for the operating frequencies, it is clear that the highest acoustic wave frequency is obtained for highest plates length and the least is obtained for smaller stack length. In this case, the acoustic pressure has an optimum corresponds to Ls = 7.5 mm with approx. 4.7 kPa and 582 Hz. It is also noticed that the time required by the system to reach the limit cycle differ from one length to another. For short stacks, the steady state occurs quickly than long stacks. It takes 0.25 s, 0.3 s, 0.4 s, and 0.7 s for the lengths 5, 7.5, 10, and 12.5 mm respectively. For Ls ≤ 2.5 mm or Ls ≥ 15 mm, the self-excited progress of the thermoacoustic engine becomes difficult or even impossible to occur. It is because that the short length of plates advance to a direct contraction of the fluid among the ends of the plates and the large stack leads to energy dissipation caused by the interaction of the gas particles and the stack walls. It is observe that for attaining the highest-pressure amplitude, the size of plates must admit a particular optimal value. It is acknowledged that a significant temperature gradient across the stack profits the evolution of the thermoacoustic engine. To obtain the adequate temperature gradient along the stack, two possibilities exist; the first is to increase the temperature of the hot end of the plates, consequently increasing the temperature gradient throughout the stack while the second possibility is to decrease the stack length. However, it should also be noted that the short length of the regenerative unit would cause a direct contraction of the oscillating fluid between the ends of the plates. This causes a gas disturbance that cannot overcome dissipation to reach the self-excited and degrading the efficiency of energy conversion of thermal exchange into acoustic energy. Therefore, the stack needs a significant length allowing sufficient acceleration of the disturbance of the working fluid, which can be easily carried out near the plates.

Effect of stack position
The aims of this analysis part is to check for the effect of varying the stack position along the wave guide length on the self-excited process of a standing-wave thermoacoustic engine. The comparative graphs for different stack center positions are shown in Figs. 9 and 10. Figures 9 and 10 illustrate the self-excited process and the amplitude of the acoustic pressure produced at the stack by applying a temperature gradient along the plates, for various locations of the stack center from the closed extremity to the opened end. From  Fig. 10a, it is clear that the amplitude of acoustical pressure increases first then falls when the stack moves away from the closed side towards the center of the wave-guide. It is clear that the pressure amplitude of the oscillations has a peak value; for the chosen positions, the optimal value is 10675 Pa corresponds to location of the stack center Xc = 15 mm away from the left end. It is also noticed that the time required by the system to achieve the steady state is practically the same for all samples expect for Xc = 15 mm which exceeds one second and still in progress. For Xc ≤ 10 mm or Xc ≥ 55 mm, the process of the self-excited progression of system becomes difficult or even impossible to occur. The disturbance of the working fluid is created at the pressure antinode when the stack is placed next the extremity of the wave-guide. Through these gas changes, the gas undergoes an expansion and compression transformations, the amplitude of the fluid velocity is reduced, which form a small acceleration course, so the initial disturbance cannot be enhanced to range the critical point of self-excited progress. Consequently, a high temperature difference between stack ends is necessary to upsurge the initial disturbance of the gas near the opened end of the resonator. With a stack shifted  towards the center of the wave-guide, the gas state changes of expansion and compression development is reduced but the acceleration course is obviously amplified. When the regenerative unit moves towards the extremity of the wave-guide, the working fluid disturbance have a much important acceleration procedure but an insufficient expansion and compression. Consequently, the energy disturbance of the working gas is a little improved in these two transformations. This process requires a high temperature difference along the stack to achieve a very powerful main gas disturbance. When plates are shifted to the closed side of the wave-guide, the gas disruption has no acceleration development here and the device cannot attain the self-excited process.

Effect of thickness and spacing between plates
The comparative graphs of the time-dependent evolution of the wave oscillations, acoustic pressure amplitude and the frequency at the steady state with various thicknesses (2l 0 ) and gaps (2y 0 ) of the parallel plates are presented in Figs. 11, 12, and 13, respectively. Five different thicknesses and gaps of plates are used throughout this study. As showed in Fig. 11a, b, it is observed that the smaller thickness of the stack plates gives the best performance in term of the acoustic pressure. Whereas, the acoustic pressure increases first until reaching an optimum value, in this case the optimal value is 4693 Pa obtained for 2y 0 = 0.5 mm, then decreases with increasing the spacing between two plates. Particularly, by enlarging the thickness of plates,  the working gas will face additional resistance from the plate walls during its passage through the stack, so viscous losses will increase, and the quantity of heat exchange by conduction from the hot side to the cold side of the plates will be promoted. It is also noticed that by increasing thickness of plates the required time for attaining the steady state increases, as the same as for the distance between plates. For example, it takes 0.3 s for 2l 0 = 0.25 mm and increases gradually until 0.6 s for 2l 0 = 0.75 mm. Concerning the frequency, it is noticed that by increasing 2l 0 the frequency upsurges gradually through small steps. However, it decreased by increasing 2y 0 with small changes. The thickness and gap of parallel plates determine the characteristic of heat exchange process between the walls of pales and the particles the working fluid. The appropriate stack parameters must induce an insufficient thermal exchange among the fluid elements and the stack walls. Therefore, significant thermal penetration depths are required for better energy conversion in thermoacoustic machines. The variation of thickness and spacing of the plates has been used to monitor several aspects that may lead to a better interpretation of the behavior of flux and heat transfer in the device. Thus, a slight thickness of the plates has a corresponding a vast spacing.
There is a less resistance through the generation of the disturbance process of the working fluid next to the plates, which relates to a slight temperature gradient. Nevertheless, the lesser thickness has a restricted heat storage and release capacity, which engender a contraction of acceleration progression. The results showed a decisive evidence that, obviously, these parameters negatively affect the thermoacoustic process. For an optimal design of a stack composed of parallel plates, small variations in thickness and spacing of the plates must be taken into account, as we have seen above simple variations (millimeters) causes an excessive variation of the pressure amplitude.

Effect of stack temperature difference
In this part, the effect of stack temperature gradient on thermoacoustic process is investigated. Figures 15 and  16 illustrate the comparative graphs of the absolute pressure, the frequency and the acoustic pressure amplitude for different hot side temperatures. Heat exchangers are the slightest obvious elements of thermoacoustic components. It is an essential part of the device; they allow the supply and the extraction of heat at the ends of the regenerative unit. The heat exchanger must provide an important heat transfer coefficient and small sound energy losses on thermoacoustic machines. The hot heat exchanger delivers heat to the hot extremity of plates while the ambient heat exchanger removes temperature from the other extremity of plates to ensure the appropriate temperature gradient through the plates. In this section, we replaced the heat exchanger by applying a temperature gradient through the plates. We applied different temperature on the hot side while keeping the cold side at a temperature of 300 k, as shown in Fig. 14. The timedependent process of the absolute pressure from the initial disturbance to the steady state and the acoustic pressure amplitude in the stationary state are illustrated in Fig. 15a, b respectively. In addition, the acoustic pressure amplitude attained at the left side of the waveguide and the frequency of oscillations of the waves are presented in Fig. 16a, b respectively. From figures, it is clear that the acoustic pressure amplitude and the frequency increase with the upsurge of the temperature gradient between the stack ends. Figure 16a displays that the amplitude of the acoustic pressure in the limit cycle for the temperature gradient of 700-300 K is equal to 38 Pa and it exceeds 5600 Pa for the gradient of 1000-300 K. However, it is clear that for Th = 600 K, the thermoacoustic engine could not reach self-exit, which explains that the temperature gradient across the stacking plates is less than ∇Tcrit discussed above. Also, it is noticed that increasing the hot side temperature induce a decrease of the time required for attaining the steady state, it takes 0.3 s for Th = 1000 K then decrease until 0.8 s for Th = 800 K. Due to the relatively high-energy potential produced by the stack, the increase in the temperature gradient of the plates, the exponential growth of the increase in acoustic pressure and the time interval required to reach the limit cycle increase. Thus, the thermoacoustic instability in the wave-guide of the standing-wave thermoacoustic engine can be produced by a lesser temperature gradient.

Effect of stack material
In this section, the performance of different stack materials with various heat capacities and thermal conductivities are investigated. Seven materials of different types, stainless steel, aluminum, kapton, Mylar, Tungsten, copper, and cordierite were tasted. Table 1 illustrate heat  In this section, we wanted to know how heat capacity and thermal conductivity of the stack material impact the energy conversion in the thermoacoustic process. By variant the stack material, the heat capacity and thermal conductivity of the stack will be changed. Figure 17a depict the time-dependent evolution of the absolute pressure oscillation with different stack materials. It is clear that the engines with a stack made of copper, aluminum, tungsten, and stainless reach the steady state  faster than the other materials. Also, it is noticed that the stainless is the best performing material with the highest-pressure amplitude of 23,340 Pa, followed by tungsten (23,105 Pa), aluminum (23,015 Pa), and copper (22,980 Pa). Mylar stack was the worst, with pressure amplitude lesser by 28.7% than stainless, as shown in Fig. 18a. According to the results obtained, there is some important properties for the stack to be suitable for thermoacoustic systems such as, heat capacity of the material constructing the stack is required to be higher than the heat capacity of the working gas and the thermal conductivity is desired to be smaller as possible.
Varying the heat capacity and the thermal conductivity of the stack affect the process of the heat transfer by conduction trough the plates from the hot side to the cold one and influence the amount of heat transfer by convection between the working fluid and the plates walls. On other words, increasing the amount of stack thermal conductivity and the heat capacity reduce the thermoacoustic energy conversion, as a results, the amplitude of pressure fluctuations decrease. The type of material to be chosen relies heavily on thermal conductivity, specific heat, and density. To avoid much loss of acoustic power, low thermal conductivity materials are required. Also, the specific heat capacity of the stack must be higher than that of the working fluid.

Influence of working gas
The efficiency of a standing-wave thermoacoustic engine filled with air, nitrogen, carbon dioxide and argon is studied contrastively in this part. Table 2 illustrates the essential thermo-physical parameters of the four working fluids in detail. The comparative graphs of the temporal progress of the absolute pressure from the initial disturbance to the steady state, the frequency of oscillations, and the amplitude of acoustic pressure at the steady state for the choosing working gases are given in Figs. 19 and 20. From figures, the findings reveal that the amplitude of the acoustic pressure and the resonance frequency of a given system depend on the working fluid. The acoustic pressure amplitude generated is a key factor that reflects the efficiency of the thermoacoustic machines and the power of the energy conversion capacity. Figure 19a, b shows how the absolute pressure evolution varies, from the first disturbance to the steady state, when air, nitrogen, argon, and carbon dioxide are adopted as working gases at a given temperature of 900 K under a pressure   Figure 20a shows that the acoustic pressure amplitude created in the regenerative unit is much higher when argon is used as working fluid, followed by air, nitrogen and then carbon dioxide. The acoustic pressures obtained is upper approximately by 74.8% by the use of argon instead of air. Higher pressure increases the depth of thermal penetration, which further reduces the heat transfer capacity in the stack and induce in additional heat loss. Figure 20b shows that the wave oscillations frequencies of a thermoacoustic engine depend on the used working fluids. The frequency is much lower when carbon dioxide is used. For air and nitrogen, the oscillation frequency are practically almost the same. Consequently, the decrease in the operating frequency is promising to promote the amplitude of the oscillating pressure. For thermoacoustic engines, the choice of working fluid is an essential feature to be considered as it influence the performance of the process. The thermophysical properties of the working fluid (density, thermal conductivity, heat capacity…) perform in determining the onset temperature difference of the system. The key parameters for choosing of the working fluid are its Prandtl number Pr and its number γ. The number of Prandtl Pr plays an important role and must be as small as possible in order to favor thermal exchanges between fluid and wall rather than global warming due to viscous losses. It is possible to play on these parameters using mixtures of rare gases. It depends on the  intended application. The CFD findings are summarized in Table 3.

Design optimization
The linear theory of thermoacoustics [Swift, Rott] allows a first estimate of the performances of thermoacoustic systems. However, in the case of high power systems where nonlinear phenomena become significant, this theory has been limited. These nonlinear phenomena are responsible in particular for the appearance of nonperiodic flows. Some are of thermo-convective origin, dynamic (turbulence), others of purely acoustic origin (streaming). These flows are sometimes the source of convective heat transfer, which can modify the temperature gradient both in the wave generator and in the refrigerator, thus influence the efficiency of the systems and limit the possibilities of optimization of thermoacoustic devices. The optimization procedure adapted in this work is based on a MATLAB code. It allows obtaining several information about the effect of geometric parameters and different operating conditions on engine stack efficiency, the acoustic energy generated in the stack, the required heat input of the hot heat exchanger in function of stack length as well as its position inside the engine. The simple linear expressions for the required energy flux Q over the stack, the acoustic energy W produced in the stack and the acoustic energy loss per unit surface area of the wave guide dwr/dS are expressed by the following equations [7,34]: where Γ is the normalized temperature gradient, ε s is the heat capacity ratio of plates, BR is the stack porosity, called the blockage ratio, and given by: And Λ is defined as where r h is the hydraulic radius, in case of stack of parallel plates r h = y 0 . The equations of the required energy flux, the acoustic power and the losses of acoustic energy per unit of the wave-guide zone (Eqs. (12), (13) and (14)) show that three main categories of independent variables perform in calculating the performance of thermoacoustic systems. First, Geometrical parameters, related to stack and resonator geometry that include stack length, position, cross-sectional area, plates gap and thickness and resonator length and diameter. Material parameters related to stack constructing material and thermophysical factors of the working fluid including density, heat capacity, thermal conductivity …, etc. Parameters concerning the design, including pressure amplitude and mean pressure, the frequency of resonance and the temperature gradient across the regenerative unit [35]. An enormous numbers of the design parameters are involved in the equations described above. To simplify and facilitate calculations normalization can reduce the number of independent parameters. The normalized parameters are presented in Table 4 [36]. By dividing the required heat flux, the acoustic power equations and the acoustic energy loss per unit surface zone in a quarter-wave length resonator (Eqs. (12), (13) and (14)) by the product AP m a, the normalized equations for the heat flow across the stack Q n , the acoustic energy generated in the stack W n and the acoustic energy losses in a quarter-wave length wave guide W rn are expressed as [36], where DR is the drive ratio defined as DR = p 1 /p m . To circumvent the nonlinearities phenomena, the drive ratio is required to be less than 0.03 so that the acoustic Mach and Reynolds numbers must be less than 0.1 and 500 respectively [37,38]. The Mach and Reynolds numbers represents the ratio of a fluid flow velocity to the speed of sound in the same fluid and the ratio of inertial forces to the viscous forces in flowing fluid respectively. These numbers characterizes the nature of flow, in particular (21) the nature of its regime (laminar, transient, turbulent…). The flowchart of the optimization algorithm used in this study is presented in Fig. 21. The engine thermal efficiency is expressed as To avoid large viscous dissipations, occur at vicinity of the viscous boundary layer which decreases the overall efficiency of the engine, the values for the normalized stack position the normalized stack length are assumed to be smaller than 0.5. The drive ratio and the resonance frequency are selected to be 0.03 and 580 Hz, respectively.
The combination of several stack lengths and positions are conducted through this study. Figure 22a, b shows the variations of stack thermal efficiency (η th ) against, the normalized length of plates (Lsn) at diverse stack center (24) η th = W Q = W n Q n positions (Xcn) and versus Xcn at different Lsn respectively. The graphs display that by moving away the stack position from the pressure antinode, the normalized stack length have to be decreased to obtain the efficiency peak. Though, at given stack length the efficiency peak magnitude decreases with increasing Xcn, and decreases with increasing Lsn at given Xcn. The ηth is very delicate to length of plates close to the peaks, a slightly shorter stack length results in a severe decrease in engine performance. The combination Xcn = 0.007 and Lsn = 0.0188 gives the best stack efficiency, η th = 0.24. Figure 22c, d shows the thermal stack efficiency behavior against the plate thicknesses (2l 0 ) and plate gaps (2y 0 ) respectively. From figures, it is shown that the efficiency peak is the best in the range 0.2-0.3 of Lsn. Also, it is noticed that, by increasing the plate thickness the stack efficiency decreases. However, by increasing the spacing between two plates the stack efficiency increases. The combinations that give the best stack efficiency are 2l0 = 0.25 with Lsn = 0.265 for plates thickness (η th = 0.1) and 2y 0 = 0.75 with Lsn = 0.255 for the spacing between two plates (η th = 0.083). Figure 22e presents the results illustrating the stack efficiency and the acoustic power versus the normalized stack length with different hot side temperatures. Different temperatures are applied to the hot side of plates while keeping the cold side at ambient temperature (300 k). The thermal stack efficiency peak at higher temperature difference is η th = 0.108 achieved for Lsn = 0.4. However, for the lesser temperature difference, the efficiency is η th = 0.052 achieved for Lsn = 0.22. It is noticed that the value of the stack efficiency from the shortest length to Lsn = 0.1 is not significantly influenced by the variation of temperature gradient across the stack. However, by increasing the normalized stack length and the temperature difference, the efficiency peaks move away from each other by a large and clear gap. To determine accurately the effect of heat capacity and thermal conductivity of the gas on the efficiency and the acoustic power of the system, comparative η th and Wn graphs of samples with different working gases are shown in Fig. 22f. The investigation is conducted with air, helium, nitrogen, argon, and carbon dioxide. According to figures, it is observed that argon gives the highest efficiency followed by helium, nitrogen, air, and carbon dioxide. It is clear that the efficiency peak for chosen gases is obtained for different normalized stack length, mainly for helium and argon while the efficiency peak is achieved at nearly ranges of Lsn for air, nitrogen, and carbon dioxide. The maximum values of η th are 0.092, 0.076, 0.073, 0.07, and 0.04 for argon, helium, nitrogen, air, and carbon dioxide respectively. By choosing the argon instead of air, the efficiency can be increased approximately by factor of 2.3. As mentioned above, the important parameters for the choice of the working fluid are its Prandtl number Pr and its number γ. The number of Prandtl Pr must be as small as possible in order to favor thermal exchanges between fluid and wall rather than inclusive heating due to viscous losses. It is possible to play on these parameters using mixtures of gases that depends on the intended application.