{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T11:06:35Z","timestamp":1761217595786,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2012,12,17]],"date-time":"2012-12-17T00:00:00Z","timestamp":1355702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This work shows the power of the variational approach for studying the efficiency of thermal engines in the context of the Finite Time Thermodynamics (FTT). Using an endoreversible Curzon\u2013Ahlborn (CA) heat engine as a model for actual thermal engines, three different criteria for thermal efficiency were analyzed: maximum power output, ecological function, and maximum power density. By means of this procedure, the performance of the CA heat engine with a nonlinear heat transfer law (the Stefan\u2013Boltzmann law) was studied to describe the heat exchanges between the working substance and its thermal reservoirs. The specific case of the M\u00fcser engine for all the criteria was analyzed. The results confirmed some previous findings using other procedures and additionally new results for the M\u00fcser engine performance were obtained.<\/jats:p>","DOI":"10.3390\/e14122611","type":"journal-article","created":{"date-parts":[[2012,12,17]],"date-time":"2012-12-17T12:00:58Z","timestamp":1355745658000},"page":"2611-2625","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Finite-Time Thermal Cycle Variational Optimization with a Stefan\u2013Boltzmann Law for Three Different Criteria"],"prefix":"10.3390","volume":"14","author":[{"given":"Juan","family":"Chimal-Egu\u00eda","sequence":"first","affiliation":[{"name":"Centro de Investigaci\u00f3n en Computaci\u00f3n del IPN., Av. Juan de Dios B\u00e1tiz s\/n U.P. Zacatenco CP 07738, DF, Mexico"}]},{"given":"Norma","family":"S\u00e1nchez-Salas","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Escuela Superior de F\u00edsica y Matem\u00e1ticas del IPN, Edif. 9 U.P. Zacatenco, CP 07738, DF, Mexico"}]},{"given":"Marco","family":"Barranco-Jim\u00e9nez","sequence":"additional","affiliation":[{"name":"Departamento de Formaci\u00f3n B\u00e1sica, Escuela Superior de C\u00f3mputo del IPN, Av. Miguel Bernard s\/n Esq. Juan de Dios B\u00e1tiz, U.P. Zacatenco, DF 07738, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2012,12,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3898","DOI":"10.1063\/1.437127","article-title":"On the efficiency of rate processes. Power and efficiency of heat engines","volume":"69","author":"Procaccia","year":"1978","journal-title":"J. Chem. 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