Strategic design and optimization of ultra-high-frequency regenerators for miniature pulse tube cryo

Abstract

Keywords: Miniature Pulse Tube Cryocooler; REGEN3.3; Regenerator; Ultra-High Frequency

Introduction

The development of miniature electronic technologies in space applications, like remote sensing, data transmission, infrared imaging, etc. requires miniature cryocoolers for their thermal management at temperatures ranging from 60K - 80 K with small payloads up to 1 W. Stirling Cryocoolers have effectively cooled infrared sensors for years, leading to advancements that limit the use of pulse tube cryocoolers (PTC) [1]. However, the mechanical displacer at the cold end of the Stirling cryocooler produces vibration and thus reduces longevity. In the past few years, there has been considerable progress in the design of PTCs. The combination of pulse tube, inertance tube, and surge volume in PTC design allows for the removal of mechanical components at the cold end, as shown in Figure 1. By removing this mechanical element, the PTC experiences reduced vibration, enhancing reliability and extending operational lifespan. These characteristics are particularly advantageous for space applications, where stability, durability, and longevity are paramount. The regenerator is typically framed of different structures and geometries in a thin-walled metallic tube. The hot and cold gases flow periodically within the regenerator matrices. The matrix stores the heat released by the hot gas during the hot blow period and rejects the same stored heat to the cold gas during the cold blow period, thus completing the thermal cycle. One of the most challenging tasks in the PTC is scaling down compared to the Stirling cryocooler while maintaining the same high efficiency. Miniaturization of the PTC alters the governing parametric quantities that apply to conventional PTCs. One of the limiting factors for the miniaturization of the PTC is the reduction of the working gas volume, and the effectual solution to compensate is to enhance the power density by increasing the operating frequency and filling higher charge pressure[2, 3]. Miniature Pulse Tube Cryocoolers (MPTC) operating at ultra-high frequencies exceeding 100 Hz are designed to enable compact dimensions while maintaining high efficiency and performance. The system’s regenerator significantly affects

the miniature cryocooler’s overall thermal performance at these frequencies. Several researchers have attempted the miniaturization of the pulse tube cryocoolers. Vanapalli et al. [4] optimized the regenerator of their MPTC using REGEN 3.2 software tools, resulting in dimensions of 9.02 mm inner diameter and 30 mm length. They attributed the system’s high efficiency to using a denser wire mesh screen (SS 635 #) within the regenerator. However, the system’s ability to achieve the theoretical high frequency was hindered by being driven by a relatively larger commercially available compressor. Radebaughet al. [5] designed the regenerator for their MPTC using REGEN 3.2 and SAGE 4.0 software tools for a frequency of 150 Hz, with an operating charge pressure of 5.0 MPa and a pressure ratio of 1.3, aiming for a cooling power of 1W@80K. However, in experimental trials, they achieved only a pressure ratio of 1.13 instead of the designed 1.30. Despite this discrepancy, the system reached a no-load temperature of 97.5 K within approximately 100 seconds. The researchers concluded that the regenerator design would likely perform even better at increased frequencies and pressure ratios. Wang et al. [6] conducted an analytical and experimental investigation on an MPTC operating at frequencies of 100 Hz and 280 Hz. They observed that as the frequency increased from 100 Hz to 280 Hz, the performance of the PTC deteriorated, resulting in a reduction in cooling power from 2.8 W@77 K to 1W@ 80 K. This degradation was attributed to the unsuitability of the regenerator matrix material. Mohan and Atrey [7] designed and developed an MPTC with a total pulse tube and regenerator volume of 1.72 cc, operating at a very high frequency of 148 Hz. Ouyang et al. [8] developed an MPTC operating at a frequency of 118 Hz with an overall cryocooler weight of 930 grams with a cold finger external diameter of 10 mm. The cooling power achieved by the system is 1.24 W @ 80K with an input power of 35 W. A 2-D regenerator model is developed to optimize hybrid stainless steel wire mesh structures for Stirling-type PTC. This study examines the characteristics of heat transfer and pressure drop, supported by experimental data to confirm the

theoretical outcomes [9]. Shaoshuai Liu et al. [10] examine how operating parameters like charge pressure and frequency affect the cooling efficiency of PTCs. Their study shows that optimizing these frequency parameters of 51 and

52. Hz and average pressures between 2.5 MPa and 3.5 MPa

can greatly minimize regenerator losses and optimize phase angles, leading to increased cooling efficiency. They report achieving a cooling power of 10W at 80 K with a (COP) coefficient of performance of 16.0% and compressor efficiencies of up to 88%. Garg et al. [11] investigated the effect of porous regenerators on miniature Stirling cryocooler efficiency, finding that lower porosity increases power consumption due to higher pressure drops, while higher porosity reduces power input but impairs thermal performance. Zhao et al. [12] developed a CFD model to study the impact of frequency and pulse tube length on a micro coaxial Stirling-type PTC, highlighting the need for further exploration of regenerator geometries to optimize performance and compactness. Malwe et al. [13] investigated the performance and exergy of a PTC at 80 K, testing charge pressures of 12, 14, and 16 bar. Results indicated that the refrigerating effect increased with higher charge pressure. For optimal performance at 70 K and a 15 W cooling load, 16 bar was identified as optimal, with a pulse tube diameter of 9.6 mm and length of 128.5 mm. Further studies on higher charge pressures and additional parameters are needed for miniaturization. Abraham and Kuzhiveli [14] investigated the effect of stainless-steel mesh characteristics on PTC performance. They found that using multi-mesh configurations improved efficiency and heat transfer. Additionally, incorporating multi-mesh regenerators with an inertance tube and bounce space as a phase shifter increases the COP. This research highlights the critical importance of mesh configuration in optimizing PTC performance. Guo and Zhu [15] highlighted the crucial role of the regenerator in their study of a micro-PTC with a displacer phase shifter. They determined that adjusting the regenerator’s design is essential for maintaining efficient cooling power across different pulse tube diameters, showcasing its essential contribution to improving cryocooler performance. Guo and Zhu [16] conducted a numerical study examining the regenerative impact of heat exchangers in PTCs. Their findings indicate that the regenerative effect diminishes expansion work, as measured by enthalpy flow, thereby decreasing system performance, particularly at higher pressure ratios. Feng et al. [17] developed a micro pulse tube cryocooler featuring a regenerator with a diameter of 10 mm and a length of 42 mm. This system, weighing 598 grams, achieves a cooling power of 1.2 W at 80 K, utilizing a compressor input power of 45 W within a remarkably short duration of just 3 minutes. Zhao et al. [18] developed a CFD model for a coaxial Stirling-type PTC operating at 125 Hz, analyzing how operating frequency affects fluid dynamics and thermal transfer in the regenerator. Li et al. [19] discuss a micro coaxial type PTC optimized for infrared detection at an optimal frequency of 175 Hz. Their study emphasizes specific

geometrical configurations, such as regenerator length, larger dimensions, and matrix type, while considering diameter. However, their focus on charge pressure limits a comprehensive understanding of the system’s overall performance. This highlights the need for a more systematic approach to explore the interactions among various design elements, potentially enhancing insights into the cryocooler’s efficiency and effectiveness. Min et al. [20] examined a specific regenerator design incorporating a middle heat exchanger through numerical and experimental approaches but did not investigate a wider range of geometrical configurations. The use of fixed operational parameters limits the research and does not provide a systematic analysis of how different regenerator lengths and configurations affect overall efficiency. Wang et al. [21] developed a coaxial-type single-stage PTC that operates at a frequency of 102 Hz, which may not be suitable for all applications. While this design is lighter than traditional models, its weight of 4.4 kg poses challenges for ultra-lightweight in space applications. Zhiet al. [22] conducted an experimental study on an MPTC, offering valuable insights into its refrigeration performance. However, the investigation primarily focuses on the cryocooler’s performance at a fixed charge pressure of 4 MPa, with input electric powers of 30 W and 68 W, and a frequency of 115 Hz. The highest cooling capacity achieved was 1.73 W at a temperature of 77 K. This limited focus restricts the exploration of how variations in other geometrical and operational parameters impact overall system performance. Several studies have investigated regenerators and their influence on cryocooler performance with REGEN. REGEN 3.3 is a software tool used for designing and optimizing to achieve the desired heat lift and cold end temperature. Pfotenhauer et al. [23] conducted optimization of a regenerator operating at the frequency (30 Hz to 60 Hz), warm end temperature of 300 K, cold end temperatures (60 K, 80 K), cooling power (100 W to 900 W), area (0.015 to 0.04 m2) and length (0.04 to 0.06 m) using software tool REGEN 3.2. However, their results are restricted to a fixed average pressure (2.0 MPa) and pressure ratio (1.2). Radebaugh et al. [24] investigated the performance of rare-earth materials in regenerators, analyzing porosities from 0.1 to 0.38 at 30 Hz and pressures of 0.3 MPa to 1.5 MPa using He-3 and He-4. They compared parallel holes and packed spherical ball structures, focusing on pressure loss and COP at 4K. This study highlights the importance of porosity and structural design in optimizing regenerator performance. Wang et. al. [25] performed a comparative analysis using REGEN 3.2 to assess the performance of various regenerator filler materials, including steel wire screens (500 #), lead screens (500 #), and lead spheres at a high frequency of 40 Hz and a temperature of 35 K. They found that lead spheres, which have a greater volumetric specific heat capacity, exhibited the poorest performance of the three materials due to challenges related to heat transfer depth and pressure drop. In contrast, the SS 500 # wire

screen satisfied the design requirements for the matrix. Thota et al. [26] designed the PTC aimed for operation at a frequency of 100Hz for the payload of 1 W @ 80 K. They employed REGEN 3.3 and SAGE software for performance analysis at both component and system levels. Pfotenhauer et al. [27] indicated that the performance of a cryocooler regenerator can be broadly optimized, with the potential for higher COP using denser SS 400# mesh and charge pressures above 2 MPa. Srinivasan et al.’s [28] comparison of ANSYS and REGEN for regenerator pressure drop yielded consistent results, confirming REGEN’s reliability for the Stirling Cryocooler regenerator design. REGEN’s precise prediction of pressure drop, validated across different inlet pressures, supports its effectiveness for cryocooler regenerator analysis and design. Srinivasan et al. [29] designed and developed a prototype regenerator using the Direct Metal Laser Sintering technique for a cryocooler. Yadav and Ramana [30] investigated the influence of a regenerator metal matrix on the operation of an MPTC operating at ultra-high frequencies. Desai et al. [31] conducted numerical studies on layered regenerator configurations in PTCs, focusing on low frequencies. Their findings demonstrated that a hybrid configuration outperformed traditional uniform regenerator matrix materials, highlighting its improved effectiveness. Chen et al. [32] introduce a novel approach using a 1D-CNN to predict regenerator performance in cryogenic refrigerators, streamlining calculations and achieving high accuracy in power, pressure ratio, and COP prediction. This method improves efficiency and promises to transform cryogenic refrigerator design with agile and precise optimization capabilities. The literature emphasizes the need to optimize regenerators to enhance the performance of PTCs, primarily focusing on low frequencies. Numerous research studies have investigated the performance of regenerators and their impact on cooling performance, with results reported using tools such as REGEN and computational fluid dynamics (CFD). While there have been some improvements using CFD and experimental analysis of MPTCs, there is still a lack of understanding regarding geometrical and operating characteristics at ultra-high frequencies. The research gaps include the limited exploration of cryocooler performance at frequencies above 100 Hz, as most existing studies concentrate on lower frequencies. Additionally, the scarcity of resistance coefficient values based on experimental pressure drop data for ultra-high frequencies hinders the validation of CFD simulations, thus limiting their reliability for parametric studies. In contrast, REGEN software proves valuable for modeling the complex thermodynamic cycles of regenerative systems, such as pulse tubes and Stirling cryocoolers. Unlike CFD, REGEN does not directly simulate compression and expansion but uses boundary conditions at the regenerator’s inlet and outlet to account for these effects. This approach allows REGEN to effectively capture dynamic processes, providing insights into cycle efficiency and cooling capacities while also requiring

less computational time and not relying on experimental hydrodynamic data, making it an efficient tool for predicting regenerator performance and advancing miniaturization efforts with an improved COP. The present work aims to address existing gaps by investigating the underexplored challenges of ultra-high frequency operation in MPTCs. The novelty of this work lies in its focus on its systematic numerical optimization of ultra-high frequency regenerators for MPTCs. This area has been underexplored in previous research. By exploring this gap, the study provides new insights and optimization strategies, offering valuable guidelines for designing more efficient and compact systems. This investigation employs state-of-the-art steel screens (SS 635#) as the regenerator matrix structure for an MPTC, focusing on systematically analyzing various operating parameters. These parameters encompass ultra-high frequencies ranging from 100 Hz to 200 Hz, charge pressures from 3.5 MPa to 7.0 MPa, and pressure ratios varying from 1.15 to 1.3. The primary focus is on evaluating their collective influence on the cooling performance of the cryocooler. Additionally, this study aims to develop a roadmap for an ultra-high frequency regenerator for the MPTC.

Numerical Methodology

The REGEN 3.3 [33] is an industry-standard one-dimensional numerical software developed by the National Institute of Standards & Technology (NIST) for designing and optimizing cryocooler regenerators. Figure 2 and Figure 3 depict the main screen and input screens of REGEN 3.3, which accept various geometries and operating input parameters.

incorporating the friction factors added to the momentum and energy equations. REGEN 3.3 is a full implicit model that solves non-linear equations for pressure, temperature, and mass flux at the grid points by Newton iteration simultaneously. REGEN 3.3 [33] numerical model solves the following conservation equations for the fluid flow: Mass Conservation Equation (1) Momentum Equation

(2) Where f(ρ,T,v) f(p,T,v) is the friction term Figure 3. Input screen of REGEN 3.3 software.

REGEN 3.3 software tools solve temperature-dependent thermal properties and provide good accuracy and stability making it the first choice for regenerator optimization by academicians and industries. REGEN 3.3 accommodates various input variables that users can freely define. These variables cover many parameters related to the regenerative cryocooler system, including temperature-dependent thermal properties, heat transfer correlations, fluid properties, geometrical and operational parameters, and others. It can efficiently process large amounts of data and perform numerical calculations to accurately model system behavior under various operating conditions. REGEN 3.3 provides comprehensive data output, including key performance indicators such as COP, pressure drop, heat transfer rates, temperature profiles, and other relevant parameters. This extensive output allows users to thoroughly analyze simulation results in detail and make well-informed decisions regarding system optimization. The comparison between results of REGEN 3.3 and Sage (used for system-level simulation software tool) are compared in terms of different parameters viz. hot end pressure ratio, phase angle, cooling power to PV input power over a range of temperature from

50. K to 150 K. The study concluded that there was excellent

agreement between the simulation results obtained from REGEN and Sage. This finding provides high confidence in the thermodynamic modeling for cryocooler regenerators, indicating the reliability and accuracy of both software tools for such applications [34]. The results obtained from REGEN 3.2 software on high-frequency PTC are also validated with the experimentation [35]. The mass, momentum, and energy equations for conservation are solved by REGEN 3.3. The effect of the porous media in the mathematical model is considered by

(4) Where the matrix material thermal term D(x, T) is defined by (5) The thermal term has the form (6) Where H (p,v) is the heat transfer coefficient The determination of the COP involves normalizing the net refrigeration power concerning the PV work occurring at the hot end. (7) NTCAD represents the net cooling power of the regenerator, while PVWK0T denotes the PV work at the hot end of the regenerator. The NTCAD is determined by calculating the PV work term at the cold end of the regenerator (PVWK1). This is calculated from integral, (8) Here, v(L,t) and ρ(L,t) represent the instantaneous velocity and density of the gas at the cold end of the regenerator, respectively. pd(t) denotes the instantaneous dynamic

pressure, while ρc indicates the gas density at the cold end temperature Tc The NTCAD is found by NTCAD = GRCAD – RGLOSS – HTFLUX – TUBECD (9) Where GRCAD is the gross cooling power for losses in the expansion space and is given by GRCAD = (PVWK1-PRLOSS)*COOLING_MULT (10) PRLOSS (W) serves as the correction term utilized to assess the influence of pressurization on the enthalpy flux of the gas and obtained by

and COOLING_MULT is the factor used to estimate a reduced cooling power produced by a non-isothermal expansion process. RGLOSS is the loss due to the regenerator’s ineffectiveness and it is given by RGLOSS = ENTFLX – PRLOSS

ENTFLX denotes the integral average of the enthalpy flux at the cold side of the regenerator throughout one cycle and is given by (13) HTFLUX represents the heat transfer resulting from thermal conduction at the cold end of the regenerator. While TUBECD refers to the thermal conduction occurring through the tube that housed the regenerator matrix. The tube material is assumed to be stainless steel, exhibiting a steady-state temperature profile. Thermal

exchange between the tube and the matrix is disregarded in the analysis. To calculate TUBECD the following formula is employed, (14) Where σ (T) is the thermal conductivity of Stainless Steel. The regenerator is constructed using a porous material with lower thermal, needing a correction factor for porosity in REGEN software. Calculating the hot end PV work of the regenerator follows a methodology similar to equation 8 but with velocity and density evaluated at the hot end (x=0). To determine the pressure at the hot end, the pressure drop is added to the pressure at the cold end for a comprehensive analysis. The pressure drop is calculated through the integration of the pressure gradient. Boundary Conditions and Solution Procedure REGEN 3.3 is component-level design software, and thus it incorporates other components of the system, such as the compressor, pulse tube, and phase shift mechanisms modeled indirectly with the appropriate boundary conditions at both ends of the regenerator. REGEN 3.3 models presume sinusoidal mass flux at both ends (inlet and outlet) of the regenerator as an appropriate boundary condition. The regenerator numerical model is depicted in Figure 4. The phase angle between the pressure and mass flow plays a crucial role in the performance of the regenerator, which in turn affects the overall efficiency of the cryocooler. Therefore, the mass flow rate (s) and phase angle at the cold end (s) for a fixed cold end PV work value were calculated. Input parameters needed in REGEN 3.3 are gas inflow at the hot end and cold end temperature, pressure ratio, average pressure, operating frequency, flow amplitude, hydraulic diameter, cold end phase angle, and the geometrical parameters of the matrices i.e. matrix material, length of regenerator, porosity, cross-sectional area. Input

Validation of the Model To validate the numerical procedure in REGEN 3.3, we conducted a comparison with previously published studies. Initially, we simulated a model using the same numerical data as Pfotenhauer et al. [27] to verify accuracy. Additionally, we simulated the regenerator model using the same data from Garaway et al. [36], who provided regenerator dimensions and experimentally tested the results. The spatial and temporal data comprised 41 mesh points, 200 time steps per cycle, and 30,000 cycles. The simulation results, obtained with the specified geometrical and operating data, agree with the published results and are presented in Table 1.

Figure 5. Systematic procedure for optimizing regenerator geometry in REGEN 3.3.

parameters are varied for the simulated model to obtain the maximum COP of the regenerator. A detailed discussion of the methodology employed by REGEN 3.3 is available in the software’s user guide [33]. To systematically optimize the regenerator using REGEN 3.3, we employ the algorithm illustrated in Figure 5 following the methodology demonstrated in the previously published work [27]. All the simulations are carried out for various mass flows and phase angles at the cold end for fixed regenerator dimensions i.e. diameter and length. Further, the simulation case runs are repeated for different regenerator lengths to predict the optimal coefficient of performance. The same iterative procedure is repeated for all the geometrical and operating parameters to determine the optimal values of these variables for the maximum COP of a regenerator. The computational time required for each simulation in REGEN 3.3, given the above-selected range of parameters, is approximately 4 to 5 hours on a personal computer.

Regenerator Analysis In this investigation, we examined the regenerator of an MPTC to enhance compactness and improve performance. We utilized REGEN 3.3 to optimize geometric dimensions and various operating parameters. The SS 635# wire mesh screen was chosen as the regenerator filler material due to its optimal pore size, porosity, and reliability across the cryogenic temperature range of 300 K to 80 K, as detailed in Table 2. However, the hydraulic diameter of the SS 635# screen limits the MPTC system’s operating frequency to 200 Hz due to thermal penetration depth constraints. As a result, we fixed the parameters of the wire mesh screen and focused on optimizing other parameters within this constraint. Geometric dimensions and additional fixed and variable parameters are listed in Table 3. Thermo-physical properties of the working gas (He-4) and the matrix materials selected as temperature-dependent properties. Based on the previously published data, the conductivity factor for the SS 635 # wire mesh screen was 0.13 for all the cases. One of the assumptions for all the simulated case runs is that the compressor impedance matched with the system. Grid and Time Independence Study A numerical model can reach a cyclic steady state after completing a simulation. To ensure the results’ accuracy,

Table 1. Validation results for cryocooler regenerator Parameters

Ntcad

Table 3. Operating parameters for optimization of regenerator Sr. No.

Figure 6. COP as a function of Number of Mesh Points for D= 4 mm and L = 30 mm, Ncycles = 10,000, mc = 0.0004 Kg/s case 1: (F =100 Hz) (θc) = -25º Case 2: (F =200 Hz) (θc) = -35º.

the user needs to define certain parameters such as the number of iterations, grid points, and time steps used in each cycle. The regenerator simulation was started with an initial guess of 41 grid points, 100-time steps per cycle, and 1000 iteration conditions. Figure 6 displays a mesh independence study with two different sets of conditions. In both of the two cases, the diameter is 4 mm and length is 30 mm, the mass flow at the cold end is (mc) = 0.0004 kg/s, the number of cycles is 10,000, the number of time steps per cycle is 250, and the midpoint temperature ratio is 0.45. The value of phase angle at the cold end and frequency for Case 1 and 2 are (θc) = -25º, ƒ1=100 Hz & (θc) = -35º, ƒ2= 200 Hz, respectively. The study reveals that very little variation is observed in both cases with higher grid points over 81. The variation obtained in COP is 0.0115 % & 0.0145 % for case no. 1 & 2

respectively. Hence for all the further simulations, 81 mesh points are selected. Figure 7 displays how the COP of the system changes with the number of time steps per cycle. The analysis was performed on a regenerator with a diameter of 4 mm, length of 25 mm, a frequency of 100 Hz, 10,000 numbers of cycles, and a midpoint temperature ratio of 0.45. When the time step is varied from 100 to 250, COP variation was found

Conclusion

The numerical investigations in the present study focused on the design and optimization of the regenerator, to improve performance at ultra-high frequencies and

maximize the coefficient of performance. This study presents a novel approach by addressing the underexplored challenges of ultra-high frequency operation in miniature pulse tube cryocoolers, offering new insights and optimization strategies essential for developing more efficient and compact systems. The study systematically examined the effects of regenerator structural dimensions and operating parameters on the coefficient of performance. The key geometrical parameters considered in this paper are the diameter and length of the regenerator, while the operating parameters include ultra-high frequencies (100 Hz-200 Hz), charge pressures (3.5 MPa-7.0 MPa), and cold-end pressure ratios (1.15-1.3). In the study, the parameters are optimized to maximize the coefficient of performance and achieve 1 W cooling at 80 K. The main findings are summarized as follows: Regenerator Dimensions and Frequency Effects:

1. At 100 Hz, the optimal regenerator dimensions are a

diameter of 4 mm and a length of 25 mm, which results in a coefficient of performance of 0.0819.

2. At 200 Hz, with the same diameter but a reduced length

of 20 mm, the coefficient of performance reduces to 0.07491. This trend indicates that the regenerator performance deteriorates with increasing operating frequency, even when the dimensions are optimized.

3. Increasing the frequency from 100 Hz to 200 Hz, while

maintaining optimal regenerator dimensions, results in a 9.768% decrease in the coefficient of performance and a 20% reduction in the regenerator length. Thus, we can conclude that the working frequency has little influence

on the optimal diameter, while the regenerator length can be reduced under certain conditions. Effect of Charge Pressure:

1. With optimized diameter and length, the coefficient

of performance improves by 28.65 % at a frequency of 100 Hz when the charge pressure is increased from 3.5 MPa to 7.0 MPa and increases by 26.99 % at 200 Hz. However, charge pressure has little effect on the regenerator’s optimum dimensions. Effect of Pressure Ratio:

1. For an optimal diameter and length, the coefficient

of performance increased by 80.75% with an increase in pressure ratio from 1.15 to 1.3. The higher the cold end pressure ratio, the shorter the optimal length of the regenerator, and the smaller the path.

2. The performance of the regenerator of a miniature pulse

tube cryocooler improves with the increased charge pressure and pressure ratio. The optimized regenerator dimension is independent of the charge pressure and dependent on the cold end pressure ratio. To achieve the best performance of the regenerator in a miniature pulse tube cryocooler, it is essential to select the optimal regenerator size and strategically increase both charge pressures and pressure ratios. By improving thermal performance in compact designs, our results address the requirements for high efficiency and reduced size, weight, and power requirements essentials for cryogenic cooling in satellite applications, particularly for infrared sensors used in Earth Observation and surveillance. This work enhances the reliability and effectiveness of the system, which are critical for ensuring the optimal functionality of satellite instruments. Moreover, these results can also be used for ultra-high frequency Stirling cryocoolers. The most prominent feature of the present study is the capability to design an ultra-high-frequency regenerator that operates at high efficiency while maintaining compact dimensions. Due to the high cost and complexity of prototype testing at these frequencies, this research relied on numerical simulations using REGEN 3.3. Future work will focus on validating these findings through experimental testing as resources and facilities become available.

Nomenclature

Cross-sectional Area, m2 Diameter, m Hydraulic diameter, m Wire diameter, m Tube thickness, m Tube Length, m Heat capacity, J/m3k Total Energy, J/m3 Heat transfer rate between the gas and the matrix, W / m.K Thermal conductivity of the matrix material, W / m.K

p(t) Pr pd(t) Tc Th Tm t x v Greek symbols ρ (p, T) τ ϕ σ (T) Subscripts m c h

Pressure, Pa Pressure ratio Instantaneous dynamic pressure Cold-end gas inflow temperature, K Hot-end gas inflow temperature, K Temperature of the matrix material, K Time, sec Spatial coordinate, 0Velocity of the gas, m/s Velocity of the gas, 0 ≤ x ≤ L, m Density of the working gas, kg/m3 Period of oscillation, s Porosity thermal conductivity of stainless steel Refers to matrix Refers to cold end temperature of gas Refers to the hot end temperature of the gas

Pulse Tube Cryocooler Miniature Pulse Tube Cryocooler National Institute of Standards & Technology COP Coefficient of Performance CFD computational fluid dynamics

Ntcad

Net Cooling Power pvwk0T Hot end pV work PVWK1 Cold end pV work

Grcad

Gross cooling power rgloss Loss due to regenerator ineffectiveness

Prloss

Correction term to estimate the effect of pressurization on the enthalpy flux of gas TUBE_H Tube thickness COOLING_MULT Multiplication Factor to reduce gross refrigeration power

Entflux

Integral average of Enthalpy Flux at the right end of the regenerator over one cycle

Tubecd

Thermal Conduction through the tube containing the regenerator matrix

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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