本研究基於先前實驗室所開發之三級史特靈致冷器的熱力模型，並導入非支配排序遺傳演算法二代。首先建立實驗平台，於不同壓力與轉速下測試，驗證熱力模型的準確性，且基準組的最低致冷溫度為158.57 K。在最佳化分析方面，目標函數為最低第三級致冷頭溫度與最小指示功率。取樣階段採用拉丁超立方抽樣，並利用高斯過程回歸建構代理模型，接著導入非支配排序遺傳演算法二代搜尋柏拉圖前緣。
首先，最佳化過程為三變數最佳化，針對各級再生器孔隙率進行最佳化分析，結果顯示最低溫可達130.64 K，更進一步進行九變數最佳化，針對各級再生器之孔隙率、半徑與長度兩階段探討結果顯示最低溫可達112.23 K。後續針對三變數最佳化之最低溫度解進行實驗驗證，驗證結果成功降至150.16 K，相較於基準組獲得了8.41 K的降溫，甚至於3 bar與700 rpm，可達目前致冷器最低溫111.52 K。
本研究證實，使用熱力模型並結合高斯過程回歸、非支配排序遺傳演算法二代與響應曲面法，能應用於史特靈致冷器設計，更為未來多級史特靈致冷器的幾何配置與性能提升，提供系統性的技術參考。

To enhance the performance of a three-stage Stirling cooler, this study develops a comprehensive thermodynamic model optimized via NSGA-II. Experimental testing was conducted at 2 bar and 700 rpm to validate the model's accuracy, with the original system achieving a minimum temperature of 158.57 K. During optimization, LHS was combined with the Maximin distance criterion to ensure efficient exploration of the design space, followed by GPR-based surrogate modeling and NSGA-II searching.
The optimization was performed in two stages: a three-variable design focusing on regenerator porosity, and a nine-variable design incorporating porosity, radius, and length. The results indicate that the minimum temperature reached 130.64 K and 112.23 K, respectively. Subsequent experimental validation of the three-variable design achieved 150.16 K, marking an 8.41 K improvement over the original configuration. At 3 bar and 700 rpm, the optimized cooler reached a minimum third-stage temperature of 111.52 K. This research confirms that integrating thermodynamic modeling with GPR, NSGA-II, and RSM provides a systematic technical framework for the geometric configuration and performance enhancement of multi-stage Stirling coolers.

[1] D. M. Berchowitz and Y. Kwon, "Environmental Profiles of Stirling-Cooled and Cascade-Cooled Ultra-Low Temperature Freezers," Sustainability, vol. 4, no. 11, pp. 2838-2851, 2012.
[2] M. Montelatici, "Numerical and Experimental Analysis of a Stirling Refrigerator for Innovative Ultra Low Temperature Applications, " University of Florence, FLORE Repository, 2019.
[3] V. V. Kishor Kumar, "Hydrodynamic and Heat Transfer Characteristics of Miniature Stirling Cryocooler Regenerators — A Review," International Journal of Air-Conditioning and Refrigeration, vol. 29, no. 04, Art. no. 2130006, 2021.
[4] W. L. Zhao, H. L. Li, H. Sun, W. Huang, R. Z. Li, J. Huan, J. Chen, Y. X. Zhang, and R. J. Xu, "Overview of Micro-Rotary Stirling Cryocoolers for HOT IR Detectors," Infrared Technology, vol. 45, no. 02, pp. 195–201, 2023.
[5] W. Chen, M. J. DiPirro, I. M. McKinley, C. Cho, and H. Tseng, "Active Cryocooling Needs for NASA Space Instruments and Future Technology Development," Cryogenics, vol. 141, Art. no. 103877, 2024.
[6] F. Kharadi, K. A., V. Bhojwani, P. Dixit, N. J. Kanu, and N. Jain, "Experimental Study of the Operating Parameters on the Performance of a Single-Stage Stirling Cryocooler Cooling Infrared Sensor for Space Application," Aircraft Engineering and Aerospace Technology, vol. 96, no. 8, pp. 1083-1091, 2024.
[7] D. M. Berchowitz, "Maximized Performance of Stirling Cycle Refrigerators," in Proc. IIR-Gustav Lorentzen Conf. Natural Working Fluids, Oslo, Norway, pp. 422–429, 1998.
[8] D. Erol, "An Experimental Comparative Study of the Effects on the Engine Performance of Using Three Different Motion Mechanisms in a Beta-Configuration Stirling Engine," Energy, vol. 293, Art. no. 130660, 2024.
[9] J. Li, Q. S. Liu, P. Yang, and Y. W. Liu, "Application and Validation of Stirling Cryocoolers in Domestic Refrigerators," Journal of Refrigeration, vol. 45, no. 1, pp. 137–144, 2024.
[10] A. Yasuda, K. Otsuka, S. Tsunematsu, Y. Hiratsuka, and K. Kanao, "Improvement of the Two-Stage Stirling Cooler Below 20 K," IOP Conf. Series: Materials Science and Engineering, vol. 1301, 012023, 2024.
[11] M. Crook, M. Hills, S. Brown, S. Cleary, P. Iredale, N. Hardy, and H. Korswagen, "CryoBlue — A Low-Vibration 50 K Stirling Cryocooler," 20th International Cryocooler Conference, On Line Virtual Conference, 2018.
[12] Raphael Paul, Abdellah Khodja, Andreas Fischer and Karl Heinz Hoffmann, "Cooling Cycle Optimization for a Vuilleumier Refrigerator," Entropy, vol. 23, no. 12, 1562, 2021.
[13] A. L. Tarish, N. T. Alwan, B. M. Ali, O. M. Ali, S. R. Aslan, and O. R. Alomar, "Design and Performance Analysis of a Joule-Thomson Cryocooler Systems," Energy Reports, vol. 11, pp. 4572-4586, 2024.
[14] C. Dong, S. Liu, X. Sha, W. Yin, Z. H. Jiang, and Y. N. Wu, "Development of a 2 K Joule-Thomson Cryocooler with ⁴He," in Advances in Cryogenic Engineering, vol. 70, M. K. D. et al., Eds. Cham, Switzerland: Springer, pp. 649–655, 2024.
[15] G. Le Tetû, C. Fluhr, B. Dubois, J. Paris, R. Hostein, and V. Giordano, "Ultra-Stable Microwave Cryogenic Oscillator Operated with a Gifford-McMahon Cryocooler," Cryogenics, vol. 135, Art. no. 103745, 2023.
[16] K. D. Timmerhaus and R. P. Reed, Eds., Cryogenic Engineering: Fifty Years of Progress. New York, NY, USA: Springer, 2007.
[17] R. F. Barron, Cryogenic Systems, 2nd ed., Monographs on Cryogenics, vol. 1. New York, NY, USA: Oxford University Press, 1985.
[18] S. K. Garg, B. Premachandran, M. Singh, S. Sachdev, and M. Sadana, "Effect of Porosity of the Regenerator on the Performance of a Miniature Stirling Cryocooler," Thermal Science and Engineering Progress, vol. 15, Art. no. 100442, 2020.
[19] J. M. Pfotenhauer and X. Zhi, "Pulse Tube Cryocoolers," in Cryogenic Engineering, New York, NY, USA: Springer, pp. 45–89, 2021.
[20] Y. Xu, D. Sun, X. Qiao, Y. S. W. Yu, N. Zhang, J. Zhang, and Y. Cai, "Operating Characteristics of a Single-Stage Stirling Cryocooler Capable of Providing 700 W Cooling Power at 77 K," Cryogenics, vol. 83, pp. 78–84, 2017.
[21] D. Sun, X. Qiao, D. Yang, and Q. Shen, "Experimental Study on a Two-Stage Large Cooling Capacity Stirling Cryocooler Working Below 30 K," Cryogenics, vol. 129, Art. no. 103619, 2023.
[22] B. Zhang, S. Fang, S. Zhang, and R. Zhang, "Multi-Objective Optimization of Motion Characteristics of Gamma-Type Stirling Engine Based on Response Surface Method," Energy Sources, vol. 47, no. 7, pp. 5041–5058, 2025.
[23] S. Popesku, "Experimental and Numerical Study of Beta-Type Stirling Cryocooler," M.S. thesis, Dept. of Aeronaut. and Astronaut. Eng., National Cheng Kung University, Tainan, Taiwan, 2023.
[24] X. Tilot-Lu, "Experimental and Numerical Study of a Three-Stage Beta-Type Stirling Cooler," M.S. thesis, Intl. Master's Prog. in Energy Eng., National Cheng Kung University, Tainan, Taiwan, 2024.
[25] H. Kuehl, "Numerically Efficient Modelling of Non-Ideal Gases and Their Transport Properties in Stirling Cycle Simulation," in Proceedings of the 17th International Stirling Engine Conference and Exhibition, UK, pp. 24–26, 2016.
[26] E. D. Marquardt, J. P. Le, and R. Radebaugh, "Cryogenic Material Properties Database," in Cryocoolers 11, pp. 681–687, 2002.
[27] IPM無刷直流馬達及BL系列驅動器使用說明書, 第八版,愛德利科技, 2018.
[28] KTR Systems, "DATAFLEX® torque measuring shafts: Installation and operating instructions," KTR-N 46010 EN, KTR Systems, 2024.
[29] J. H. Holland, Adaptation in Natural and Artificial Systems. Ann Arbor, MI, USA: University of Michigan Press, 1992.
[30] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. Boston, MA: Addison-Wesley, 1989.
[31] H. Nejati SharifAldin and F. NayebiPour, "Geometry-Aware and Approximation-Free Dijkstra Optimization for Discrete and Continuous Pareto Fronts for Task Offloading in Edge Computing," Internet of Things, vol. 36, p. 101888, 2026.
[32] I. Benchelih, Multi-Objective Multi-Depot Electric Vehicle Routing Problem with Time Windows and Vehicle-to-Vehicle (V2V) Charging Constraints, M.S. thesis, ESSA Tlemcen, Tlemcen, Algeria, 2025.
[33] G. E. P. Box and K. B. Wilson, "On the experimental attainment of optimum conditions," Journal of the Royal Statistical Society: Series B, vol. 13, no. 1, pp. 1–38, 1951.
[34] Y. Li, J. Li, G. Chen, T. Ba, L. He, and Y. Liu, "Parameter Optimization and Performance Enhancement of Anode Fishbone-Rib Flow Channels in Proton Exchange Membrane Fuel Cells Based on LHS-NSGA-II," Renewable Energy, vol. 267, p. 125748, 2026.
[35] M. D. McKay, R. J. Beckman, and W. J. Conover, "A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code," Technometrics, vol. 21, no. 2, pp. 239–245, 1979.
[36] J. L. Loeppky, J. Sacks, and W. J. Welch, "Choosing the Sample Size of a Computer Experiment: A Practical Guide," Technometrics, vol. 51, no. 4, pp. 366–376, 2009.
[37] M. D. Morris and T. J. Mitchell, "Exploratory Designs for Computational Experiments," Journal of Statistical Planning and Inference, vol. 43, no. 3, pp. 381–402, 1995.
[38] M. E. Johnson, L. M. Moore, and D. Ylvisaker, "Minimax and Maximin Distance Designs," Journal of Statistical Planning and Inference, vol. 26, no. 2, pp. 131–148, 1990.
[39] R. B. Gramacy and H. K. Lee, "Cases for the Nugget in Modeling Computer Experiments," Statistics and Computing, vol. 22, no. 3, pp. 713–722, 2012.
[40] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, "A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182–197, 2002.
[41] M. Ghoroqi, P. Ghoddousi, A. Makui, A. A. Shirzadi Javid, and S. Talebi, "Integration of Resource Supply Management and Scheduling of Construction Projects Using Multi-Objective Whale Optimization Algorithm and NSGA-II," Soft Computing, vol. 28, pp. 6983–7001, 2024.
[42] R. T. Silvestrini, D. C. Montgomery, and B. Jones, "Comparing Computer Experiments for the Gaussian Process Model Using Integrated Prediction Variance," Quality Engineering, vol. 25, no. 2, pp. 164–174, 2013.
[43] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, "Scalable Test Problems for Evolutionary Multiobjective Optimization," in Evolutionary Multiobjective Optimization, London, UK: Springer, pp. 105–145, 2005.