Simulation studies on the performance comparison of thermoacoustic prime mover with various resonator geometries and different stack materials

Abstract

The present investigation deals with analysing the performance of a thermoacoustic prime mover (TAPM) measured in terms of frequency, pressure amplitude, and onset temperature with the help of the renowned simulation tool, DeltaEC. Various resonator geometries, such as straight, tapered, coiled, resonator fitted with compliance and twin TAPM, were aligned in the simulation setup with nitrogen as a fluid medium and operated at a pressure of 10 bars. The operational and geometrical parameters of the experimental setup in DeltaEC were kept constant except the resonator geometry and stack material made of stainless steel and nickel. In the literature, no significant studies evaluating the performance of TAPM with different resonator geometry have been published. This study is the first of its kind to analyse TAPM’s performance using five distinct resonator geometry. In the present simulation, it was observed that the TAPM of the resonator fitted with compliance generated oscillations with low frequency and high-pressure amplitude compared to the simulation setup with other resonators’ geometry with a stack made of stainless steel. Similarly, it was observed from the simulation results that the twin TAPM generated oscillations at a lower onset temperature than others with both stack materials. Among the chosen resonators, the TAPM fitted with compliance produces high pressure amplitude oscillations with a slight increase in onset temperature compared to the twin TAPM with the stack made of stainless steel, which will be used as a drive for cryocoolers.

Keywords:

• Coiled
• tapered
• compliance
• twin TAPM
• pressure amplitude
• onset temperature

REVIEWING EDITOR:

• Pham D T, University of Birmingham, Birmingham, UK

SUBJECTS:

• Thermodynamics
• Heat Transfer
• Mechanical Engineering
• Power & Energy
• Acoustical Engineering

1. Introduction

The surplus of energy in the form of heat emitted around the world from various sources, such as solar, industrial boilers, automobiles, and so on, is being utilized for several applications, such as power production, water heating, steam generation, and solar heat-driven cooling. Thermoacoustics is one of the novel blends of thermodynamics and acoustics that produce acoustic waves by consuming heat energy obtained from the above-mentioned sources and vice versa. Investigations are being carried out by researchers to design a TAPM with optimized geometrical and operational parameters. The perfect design procedure starts with the selection of working fluids and the geometrical parameters of the TAPM. The present work deals with the analysis of TAPM’s performance by varying the geometry of the resonator with different stack materials using the simulation software DeltaEC. The working fluid and its operational pressure, plate thickness, plate spacing, stack position, and stack length used in this simulation were selected from the author’s previously published research (Hariharan et al. 2013a, 2013b, 2013c).

Yu et al. (2007) developed a thermoacoustic Stirling heat engine with a tapered resonator to achieve the high-pressure ratio. The fabricated geometry developed a pressure ratio of 1.40 with the lowest onset temperature difference of 73 °C with various operating pressures of different working fluids. The results obtained from the experimental results were validated with Computational Fluid Dynamics (CFD). Zink et al. (2010) investigated the performance of thermoacoustic engine with straight and coiled resonator using CFD. The CFD results indicated that the curvature introduced caused the pressure oscillations to amplify less in coiled resonator compared to straight resonator but the frequency of the oscillations remains constant for both the resonator. Eldeeb et al. (2011) demonstrated the efficiency of TAPM by varying the resonator geometries in the form of truncated pyramids, plate spacing, heat input, and stack length using DeltaEC. The truncated square pyramid resonator was designed, and the stack made of Celcor was used in conducting the prime mover experiments. The efficiency of 30.61% was obtained with the developed geometry. Hariharan et al. (2013a, 2013b, 2013c) examined the performance of twin TAPM using DeltaEC software and thermoacoustic refrigerator using response surface methodology by varying different geometrical and operating parameters in their experimental setup. From the results, they found that the experimental and simulated values were just deviating within the range of 10%, which showed that the DeltaEC served as an efficient tool to analyse the performance of thermoacoustic devices. Konaina et al. (2014) developed software for sizing thermoacoustic coolers with various governing equations. They also developed a design strategy for the prime mover to determine the impact of physical parameters on its performance. Results obtained from the developed strategy were compared with standard DeltaEC software. Yang et al. (2014) determined the optimum structure of a phase adjuster that influences onset temperature, frequency, and pressure amplitude using response surface methodology. The optimal parameters were simulated using DeltaEC, and the results obtained were compared with the response surface methodology.

Kalra et al. (2015) analysed the performance of TAPM in terms of various parameters, especially acoustic power and efficiency using DeltaEC software, by varying the stack length, resonator, hot and cold heat exchangers, working fluids with various compositions. They found that the pressure amplitude increases with an increase in pressure for the denser gas argon, whereas the acoustic power and efficiency exhibit the same trend with a decrease in pressure for working gases. Researchers carried out several experiments by varying the operational and geometric parameters for analysing the performance of TAPM. Chen et al. (2017) performed a numerical analysis on the performance of a two-stage traveling wave thermoacoustic engine using DeltaEC. According to his investigation, the simulation results produced a thermal efficiency of 40.2% with a relative Carnot efficiency of 60.2% and a maximum pressure ratio of 1.37. Balonji et al. (2019) developed an adjustable thermoacoustically driven engine using DeltaEC, where the performance studies were observed by dividing the geometry of resonator into two portions. Performance in terms of acoustic power, temperature difference across the stack, and frequency of the oscillations were measured. Among the parameters, the resonator’s geometry and its length are two of the key parameters in improving the performance of TAPM. However, the frequency of acoustic waves is calculated from resonator length, and the strength of acoustic waves is estimated based on the resonator’s geometry and the stack material to a certain extent. Imrul Kayes and Ashiqur Rahman (2023) enhanced the performance of a wet thermoacoustic engine by placing a tightly packed mesh screen matrix between the parallel stack, which enhanced the heat transfer area and also prevented the production of higher harmonics. They measured the onset temperature difference, acoustic power, harmonic ratio, and fundamental frequency by varying the plate spacing, mesh packing density, mesh number, and mesh packing length. From the optimum design developed, the maximum acoustic power was generated at a low onset of 13 °C by decreasing the higher harmonics to 58%. Guo et al. (2023) investigated the performance of a twin standing-wave thermoacoustic engine (TSWTAE) in terms of augmenting the acoustic power output by altering the operating conditions, such as working fluids, operating pressures, and geometric shapes like the resonator and buffer. Experimental results were validated with the results obtained from DeltaEC and CFD. Due to an increase in the molecular weight of working fluids and their operating pressures, they found that there was an increase in acoustic oscillation pressure of 82.03% and a conversion efficiency of 32.60% in the tapered conical resonator compared to the narrow cylindrical resonator chosen for their investigation. The optimized performance of TSWTAE was observed by charging pressure of 0.6 MPa helium gas for its geometry.

Based on the above-cited literature, it was found that the operational parameters of TAPM were varied, and its performance was perfectly predicted by DeltaEC. Several numerical simulation studies have been reported based on Computational Fluid Dynamics (De Meglio & Massarotti, 2022), Galerkin method (Yao et al., 2023), response surface methodology (Yang et al., 2014) and so on (Bhatti et al., 2023). By varying the geometrical parameters such as stack position, stack length, resonator length, plate spacing, plate thickness, as well as the operational parameters such as various working fluids (Wang & Hu, 2023; Hariharan et al. 2013a, 2013b, 2013c) and the pressure of the working fluid, the performance of TAPM was also carefully analysed. Among the aforementioned parameters, no major research comparing the performance of resonator geometry has been published. The present simulation study has taken this factor as a unique parameter, and the best geometry of a resonator among the types such as coiled, straight, straight with compliance, tapered, and twin TAPM was analysed with different stack materials such as stainless steel and nickel, which generate high pressure amplitude oscillations with less onset temperature, using DeltaEC as a simulation tool. The optimized TAPM with the suitable resonator geometry and stack material, which produces high pressure amplitude oscillations, may be used as a drive for cryocoolers in space applications, refrigerators, electric power generation (Timmer et al., 2018), microelectronics cooling, and in several other applications (Chen et al., 2021).

2. Simulation using DeltaEC

Los Alamos National Laboratory, USA, developed exclusive simulation software for thermoacoustics called “Design Environment for Low Amplitude Thermoacoustic Energy Conversion” or “DeltaEC”. Various forms of TAPMs, thermoacoustic refrigerators (Nsofor & Ali, 2009), loudspeaker driver thermoacoustic refrigerators (Tijani, 2001), pulse tube refrigerators, etc could be simulated, and the performance of thermoacoustic devices was easily assessed using DeltaEC. The continuity (Equation (1)(1) $$\frac{\partial \rho }{\partial t}+\nabla .\left(\mathrm{\rho }\mathrm{v}\right)=0$$(1) ), momentum (Equation (2)(2) $$\rho \left[\frac{\partial v}{\partial t}+(v.\nabla )v\right]=-\nabla p+\mu {\nabla }^{2}v+\left(\xi +\frac{\mu }{3}\right)\nabla \left(\nabla .v\right)$$(2) ), energy (Equation (3)(3) $$\frac{\partial }{\partial t}\left(\rho \epsilon +\frac{1}{2}\rho {\nu }^2\right)=-\nabla .\left[-k\nabla T-v.\Sigma +\left(\rho h+\frac{1}{2}\rho {\nu }^2\right)v\right]$$(3) ), and wave (Equation (4)(4) $$\left[1+\frac{\left(\gamma -1\right)}{\left(1+{\epsilon }_s\right)}{f}_{\kappa }\right]p_1+\frac{a^2{\rho }_m}{{\omega }^2}\frac{d}{\mathit{\text{dx}}}\left(\frac{1-f_{\nu }}{{\rho }_m}\frac{d{p}_1}{\mathit{\text{dx}}}\right)+\frac{a^2}{{\omega }^2}\frac{f_{\kappa }-f_{\nu }}{\left(\sigma -1\right)\left(1+{\epsilon }_s\right)}\frac{1}{T_m}\beta \frac{d{T}_m}{\mathit{\text{dx}}}\frac{d{p}_1}{\mathit{\text{dx}}}=0$$(4) ) equations of the thermoacoustic system were modified into ordinary differential equations for pressure amplitude (Equation (5)(5) $$\frac{d{p}_1}{\mathit{\text{dx}}}=\frac{-j{\rho }_m\omega ⟨u_1⟩}{\left(1-f_{\nu }\right)}$$(5) ), velocity amplitude (Equation (6)(6) $$\frac{d⟨u_1⟩}{\mathit{\text{dx}}}=\frac{-j\omega }{{\rho }_m{a}^2}\left[1+\frac{\left(\gamma -1\right)}{\left(1+{\epsilon }_s\right)}{f}_{\kappa }\right]p_1+\frac{\left(f_{\kappa }-f_{\nu }\right)}{\left(1-\sigma \right)\left(1+{\epsilon }_s\right)\left(1-f_{\nu }\right)}\beta \frac{d{T}_m}{\mathit{\text{dx}}}⟨u_1⟩$$(6) ), and temperature gradient (Equation (7)(7) $$\frac{d{T}_m}{\mathit{\text{dx}}}=\frac{\left\{\frac{\dot{E_2}}{A_g}-\frac{1}{2}\mathit{\text{Re}}\left[p_1〈u_1^{\mathrm{*}}〉\left(1-\frac{\left(f_k^{\mathrm{*}}-f_v^{\mathrm{*}}\right)}{\left(1+\sigma \right)\left(1+{\epsilon }_s\right)\left(1-f_v^{\mathrm{*}}\right)}\right)\right]\mathrm{ }\right\}}{\frac{{\rho }_m{C}_{p{\left|〈u_1〉\right|}^2}}{2\omega \left(1-\sigma \right)\left(1+{\epsilon }_s\right){\left|1-f_v^{\mathrm{*}}\right|}^2}\mathrm{ }\mathit{\text{Im}}\mathrm{ }\left[f_v^{\mathrm{*}}+\frac{\left(f_k^{\mathrm{*}}-f_v^{\mathrm{*}}\right)\left(1+\frac{{\epsilon }_s{f}_v}{f_k}\right)}{\left(1+\sigma \right)\left(1+{\epsilon }_s\right)}\right]-K-\frac{A_s}{A_g}{K}_s}$$(7) ) (Tijani, 2001), which were expressed below and numerically solved by codes of DeltaEC (http://www.lanl.gov/thermoacoustics/UsersGuide.pdf), using fourth-order Runge–Kutta method. (1) $$\frac{\partial \rho }{\partial t}+\nabla .\left(\mathrm{\rho }\mathrm{v}\right)=0$$(1) (2) $$\rho \left[\frac{\partial v}{\partial t}+(v.\nabla )v\right]=-\nabla p+\mu {\nabla }^{2}v+\left(\xi +\frac{\mu }{3}\right)\nabla \left(\nabla .v\right)$$(2) (3) $$\frac{\partial }{\partial t}\left(\rho \epsilon +\frac{1}{2}\rho {\nu }^2\right)=-\nabla .\left[-k\nabla T-v.\Sigma +\left(\rho h+\frac{1}{2}\rho {\nu }^2\right)v\right]$$(3) (4) $$\left[1+\frac{\left(\gamma -1\right)}{\left(1+{\epsilon }_s\right)}{f}_{\kappa }\right]p_1+\frac{a^2{\rho }_m}{{\omega }^2}\frac{d}{\mathit{\text{dx}}}\left(\frac{1-f_{\nu }}{{\rho }_m}\frac{d{p}_1}{\mathit{\text{dx}}}\right)+\frac{a^2}{{\omega }^2}\frac{f_{\kappa }-f_{\nu }}{\left(\sigma -1\right)\left(1+{\epsilon }_s\right)}\frac{1}{T_m}\beta \frac{d{T}_m}{\mathit{\text{dx}}}\frac{d{p}_1}{\mathit{\text{dx}}}=0$$(4) (5) $$\frac{d{p}_1}{\mathit{\text{dx}}}=\frac{-j{\rho }_m\omega ⟨u_1⟩}{\left(1-f_{\nu }\right)}$$(5) (6) $$\frac{d⟨u_1⟩}{\mathit{\text{dx}}}=\frac{-j\omega }{{\rho }_m{a}^2}\left[1+\frac{\left(\gamma -1\right)}{\left(1+{\epsilon }_s\right)}{f}_{\kappa }\right]p_1+\frac{\left(f_{\kappa }-f_{\nu }\right)}{\left(1-\sigma \right)\left(1+{\epsilon }_s\right)\left(1-f_{\nu }\right)}\beta \frac{d{T}_m}{\mathit{\text{dx}}}⟨u_1⟩$$(6) (7) $$\frac{d{T}_m}{\mathit{\text{dx}}}=\frac{\left\{\frac{\dot{E_2}}{A_g}-\frac{1}{2}\mathit{\text{Re}}\left[p_1〈u_1^{\mathrm{*}}〉\left(1-\frac{\left(f_k^{\mathrm{*}}-f_v^{\mathrm{*}}\right)}{\left(1+\sigma \right)\left(1+{\epsilon }_s\right)\left(1-f_v^{\mathrm{*}}\right)}\right)\right]\mathrm{ }\right\}}{\frac{{\rho }_m{C}_{p{\left|〈u_1〉\right|}^2}}{2\omega \left(1-\sigma \right)\left(1+{\epsilon }_s\right){\left|1-f_v^{\mathrm{*}}\right|}^2}\mathrm{ }\mathit{\text{Im}}\mathrm{ }\left[f_v^{\mathrm{*}}+\frac{\left(f_k^{\mathrm{*}}-f_v^{\mathrm{*}}\right)\left(1+\frac{{\epsilon }_s{f}_v}{f_k}\right)}{\left(1+\sigma \right)\left(1+{\epsilon }_s\right)}\right]-K-\frac{A_s}{A_g}{K}_s}$$(7)

The simulated model of various forms of TAPMs is presented in Figure 1. BEGIN, DUCT, HX, STKSLAB, COMPLIANCE, and HARDEND are the major segments used to design a TAPM. Different forms of stack geometry are available, such as STKSLAB, STKPINS, STKRECT, STKSCREEN, STKDUCT, STKCONE, and STKCIRC, which are considered the heart of the TAPM.

Figure 1. Modeling of various forms of TAPM in DeltaEC (a) TAPM with straight resonator, (b) TAPM with coiled resonator, (c) TAPM with tapered resonator, (d) TAPM of resonator with compliance, and (e) twin TAPM.

The geometrical data of all the segments and the properties of gas were substituted into the simulated model as input in the DeltaEC program to obtain the results. The logistical segment BEGIN has parameters like pressure, frequency, temperature, volumetric flow rate, and pressure amplitude, where temperature, pressure amplitude, and frequency were set as the guesses. These guesses were considered the output of the simulated model for this present investigation. DeltaEC calculated the outputs after several iterations from the assumed values until they reached consistent values. The diameter of the resonator was taken as 0.038 m, and the corresponding area and perimeter of the resonator were substituted in DUCT. The length of the resonator was taken as 1 m for all forms of TAPM with different resonator geometry. Sequentially, a hot heat exchanger, stack in a parallel plate fashion, and a cold heat exchanger were arranged to develop a simulated TAPM as shown in Figure 1. The schematic representation of TAPM with various geometries is shown in Figure 2. The cross-sectional area, blockage ratio, length, half plate thickness, and half plate spacing of heat exchangers and stack were substituted in HX (Heat Exchanger – hot and cold heat exchangers) and STKSLAB (parallel plate stack), respectively (http://www.lanl.gov/thermoacoustics/UsersGuide.pdf). The half plate thickness, half plate spacing, and blockage ratio of the stack and heat exchangers were substituted as 0.0005 m, 0.0005 m, and 0.5, respectively, whereas the length of the stack and heat exchangers was kept at 0.05 m, and 0.02 m, respectively. The geometrical parameters of TAPM with various resonator geometries are presented in Table 1. Stainless steel and nickel were chosen as the materials of construction for the stack in the present simulation. Further, copper was chosen for hot and cold heat exchangers. The simulation was performed for various resonator geometries with different stack materials. As the system was a closed half-wavelength TAPM, the segment HARDEND was used for all the present simulation models, beyond which there was no flow, and the real and imaginary parts of specific impedance were set as targets. COMPLIANCE and CONE segments present in Figure 1 represent the buffer volume and tapered resonator, respectively.

Figure 2. Schematic representation of various forms of TAPM (a) TAPM with straight resonator, (b) TAPM with coiled resonator (c) TAPM with tapered resonator, (d) TAPM of resonator with compliance and (e) twin TAPM.

Table 1. Geometrical parameters of TAPM with various geometries.

Operational parameters	Various resonator geometries
	Straight	Tapered	Coiled	Resonator fitted with compliance (buffer)	Twin TAPM
Stack position (m)	0.08	0.08	0.08	0.08	0.08
Stack length (m)	0.05	0.05	0.05	0.05	0.05
Resonator length (m)	1	1	1	1	1
Plate spacing (m)	0.0005	0.0005	0.0005	0.0005	0.0005
Plate thickness (m)	0.0005	0.0005	0.0005	0.0005	0.0005
Tapered diameter at the output (m)	NA	0.068	NA	NA	NA
Coiled at the position (m) (resonator coiled as right angle to the geometry of simulation setup)	NA	NA	0.5	NA	NA
Compliance diameter (m)	NA	NA	NA	0.11	NA
Working fluid	Nitrogen
Working pressure (bar)	10

3. Results and discussion

Simulations were performed for various resonator geometry types, such as coiled, straight, straight with compliance, tapered, and twin TAPM, with different stack materials, while maintaining all other geometrical parameters as constants. Performance in terms of frequency, onset temperature, and pressure amplitude of acoustic waves was measured using the simulation tool DeltaEC.

3.1. Effect of onset temperature for various resonator geometries

One of the main factors to determine the performance of TAPM is onset temperature. When the quantity of thermal energy transformed into acoustical energy exceeds the quantity being lost due to thermal and viscous processes on the surfaces of the various elements, oscillations begin to occur. To initiate self-oscillation, there must be a greater temperature differential observed in relation to the heat exchangers, which facilitates the convection of heat away from the stack. The effect of onset temperature on various TAPMs with different resonator geometries and different stack materials is shown in Figure 3. It is observed from Figure 3 that, compared to all other forms of the prime mover, twin TAPM consumes less heat to produce the oscillations. Also, the prime mover with straight resonator produces oscillations with high heat consumption. Due to the frequency shift, the resonator geometry with compliance also consumes less heat next to the twin TAPM to generate the oscillations. The increasing order of onset temperature for the present simulation is twin TAPM, TAPM of the resonator with compliance, TAPM with the tapered resonator, TAPM with the coiled resonator, and TAPM with the straight resonator. It is also observed from Figure 3 that the onset temperature of both the stack materials, nickel and stainless steel, was almost the same.

Figure 3. Effect of various resonator geometries with different stack materials on onset temperature.

3.2. Effect of frequency for various resonator geometries

The resonator plays a major role in predicting the frequency of oscillations generated by the TAPM. From Figure 4, it is observed that the resonance frequency of the TAPM is lower for the resonator geometry with compliance due to the frequency jumping, whereas the frequency is higher for the prime mover with a tapered resonator (Tang et al., 2006). It is clear from the plot that resonance frequency is not only dependant on the length of the resonator but also on the geometry of resonator, especially the resonator attached to compliance (buffer), which has a significant effect on decreasing the frequency. The increasing order of frequency for the present simulation is TAPM of the resonator with compliance, twin TAPM, TAPM with a coiled resonator, TAPM with a tapered resonator, and TAPM with a straight resonator. Also, it is observed from Figure 4 that the frequency of the oscillations was the same for both the stack materials, stainless steel and nickel, as the frequency depends mainly on the resonator.

Figure 4. Effect of various resonator geometries with different stack materials on frequency.

3.3 Effect of pressure amplitude for various resonator geometries

It is observed from Figure 5 that the TAPM of a resonator with compliance generates oscillations with high pressure amplitude compared to other resonator geometries, and resonator with a coiled geometry generates low pressure amplitude oscillations (Zink et al., 2010). Next to the TAPM of resonator with compliance, the tapered resonator produces the high-pressure amplitude oscillations. The use of a tapered resonator suppresses harmonic mode oscillations, nonlinear effects, and viscous loss due to large interface area (Tang et al., 2006). An increase in pressure amplitude is achieved by an increase in the gas-solid interface for a fixed resonance tube length by varying the dimensions of the resonator geometry. The increasing order of pressure amplitude for the present simulation is TAPM with a straight resonator, TAPM with a coiled resonator, TAPM with a tapered resonator, twin TAPM, TAPM of the resonator with compliance. From Figure 5, it is observed that the pressure amplitude of the oscillations generated from all forms of prime mover was less for stack materials made of nickel than stainless steel. The material’s heat capacity must always be greater than the working fluid’s, and its thermal conductivity must be as low as feasible, for an efficient thermoacoustic conversion to take place. The stack’s heat capacity and thermal conductivity vary along with the construction material. This in turn impacts the quantity of heat transfer by convection between the working fluid and the plate walls as well as the process of heat transfer by conduction between the plates from the hot side to the cold one. To avoid loss of acoustic power, materials with low thermal conductivity are required. Also, the specific heat capacity of the stack must be higher than that of the working fluid (Bouramdane et al., 2023). Thus, the pressure amplitude is higher for a stack made of stainless steel than nickel because the stainless steel has a low thermal conductivity than nickel, which produces high pressure amplitude oscillations.

Figure 5. Effect of various resonator geometries with different stack materials on pressure amplitude.

4. Conclusion

In general, TAPMs operating with minimum frequency and maximum pressure amplitude were selected by the researchers to drive cryocoolers. The present simulation of TAPM was performed by varying the resonator’s geometry and stack material using 10 bars of nitrogen as a working fluid. The performance of TAPM was analysed in terms of onset temperature, frequency, and pressure amplitude. It was found that twin TAPM generates oscillations with minimum onset temperature, while TAPM of a resonator with compliance produces oscillations with less frequency and high-pressure amplitude with stainless steel as the stack material compared to nickel. As a result, the present simulation confirmed that the TAPM of the resonator fitted with the compliance having a stainless steel stack drives the cryocoolers with better efficiency as the cryocoolers are operated with oscillations of high-pressure amplitude and less frequency. The present results will pave the way to carry out more research in the resonator geometry fitted with compliance by varying the shape of the compliance, twin TAPM fitted with compliance in the middle of the resonator, and many more. The optimized design of TAPM developed using the present numerical method may be practically designed and used as a drive for cooling microelectronics, space applications, and electric power generation.

Authors contribution Statement

N. M Hariharan: Design of work, Data acquisition, Data analysis, Data interpretation and Drafting the paper. P. Sivashanmugam: Final approval of the version to be communicated. P SaiPreethi: Drafting and reviewing the paper. V. T. Perarasu: Data analysis & interpretation of simulation results. Perumal Asaithambi: Reviewing the paper for final version.

The authors didn’t obtain any funds to carry out this present simulation study.

Nomenclature
ρ	=	density (kg/m3)
v	=	velocity (m/s)
t	=	time (s)
v	=	vector velocity component (m/s)
p	=	pressure (Pa)
μ	=	dynamic viscosity (Pa. s)
ξ	=	second viscosity (Pa. s)
ε	=	internal energy per unit mass (J/kg)
k	=	thermal conductivity (W/m K) in Eq. (3)(3) $$\frac{\partial }{\partial t}\left(\rho \epsilon +\frac{1}{2}\rho {\nu }^2\right)=-\nabla .\left[-k\nabla T-v.\Sigma +\left(\rho h+\frac{1}{2}\rho {\nu }^2\right)v\right]$$(3)
Σ	=	viscous stress tensor (N/m2)
T	=	temperature (K)
h	=	enthalpy per unit mass (J/kg)
γ	=	adiabatic index
fk	=	spatially averaged thermal function
fv	=	spatially averaged viscous function
ω	=	angular frequency (s−1)
a	=	sound speed (m/s)
σ	=	Prandtl number
p1	=	pressure amplitude (Pa)
pm	=	mean pressure (Pa)
β	=	thermal expansion coefficient (K−1)
Tm	=	mean temperature (K)
j	=	imaginary unit
Ė2	=	second order total power (W)
Ag	=	cross sectional area available for gas (m2)
$u_1^{*}$	=	gas particle velocity amplitude (m/s)
fk*	=	complex conjugates of spatially averaged thermal function
fv*	=	complex conjugates of spatially averaged viscous function
εs	=	stack heat capacity ratio (no unit)
K	=	thermal conductivity of gas (W/m K)
Ks	=	thermal conductivity of solid (W/m K)
As	=	cross-sectional area available for stack (m2)

Acknowledgements

The author is grateful to Dr. S. Kasthurirengan, Professor Emeritus, Centre for Cryogenic Technology, Indian Institute of Science, Bangalore for their valuable advice and contribution towards this numerical analysis research.

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

All the necessary data pertaining to the simulations has been included in the manuscript in the form of tables and figures. Additional information will be provided on request.