{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T00:12:47Z","timestamp":1780531967778,"version":"3.54.1"},"reference-count":51,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T00:00:00Z","timestamp":1602547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51779262"],"award-info":[{"award-number":["51779262"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Variation trends of dimensionless power density (PD) with a compression ratio and thermal efficiency (TE) are discussed according to the irreversible Atkinson cycle (AC) model established in previous literature. Then, for the fixed cycle temperature ratio, the maximum specific volume ratios, the maximum pressure ratios, and the TEs corresponding to the maximum power output (PO) and the maximum PD are compared. Finally, multi-objective optimization (MOO) of cycle performance with dimensionless PO, TE, dimensionless PD, and dimensionless ecological function (EF) as the optimization objectives and compression ratio as the optimization variable are performed by applying the non-dominated sorting genetic algorithm-II (NSGA-II). The results show that there is an optimal compression ratio which will maximize the dimensionless PD. The relation curve of the dimensionless PD and compression ratio is a parabolic-like one, and the dimensionless PD and TE is a loop-shaped one. The AC engine has smaller size and higher TE under the maximum PD condition than those of under the maximum PO condition. With the increase of TE, the dimensionless PO will decrease, the dimensionless PD will increase, and the dimensionless EF will first increase and then decrease. There is no positive ideal point in Pareto frontier. The optimal solutions by using three decision-making methods are compared. This paper analyzes the performance of the PD of the AC with three losses, and performs MOO of dimensionless PO, TE, dimensionless PD, and dimensionless EF. The new conclusions obtained have theoretical guideline value for the optimal design of actual Atkinson heat engine.<\/jats:p>","DOI":"10.3390\/e22101150","type":"journal-article","created":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T08:55:53Z","timestamp":1602579353000},"page":"1150","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":53,"title":["Four-Objective Optimization of Irreversible Atkinson Cycle Based on NSGA-II"],"prefix":"10.3390","volume":"22","author":[{"given":"Shuangshuang","family":"Shi","sequence":"first","affiliation":[{"name":"Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan 430205, China"},{"name":"School of Mechanical &amp; Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yanlin","family":"Ge","sequence":"additional","affiliation":[{"name":"Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan 430205, China"},{"name":"School of Mechanical &amp; Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9012-6736","authenticated-orcid":false,"given":"Lingen","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan 430205, China"},{"name":"School of Mechanical &amp; Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huijun","family":"Feng","sequence":"additional","affiliation":[{"name":"Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan 430205, China"},{"name":"School of Mechanical &amp; Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,13]]},"reference":[{"key":"ref_1","unstructured":"Andresen, B. (1983). Finite-Time Thermodynamics, University of Copenhagen. Physics Laboratory II."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1021\/ar00104a001","article-title":"Thermodynamics for processes in finite time","volume":"17","author":"Andresen","year":"1984","journal-title":"Acc. Chem. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1063\/1.362674","article-title":"Entropy generation minimization: The new thermodynamics of finite-size device and finite-time processes","volume":"79","author":"Bejan","year":"1996","journal-title":"J. Appl. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1515\/JNETDY.1999.020","article-title":"Finite time thermodynamic optimization or entropy generation minimization of energy systems","volume":"24","author":"Chen","year":"1999","journal-title":"J. Non Equilib. Thermodyn."},{"key":"ref_5","unstructured":"Chen, L.G. (2005). Finite-Time Thermodynamic Analysis of Irreversible Processes and Cycles, Higher Education Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1002\/anie.201001411","article-title":"Current trends in finite-time thermodynamics","volume":"50","author":"Andresen","year":"2011","journal-title":"Angew. Chem. Int. Ed."},{"key":"ref_7","unstructured":"Chen, L.G., and Li, J. (2020). Thermodynamic Optimization Theory for Two-Heat-Reservoir Cycles, Science Press."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Medina, A., Curto-Risso, P.L., Calvo-Hern\u00e1ndez, A., Guzm\u00e1n-Vargas, L., Angulo-Brown, F., and Sen, A.K. (2014). Quasi-Dimensional Simulation of Spark Ignition Engines. From Thermodynamic Optimization to Cyclic Variability, Springer.","DOI":"10.1007\/978-1-4471-5289-7"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ge, Y.L., Chen, L.G., and Sun, F.R. (2016). Progress in finite time thermodynamic studies for internal combustion engine cycles. Entropy, 18.","DOI":"10.3390\/e18040139"},{"key":"ref_10","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_11","first-page":"311","article-title":"Endoreversible thermodynamics","volume":"22","author":"Hoffmann","year":"1997","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Feidt, M. (2020). Carnot cycle and heat engine: Fundamentals and applications. Entropy, 22.","DOI":"10.3390\/e22030348"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Feidt, M., and Costea, M. (2020). Effect of machine entropy production on the optimal performance of a refrigerator. Entropy, 22.","DOI":"10.3390\/e22090913"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ma, Y.H. (2020). Effect of finite-size heat source\u2019s heat capacity on the efficiency of heat engine. Entropy, 22.","DOI":"10.3390\/e22091002"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1515\/JNETDY.2003.015","article-title":"Optimal process paths for endoreversible systems","volume":"28","author":"Hoffman","year":"2003","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_16","first-page":"981","article-title":"Progresses in generalized thermodynamic dynamic-optimization of irreversible processes","volume":"49","author":"Chen","year":"2019","journal-title":"Sci. China Technol. Sci."},{"key":"ref_17","first-page":"1223","article-title":"Progress in generalized thermodynamic dynamic-optimization of irreversible cycles","volume":"49","author":"Chen","year":"2019","journal-title":"Sci. China Technol. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Masser, R., and Hoffmann, K.H. (2020). Endoreversible modeling of a hydraulic recuperation system. Entropy, 22.","DOI":"10.3390\/e22040383"},{"key":"ref_19","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_20","doi-asserted-by":"crossref","unstructured":"Tsirlin, A., and Sukin, I. (2020). Averaged optimization and finite-time thermodynamics. Entropy, 22.","DOI":"10.3390\/e22090912"},{"key":"ref_21","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_22","doi-asserted-by":"crossref","unstructured":"Sch\u00f6n, J.C. (2020). Optimal control of hydrogen atom-like systems as thermodynamic engines in finite time. Entropy, 22.","DOI":"10.3390\/e22101066"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"1683","DOI":"10.1016\/j.enconman.2006.11.001","article-title":"Comparison of performances of air standard Atkinson and Otto cycles with heat transfer considerations","volume":"48","author":"Hou","year":"2007","journal-title":"Energy Convers. Manag."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/j.apenergy.2004.09.007","article-title":"Reciprocating heat-engine cycles","volume":"84","author":"Ge","year":"2005","journal-title":"Appl. Energy"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1179\/174602207X174540","article-title":"Performance of an endoreversible Atkinson cycle","volume":"80","author":"Ge","year":"2007","journal-title":"J. Energy Inst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1016\/j.apenergy.2005.12.003","article-title":"Performance of Atkinson cycle with heat transfer, friction and variable specific heats of working fluid","volume":"83","author":"Ge","year":"2006","journal-title":"Appl. Energy"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"887","DOI":"10.2298\/TSCI090128034G","article-title":"Finite time thermodynamic modeling and analysis for an irreversible Atkinson cycle","volume":"14","author":"Ge","year":"2010","journal-title":"Therm. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1306","DOI":"10.12693\/APhysPolA.132.1306","article-title":"Performance analysis of an Atkinson cycle engine under effective power and effective power density condition","volume":"132","author":"Gonca","year":"2017","journal-title":"Acta Phys. Pol. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"201","DOI":"10.12693\/APhysPolA.133.201","article-title":"Effect of volume ratio of heat rejection process on performance of an Atkinson cycle","volume":"133","author":"Ebrahimi","year":"2018","journal-title":"Acta Phys. Pol. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/S0196-8904(96)00195-1","article-title":"Efficiency of an Atkinson engine at maximum power density","volume":"39","author":"Chen","year":"1998","journal-title":"Energy Convers. Manag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1007\/s10765-009-0573-7","article-title":"A comparative performance analysis and optimization of irreversible Atkinson cycle under maximum power density and maximum power conditions","volume":"30","author":"Ust","year":"2009","journal-title":"Int. J. Thermophys."},{"key":"ref_33","first-page":"71","article-title":"Efficiency of Atkinson engine at maximum power density using temperature dependent specific heats","volume":"2","author":"Akash","year":"2008","journal-title":"Jordan J. Mech. Ind. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.enconman.2015.12.059","article-title":"Thermodynamic analysis and performance maps for the irreversible Dual-Atkinson cycle engine (DACE) with considerations of temperature-dependent specific heats, heat transfer and friction losses","volume":"111","author":"Gonca","year":"2016","journal-title":"Energy Convers. Manag."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.enconman.2015.01.040","article-title":"Thermodynamic and thermo-economic analysis and optimization of an irreversible regenerative closed Brayton cycle","volume":"94","author":"Sadatsakkak","year":"2015","journal-title":"Energy Convers. Manag."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1016\/j.ijthermalsci.2017.10.030","article-title":"Performance analysis and multi-objective optimization of a Stirling engine based on MOPSOCD","volume":"124","author":"Dai","year":"2018","journal-title":"Int. J. Therm. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1016\/j.energy.2017.12.028","article-title":"Multi-objective performance optimization of irreversible molten carbonate fuel cell\u2013Braysson heat engine and thermodynamic analysis with ecological objective approach","volume":"144","author":"Ahmadi","year":"2018","journal-title":"Energy"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1016\/j.enconman.2018.06.046","article-title":"Multi-objective optimization and decision making of endoreversible combined cycles with consideration of different heat exchangers by finite time thermodynamics","volume":"171","author":"Ghasemkhani","year":"2018","journal-title":"Energy Convers. Manag."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"103","DOI":"10.18201\/ijisae.2018642064","article-title":"Crow search based multi-objective optimization of irreversible air refrigerators","volume":"6","author":"Turgut","year":"2018","journal-title":"Int. J. Intell. Syst. Appl. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Abedinnezhad, S., Ahmadi, M.H., Pourkiaei, S.M., Pourfayaz, F., Mosavi, A., Feidt, M., and Shamshirband, S. (2019). Thermodynamic assessment and multi-objective optimization of performance of irreversible Dual-Miller cycle. Energies, 12.","DOI":"10.3390\/en12204000"},{"key":"ref_41","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_42","doi-asserted-by":"crossref","unstructured":"Chen, L.G., Tang, C.Q., Feng, H.J., and Ge, Y.L. (2020). Power, efficiency, power density and ecological function optimizations for an irreversible modified closed variable-temperature reservoir regenerative Brayton cycle with one isothermal heating process. Energies, 13.","DOI":"10.3390\/en13195133"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"119025","DOI":"10.1016\/j.ijheatmasstransfer.2019.119025","article-title":"Multi-objective optimization for helium-heated reverse water gas shift reactor by using NSGA-II","volume":"148","author":"Zhang","year":"2020","journal-title":"Int. J. Heat Mass Transfer."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Sun, M., Xia, S.J., Chen, L.G., Wang, C., and Tang, C.Q. (2020). Minimum entropy generation rate and maximum yield optimization of sulfuric acid decomposition process using NSGA-II. Entropy, 22.","DOI":"10.3390\/e22101065"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Wu, Z.X., Feng, H.J., Chen, L.G., and Ge, Y.L. (2020). Performance optimization of a condenser in ocean thermal energy conversion (OTEC) system based on constructal theory and multi-objective genetic algorithm. Entropy, 22.","DOI":"10.3390\/e22060641"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/j.enconman.2004.04.001","article-title":"Comparative performance analysis of irreversible Dual and Diesel cycles under maximum power conditions","volume":"46","author":"Parlak","year":"2005","journal-title":"Energy Convers. Manag."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1115\/1.2794770","article-title":"Optimum criteria on the important parameters of an irreversible Otto heat engine with the temperature-dependent heat capacities of the working fluid","volume":"129","author":"Zhao","year":"2007","journal-title":"ASME Trans. Energy Res. Technol."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1016\/S0196-8904(97)10003-6","article-title":"Heat transfer effects on the network output and efficiency characteristics for an air standard Otto cycle","volume":"39","author":"Chen","year":"1998","journal-title":"Energy Convers. Manag."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1080\/01430750.2003.9674923","article-title":"The power and efficiency characteristics for an irreversible Otto cycle","volume":"24","author":"Chen","year":"2003","journal-title":"Int. J. Ambient Energy"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"7465","DOI":"10.1063\/1.347562","article-title":"An ecological optimization criterion for finite-time heat engines","volume":"69","year":"1991","journal-title":"J. Appl. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/j.asoc.2016.05.001","article-title":"Soft computing based multi-objective optimization of Brayton cycle power plant with isothermal heat addition using evolutionary algorithm and decision making","volume":"46","author":"Arora","year":"2016","journal-title":"Appl. Soft Comput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/10\/1150\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:20:31Z","timestamp":1760178031000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/10\/1150"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,13]]},"references-count":51,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["e22101150"],"URL":"https:\/\/doi.org\/10.3390\/e22101150","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,13]]}}}