{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T12:50:41Z","timestamp":1763643041930,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T00:00:00Z","timestamp":1605830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008431","name":"Consejer\u00eda de Educaci\u00f3n, Junta de Castilla y Le\u00f3n","doi-asserted-by":"publisher","award":["Project No. SA017P17"],"award-info":[{"award-number":["Project No. SA017P17"]}],"id":[{"id":"10.13039\/501100008431","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The stability of endoreversible heat engines has been extensively studied in the literature. In this paper, an alternative dynamic equations system was obtained by using restitution forces that bring the system back to the stationary state. The departing point is the assumption that the system has a stationary fixed point, along with a Taylor expansion in the first order of the input\/output heat fluxes, without further specifications regarding the properties of the working fluid or the heat device specifications. Specific cases of the Newton and the phenomenological heat transfer laws in a Carnot-like heat engine model were analyzed. It was shown that the evolution of the trajectories toward the stationary state have relevant consequences on the performance of the system. A major role was played by the symmetries\/asymmetries of the conductance ratio \u03c3hc of the heat transfer law associated with the input\/output heat exchanges. Accordingly, three main behaviors were observed: (1) For small \u03c3hc values, the thermodynamic trajectories evolved near the endoreversible limit, improving the efficiency and power output values with a decrease in entropy generation; (2) for large \u03c3hc values, the thermodynamic trajectories evolved either near the Pareto front or near the endoreversible limit, and in both cases, they improved the efficiency and power values with a decrease in entropy generation; (3) for the symmetric case (\u03c3hc=1), the trajectories evolved either with increasing entropy generation tending toward the Pareto front or with a decrease in entropy generation tending toward the endoreversible limit. Moreover, it was shown that the total entropy generation can define a time scale for both the operation cycle time and the relaxation characteristic time.<\/jats:p>","DOI":"10.3390\/e22111323","type":"journal-article","created":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T09:46:18Z","timestamp":1605865578000},"page":"1323","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Optimization, Stability, and Entropy in Endoreversible Heat Engines"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2896-7542","authenticated-orcid":false,"given":"Julian","family":"Gonzalez-Ayala","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica Fundamental y Matem\u00e1ticas, Universidad de Salamanca, 37008 Salamanca, Spain"},{"name":"Departamento de F\u00edsica Aplicada, Universidad de Salamanca, 37008 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8930-9201","authenticated-orcid":false,"given":"Jos\u00e9 Miguel","family":"Mateos Roco","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Fundamental y Matem\u00e1ticas, Universidad de Salamanca, 37008 Salamanca, Spain"},{"name":"Departamento de F\u00edsica Aplicada, Universidad de Salamanca, 37008 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9797-4909","authenticated-orcid":false,"given":"Alejandro","family":"Medina","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Fundamental y Matem\u00e1ticas, Universidad de Salamanca, 37008 Salamanca, Spain"},{"name":"Departamento de F\u00edsica Aplicada, Universidad de Salamanca, 37008 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5058-0395","authenticated-orcid":false,"given":"Antonio","family":"Calvo Hern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Fundamental y Matem\u00e1ticas, Universidad de Salamanca, 37008 Salamanca, Spain"},{"name":"Departamento de F\u00edsica Aplicada, Universidad de Salamanca, 37008 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"050603","DOI":"10.1103\/PhysRevLett.124.050603","article-title":"Energetic Self-Optimization Induced by Stability in Low-Dissipation Heat Engines","volume":"124","author":"Guo","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"14305","DOI":"10.1038\/s41598-020-71130-7","article-title":"Thermodynamic optimization subsumed in stability phenomena","volume":"10","author":"Medina","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"062128","DOI":"10.1103\/PhysRevE.100.062128","article-title":"Optimization induced by stability and the role of limited control near a steady state","volume":"100","author":"Guo","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"150603","DOI":"10.1103\/PhysRevLett.105.150603","article-title":"Efficiency at maximum power of low-dissipation Carnot engines","volume":"105","author":"Esposito","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1119\/1.10023","article-title":"Efficiency of a Carnot engine at maximum power output","volume":"43","author":"Curzon","year":"1975","journal-title":"Am. J. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Berry, R.S., Salamon, P., and Andresen, B. (2020). How It All Began. Entropy, 22.","DOI":"10.3390\/e22080908"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gonzalez-Ayala, J., Roco, J.M.M., Medina, A., and Hern\u00e1ndez, A.C. (2017). Carnot-like heat engines versus low-dissipation models. Entropy, 19.","DOI":"10.3390\/e19040182"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"012151","DOI":"10.1103\/PhysRevE.96.012151","article-title":"Heat engines at optimal power: Low-dissipation versus endoreversible model","volume":"96","author":"Johal","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"022139","DOI":"10.1103\/PhysRevE.97.022139","article-title":"Entropy generation and unified optimization of Carnot-like and low-dissipation refrigerators","volume":"97","author":"Medina","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2068","DOI":"10.1088\/0022-3727\/34\/13\/318","article-title":"Local stability analysis of an endoreversible Curzon-Ahborn-Novikov engine working in a maximum-power-like regime","volume":"34","author":"Maya","year":"2001","journal-title":"J. Phys. D"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1282","DOI":"10.1088\/0022-3727\/38\/8\/028","article-title":"The effect of heat transfer laws and thermal conductances on the local stability of an endoreversible heat engine","volume":"38","year":"2005","journal-title":"J. Phys. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s11080-007-9065-z","article-title":"Local Stability of an Endoreversible Heat Engine Working in an Ecological Regime","volume":"14","year":"2007","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.applthermaleng.2008.03.002","article-title":"Local stability characteristics of a non-endoreversible heat engine working in the optimum region","volume":"29","author":"Huang","year":"2009","journal-title":"Appl. Therm. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"22","DOI":"10.4236\/jmp.2013.47A2004","article-title":"Local Stability of Curzon-Ahlborn Cycle with Non-Linear Heat Transfer for Maximum Power Output Regime","volume":"4","year":"2013","journal-title":"J. Mod. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Reyes-Ram\u00edrez, I., Barranco-Jim\u00e9nez, M., Rojas-Pacheco, A., and Guzm\u00e1n-Vargas, L. (2014). Global Stability Analysis of a Curzon\u2013Ahlborn Heat Engine under Different Regimes of Performance. Entropy, 16.","DOI":"10.3390\/e16115796"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.physa.2013.12.044","article-title":"Global stability analysis of a Curzon\u2013Ahlborn heat engine using the Lyapunov method","volume":"399","year":"2014","journal-title":"Physica A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1016\/j.tsep.2018.10.010","article-title":"Local stability of a generalized irreversible heat engine with linear phenomenological heat transfer law working in an ecological regime","volume":"8","author":"Chen","year":"2018","journal-title":"Ther. Sci. Eng. Prog."},{"key":"ref_18","unstructured":"Valencia-Ortega, G., Levario-Medina, S., and Barranco-Jim\u00e9nez, M.A. (2020). Local and global stability analysis of a Curzon-Ahlborn model applied to power plants working at maximum \u03ba-efficient power. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"8019","DOI":"10.3390\/e17127860","article-title":"Local Stability Analysis for a Thermo-Economic Irreversible Heat Engine Model under Different Performance Regimes","volume":"17","year":"2015","journal-title":"Entropy"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"034905","DOI":"10.1063\/1.2767622","article-title":"Local stability analysis of a class of endoreversible heat pumps","volume":"102","author":"Huang","year":"2007","journal-title":"J. Appl. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.ijrefrig.2007.07.004","article-title":"The effect of cooling load and thermal conductance on the local stability of an endoreversible refrigeratorImpact de la charge thermique et de la conductivit\u00e9 thermique sur la stabilit\u00e9 locale d\u2019un r\u00e9frig\u00e9rateur endor\u00e9versible","volume":"31","author":"Huang","year":"2008","journal-title":"Int. J. Refrig."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1016\/j.applthermaleng.2007.06.002","article-title":"Local stability analysis of an irreversible heat engine working in the maximum power output and the maximum efficiency","volume":"28","author":"Nie","year":"2008","journal-title":"Appl. Therm. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1016\/j.ijthermalsci.2007.04.004","article-title":"Local stability analysis of an irreversible Carnot heat engine","volume":"47","author":"Nie","year":"2008","journal-title":"Int. J. Therm. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1444","DOI":"10.1016\/j.enconman.2009.03.011","article-title":"Local asymptotic stability of an irreversible heat pump subject to total thermal conductance constraint","volume":"50","author":"Huang","year":"2009","journal-title":"Energy Convers. Manag."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"025002","DOI":"10.1088\/0031-8949\/82\/02\/025002","article-title":"Local stability analysis of an endoreversible Carnot refrigerator","volume":"82","author":"He","year":"2010","journal-title":"Phys. Scr."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.ijrefrig.2017.01.006","article-title":"Local stability analysis of an irreversible refrigerator working at the maximum thermo-ecological functions: A comparison. Analyse de la stabilit\u00e9 locale d\u2019un r\u00e9frig\u00e9rateur irr\u00e9versible fonctionnant au maximum de ses fonctions thermo-\u00e9cologiques: Une comparaison","volume":"75","author":"Wouagfack","year":"2017","journal-title":"Int. J. Refrig."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"12638","DOI":"10.1038\/s41598-018-30847-2","article-title":"Dynamic robustness of endoreversible Carnot refrigerator working in the maximum performance per cycle time","volume":"8","author":"Nie","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4561","DOI":"10.1063\/1.367237","article-title":"A fallacious argument in the finite time thermodynamics concept of endoreversibility","volume":"83","author":"Sekulic","year":"1998","journal-title":"J. Appl. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6557","DOI":"10.1063\/1.1415752","article-title":"Comment on \u201cA fallacious argument in the finite time thermodynamic concept of endoreversibility\u201d [J. Appl. Phys. 83, 4561 (1998)]","volume":"90","author":"Andresen","year":"2001","journal-title":"J. Appl. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/S0196-8904(00)00055-8","article-title":"On the Curzon\u2013Ahlborn efficiency and its connection with the efficiencies of real heat engines","volume":"42","author":"Chen","year":"2001","journal-title":"Energy Convers. Manag."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/S0196-8904(01)00067-X","article-title":"On the Curzon\u2013Ahlborn efficiency and its lack of connection to power producing processes","volume":"43","author":"Gyftopoulos","year":"2002","journal-title":"Energy Convers. Manag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"022130","DOI":"10.1103\/PhysRevE.98.022130","article-title":"Ecological efficiency of finite-time thermodynamics: A molecular dynamics study","volume":"98","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"60003","DOI":"10.1209\/0295-5075\/83\/60003","article-title":"Molecular kinetic analysis of a finite-time Carnot cycle","volume":"83","author":"Izumida","year":"2008","journal-title":"EPL"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"012123","DOI":"10.1103\/PhysRevE.96.012123","article-title":"Molecular kinetic analysis of a local equilibrium Carnot cycle","volume":"96","author":"Izumida","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_35","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_36","first-page":"195","article-title":"Endoreversible engines with finite-time adiabats","volume":"1","author":"Agrawal","year":"1994","journal-title":"Indian J. Eng. Mater. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1140\/epjst\/e2015-02429-4","article-title":"The maximum power effciency 1\u2212\u03c4: Research, education, and bibliometric relevance","volume":"224","author":"Hernandez","year":"2015","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_38","unstructured":"Deb, K. (2001). Multi-Objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Inc."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Venables, W.N., and Ripley, B.D. (2002). Modern Applied Statistics with S, Springer.","DOI":"10.1007\/978-0-387-21706-2"},{"key":"ref_41","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_42","doi-asserted-by":"crossref","first-page":"037102","DOI":"10.1103\/PhysRevE.63.037102","article-title":"Unified optimization criterion for energy converters","volume":"63","author":"Medina","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_43","unstructured":"Strogatz, S.H. (2014). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/11\/1323\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:34:54Z","timestamp":1760178894000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/11\/1323"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,20]]},"references-count":43,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["e22111323"],"URL":"https:\/\/doi.org\/10.3390\/e22111323","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,11,20]]}}}