{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T00:10:26Z","timestamp":1767831026329,"version":"3.49.0"},"reference-count":61,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,11,24]],"date-time":"2021-11-24T00:00:00Z","timestamp":1637712000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"German Federal Ministry of Education and Research","award":["01LY1706B"],"award-info":[{"award-number":["01LY1706B"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Vuilleumier refrigerators are a special type of heat-driven cooling machines. Essentially, they operate by using heat from a hot bath to pump heat from a cold bath to an environment at intermediate temperatures. In addition, some external energy in the form of electricity can be used as an auxiliary driving mechanism. Such refrigerators are, for example, advantageous in situations where waste heat is available and cooling power is needed. Here, the question of how the performance of Vuilleumier refrigerators can be improved is addressed with a particular focus on the piston motion and thus the thermodynamic cycle of the refrigerator. In order to obtain a quantitative estimate of the possible cooling power gain, a special class of piston movements (the AS motion class explained below) is used, which was already used successfully in the context of Stirling engines. We find improvements of the cooling power of more than 15%.<\/jats:p>","DOI":"10.3390\/e23121562","type":"journal-article","created":{"date-parts":[[2021,11,29]],"date-time":"2021-11-29T05:23:02Z","timestamp":1638163382000},"page":"1562","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Cooling Cycle Optimization for a Vuilleumier Refrigerator"],"prefix":"10.3390","volume":"23","author":[{"given":"Raphael","family":"Paul","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdellah","family":"Khodja","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andreas","family":"Fischer","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karl Heinz","family":"Hoffmann","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, 09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,24]]},"reference":[{"key":"ref_1","unstructured":"Vuilleumier, R. (1918). Method and Apparatus for Inducing Heat Changes. (1,275,507), U.S. Patent."},{"key":"ref_2","unstructured":"Carlsen, H. (1989, January 6\u201311). Development of a gas fired Vuilleumier heat pump for residential heating. Proceedings of the 24th Intersociety Energy Conversion Engineering Conference, Washington, DC, USA."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"115547","DOI":"10.1016\/j.apenergy.2020.115547","article-title":"State-wide comparative analysis of the cost saving potential of Vuilleumier heat pumps in residential houses","volume":"277","author":"Shi","year":"2020","journal-title":"Appl. Energy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"114767","DOI":"10.1016\/j.applthermaleng.2019.114767","article-title":"Multi-objective optimization of a free-piston Vuilleumier heat pump using a genetic algorithm","volume":"167","author":"Chen","year":"2020","journal-title":"Appl. Thermal Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1016\/j.tsep.2018.09.004","article-title":"A review on Vuilleumier machines","volume":"8","author":"Dogkas","year":"2018","journal-title":"Therm. Sci. Eng. Prog."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/0140-7007(94)P3711-9","article-title":"Experimental investigation of a free-piston Vuilleumier refrigemtor","volume":"18","author":"Schulz","year":"1995","journal-title":"Int. J. Refrig."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1299\/kikaib.61.713","article-title":"Small Vuilleumier Cryocooler: Comparison of Performance Test Results and Calculation","volume":"61","author":"Kawada","year":"1995","journal-title":"Trans. Jpn. Soc. Mech. Eng. B"},{"key":"ref_8","unstructured":"Matsubara, Y., and Kaneko, M. (1984, January 17\u201318). Vuilleumier cycle cryocooler operating below 8 K. Proceedings of the Third Cryocooler Conference, Boulder, CO, USA."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"012077","DOI":"10.1088\/1757-899X\/171\/1\/012077","article-title":"Experimental study on a one-stage Vuilleumier cryocooler with large pressure ratio","volume":"171","author":"Tong","year":"2017","journal-title":"IOP Conf. Ser. Mater. Sci. Eng."},{"key":"ref_10","unstructured":"Miller, W.S., and Potter, V.L. (1973). Fractional Watt Vuilleumier Cryogenic Refrigerator, Technical Report."},{"key":"ref_11","unstructured":"Russo, S.C. (1976). Study of a Vuilleumier Cycle Cryogenic Refrigerator for Detector Cooling on the Limb Scanning Infrared Radiometer, Technical Report."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"White, R. (1976). Vuilleumier Cycle Cryogenic Refrigeration, Air Force Flight Dynamics Dynamics Laboratory, Wright-Patterson Air Force Base. Technical Report.","DOI":"10.21236\/ADA027055"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1063\/1.329894","article-title":"Optimal Paths for Thermodynamic Systems: The ideal Otto Cycle","volume":"53","author":"Mozurkewich","year":"1982","journal-title":"J. Appl. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.3151964","article-title":"Maximum power configuration for multireservoir chemical engines","volume":"105","author":"Xia","year":"2009","journal-title":"J. Appl. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2125","DOI":"10.1063\/1.335977","article-title":"Optimal Paths for Thermodynamic Systems: The Ideal Diesel Cycle","volume":"58","author":"Hoffmann","year":"1985","journal-title":"J. Appl. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Stanislaw Sieniutycz, S., and de Vos, A. (2000). Optimal Piston Paths for Diesel Engines. Thermodynamics of Energy Conversion and Transport, Springer.","DOI":"10.1007\/978-1-4612-1286-7"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1016\/j.mcm.2011.05.014","article-title":"Optimizing piston velocity profile for maximum work output from a generalized radiative law Diesel engine","volume":"54","author":"Chen","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_18","first-page":"31","article-title":"Theoretical Evaluation of the Maximum Work of Free-Piston Engine Generators","volume":"42","author":"Kojima","year":"2017","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1648","DOI":"10.1016\/j.egyr.2020.06.012","article-title":"Power density analysis and multi-objective optimization for a modified endoreversible simple closed Brayton cycle with one isothermal heating process","volume":"6","author":"Tang","year":"2020","journal-title":"Energy Rep."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1515\/jnet-2014-0003","article-title":"Optimal motion trajectory for the four-stroke free-piston engine with irreversible Miller cycle via a Gauss pseudospectral method","volume":"39","author":"Lin","year":"2014","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1515\/jnet-2016-0031","article-title":"Maximum Work of Free-Piston Stirling Engine Generators","volume":"42","author":"Kojima","year":"2017","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"071002","DOI":"10.1115\/1.4029682","article-title":"Optimal Periodic Control of an Ideal Stirling Engine Model","volume":"137","author":"Craun","year":"2015","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_23","unstructured":"Craun, M.J. (2015). Modeling and Control of an Actuated Stirling Engine. [Ph.D. Thesis, University of California]. Available online: https:\/\/escholarship.org\/uc\/item\/2tk2v9kj."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Masser, R., Khodja, A., Scheunert, M., Schwalbe, K., Fischer, A., Paul, R., and Hoffmann, K.H. (2020). Optimized Piston Motion for an Alpha-Type Stirling Engine. Entropy, 22.","DOI":"10.3390\/e22060700"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Scheunert, M., Masser, R., Khodja, A., Paul, R., Schwalbe, K., Fischer, A., and Hoffmann, K.H. (2020). Power-Optimized Sinusoidal Piston Motion and Its Performance Gain for an Alpha-Type Stirling Engine with Limited Regeneration. Energies, 13.","DOI":"10.3390\/en13174564"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Paul, R., and Hoffmann, K.H. (2021). Cyclic Control Optimization Algorithm for Stirling Engines. Symmetry, 13.","DOI":"10.3390\/sym13050873"},{"key":"ref_27","unstructured":"Paul, R.R. (2020). Optimal Control of Stirling Engines. [Ph.D. Thesis, Technische Universit\u00e4t Chemnitz]."},{"key":"ref_28","unstructured":"Hofbauer, P. (2018). Four-Process Cycle for a Vuilleumier Heat Pump. (10,030,893), US Patent."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/j.applthermaleng.2018.05.028","article-title":"Performance analysis of a free-piston Vuilleumier heat pump with dwell-based motion","volume":"140","author":"Chen","year":"2018","journal-title":"Appl. Thermal Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1016\/j.apenergy.2019.03.077","article-title":"Dynamic modeling and parameter optimization of a free-piston Vuilleumier heat pump with dwell-based motion","volume":"242","author":"Chen","year":"2019","journal-title":"Appl. Energy"},{"key":"ref_31","first-page":"311","article-title":"Endoreversible Thermodynamics","volume":"22","author":"Hoffmann","year":"1997","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1515\/JNETDY.2003.015","article-title":"Optimal Process Paths for Endoreversible Systems","volume":"28","author":"Hoffmann","year":"2003","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_33","first-page":"1","article-title":"An introduction to endoreversible thermodynamics","volume":"86","author":"Hoffmann","year":"2008","journal-title":"AAPP\u2013Phys. Math. Nat. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1103\/PhysRevA.19.1272","article-title":"Optimal Configuration of a Class of Irreversible Heat Engines. I","volume":"19","author":"Rubin","year":"1979","journal-title":"Phys. Rev. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1277","DOI":"10.1103\/PhysRevA.19.1277","article-title":"Optimal Configuration of a Class of Irreversible Heat Engines. II","volume":"19","author":"Rubin","year":"1979","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1088\/0022-3727\/20\/2\/014","article-title":"Reflections on the power delivered by endoreversible engines","volume":"20","year":"1987","journal-title":"J. Phys. D Appl. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.342570","article-title":"An Optimal Endoreversible Three-Heat-Source Refrigerator","volume":"65","author":"Yan","year":"1988","journal-title":"J. Appl. Phys."},{"key":"ref_38","unstructured":"De Vos, A. (1992). Endoreversible Thermodynamics of Solar Energy Conversion, Oxford University Press."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2216","DOI":"10.1063\/1.354728","article-title":"Endoreversible thermal cycle with a nonlinear heat transfer law","volume":"74","year":"1993","journal-title":"J. Appl. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/0360-5442(89)90068-6","article-title":"The Theoretical Maximum Efficiency of Solar Converters with and Without Concentration","volume":"14","year":"1989","journal-title":"Energy"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/0379-6787(91)90021-G","article-title":"Is a solar cell an endoreversible engine?","volume":"31","year":"1991","journal-title":"Sol. Cells"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1515\/jnet-2018-0021","article-title":"Optimal Control of an Endoreversible Solar Power Plant","volume":"43","author":"Schwalbe","year":"2018","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.applthermaleng.2018.07.045","article-title":"Stochastic Novikov engine with time dependent temperature fluctuations","volume":"142","author":"Schwalbe","year":"2018","journal-title":"Appl. Thermal Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2893","DOI":"10.1063\/1.336281","article-title":"Intrinsically Irreversible Light-Driven Engine","volume":"58","author":"Watowich","year":"1985","journal-title":"J. Appl. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/BF02906311","article-title":"Optimal Paths for a Bimolecular, Light-Driven Engine","volume":"104","author":"Watowich","year":"1989","journal-title":"Il Nuovo Cim. B"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1007\/s11426-009-0172-2","article-title":"Optimal paths for a light-driven engine with a linear phenomenological heat transfer law","volume":"53","author":"Ma","year":"2010","journal-title":"Sci. China Chem."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1515\/jnet-2015-0061","article-title":"Endoreversible modeling of a PEM fuel cell","volume":"40","author":"Wagner","year":"2015","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"5303","DOI":"10.1063\/1.354253","article-title":"Performance Characteristics of Endoreversible Chemical Engines","volume":"74","author":"Gordon","year":"1993","journal-title":"J. Appl. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"015101","DOI":"10.1088\/0143-0807\/37\/1\/015101","article-title":"Chemical reactions in endoreversible thermodynamics","volume":"37","author":"Wagner","year":"2016","journal-title":"Eur. J. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1515\/jnet-2018-0076","article-title":"On the Efficiency of Electrochemical Devices from the Perspective of Endoreversible Thermodynamics","volume":"44","author":"Marsik","year":"2019","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1515\/JNETDY.2006.013","article-title":"Endoreversible Thermodynamics: A Tool for Simulating and Comparing Processes of Discrete Systems","volume":"31","author":"Muschik","year":"2006","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Gonzalez-Ayala, J., Mateos Roco, J.M., Medina, A., and Calvo Hern\u00e1ndez, A. (2020). Optimization, Stability, and Entropy in Endoreversible Heat Engines. Entropy, 22.","DOI":"10.3390\/e22111323"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Muschik, W., and Hoffmann, K.H. (2020). Modeling, Simulation, and Reconstruction of 2-Reservoir Heat-to-Power Processes in Finite-Time Thermodynamics. Entropy, 22.","DOI":"10.3390\/e22090997"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1515\/jnet-2020-0039","article-title":"Endoreversible Otto Engines at Maximal Power","volume":"45","author":"Smith","year":"2020","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1515\/jnet-2018-0003","article-title":"Novikov engine with fluctuating heat bath temperature","volume":"43","author":"Schwalbe","year":"2018","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Schwalbe, K., and Hoffmann, K.H. (2018). Performance Features of a Stationary Stochastic Novikov Engine. Entropy, 20.","DOI":"10.3390\/e20010052"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1515\/jnet-2019-0063","article-title":"Stochastic Novikov Engine with Fourier Heat Transport","volume":"44","author":"Schwalbe","year":"2019","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1515\/jnetdy-2013-0005","article-title":"The principal equation of state for classical particles, photons, and neutrinos","volume":"38","author":"Essex","year":"2013","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1515\/JNETDY.2004.002","article-title":"Can a quantitative simulation of an Otto engine be accurately rendered by a simple Novikov model with heat leak?","volume":"29","author":"Fischer","year":"2004","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"184","DOI":"10.5541\/ijot.877687","article-title":"An Endoreversible Model for the Regenerators of Vuilleumier Refrigerators","volume":"24","author":"Paul","year":"2021","journal-title":"Int. J. Thermodyn."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1093\/comjnl\/7.4.308","article-title":"A Simplex Method for Function Minimization","volume":"7","author":"Nelder","year":"1965","journal-title":"Comput. J."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/12\/1562\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:34:58Z","timestamp":1760168098000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/12\/1562"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,24]]},"references-count":61,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["e23121562"],"URL":"https:\/\/doi.org\/10.3390\/e23121562","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,24]]}}}