{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:47:48Z","timestamp":1760237268876,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,3,26]],"date-time":"2020-03-26T00:00:00Z","timestamp":1585180800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006360","name":"Bundesministerium f\u00fcr Wirtschaft und Energie","doi-asserted-by":"publisher","award":["19U15009C"],"award-info":[{"award-number":["19U15009C"]}],"id":[{"id":"10.13039\/501100006360","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Hybrid drive systems able to recover and reuse braking energy of the vehicle can reduce fuel consumption, air pollution and operating costs. Among them, hydraulic recuperation systems are particularly suitable for commercial vehicles, especially if they are already equipped with a hydraulic system. Thus far, the investigation of such systems has been limited to individual components or optimizing their control. In this paper, we focus on thermodynamic effects and their impact on the overall systems energy saving potential using endoreversible thermodynamics as the ideal framework for modeling. The dynamical behavior of the hydraulic recuperation system as well as energy savings are estimated using real data of a vehicle suitable for application. Here, energy savings accelerating the vehicle around 10% and a reduction in energy transferred to the conventional disc brakes around 58% are predicted. We further vary certain design and loss parameters\u2014such as accumulator volume, displacement of the hydraulic unit, heat transfer coefficients or pipe diameter\u2014and discuss their influence on the energy saving potential of the system. It turns out that heat transfer coefficients and pipe diameter are of less importance than accumulator volume and displacement of the hydraulic unit.<\/jats:p>","DOI":"10.3390\/e22040383","type":"journal-article","created":{"date-parts":[[2020,3,27]],"date-time":"2020-03-27T09:04:38Z","timestamp":1585299878000},"page":"383","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Endoreversible Modeling of a Hydraulic Recuperation System"],"prefix":"10.3390","volume":"22","author":[{"given":"Robin","family":"Masser","sequence":"first","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":[[2020,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1109\/MCS.2007.338280","article-title":"Control of hybrid electric vehicles","volume":"27","author":"Sciarretta","year":"2007","journal-title":"IEEE Contr. Syst. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1007\/s12239-008-0089-3","article-title":"Combined control of a regenerative braking and anti-lock braking system for hybrid electric vehicles","volume":"9","author":"Peng","year":"2008","journal-title":"Int. J. Automot. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2309","DOI":"10.1109\/TPEL.2011.2172465","article-title":"Modeling and Control of a New Three-Input DC-DC Boost Converter for Hybrid PV\/FC\/Battery Power System","volume":"27","author":"Nejabatkhah","year":"2012","journal-title":"IEEE T. Power Electr."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1109\/TIE.2006.870880","article-title":"Energy-Management System for a Hybrid Electric Vehicle, Using Ultracapacitors and Neural Networks","volume":"53","author":"Moreno","year":"2006","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s12239-009-0027-z","article-title":"Analysis of a regenerative breaking system for hybrid electric vehicles using an electro-mechanical brake","volume":"10","author":"Ahn","year":"2009","journal-title":"Int. J. Automot. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1115\/1.2896497","article-title":"Modeling of a Hydraulic Energy Regeneration System: Part I\u2014 Analytical Treatment","volume":"114","author":"Pourmovahed","year":"1992","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_7","unstructured":"Shan, M. (2009). Modeling and Control Strategy for Series Hydraulic Hybrid Vehicles. [Ph.D. Thesis, University of Toledo]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.jterra.2009.08.004","article-title":"Torque control strategy for a parallel hydraulic hybrid vehicle","volume":"46","author":"Hui","year":"2009","journal-title":"J. Terramech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.autcon.2009.10.006","article-title":"Research on the system configuration and energy control strategy for parallel hydraulic hybrid loader","volume":"19","author":"Hui","year":"2010","journal-title":"Automat. Constr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s12239-013-0012-4","article-title":"Dynamic analysis of energy storage unit of the hydraulic hybrid vehicle","volume":"14","author":"Chen","year":"2013","journal-title":"Int. J. Automot. Technol."},{"key":"ref_11","unstructured":"Rupprecht, K.R. (1988). Hydrospeicher, Experimentelle und Analytische Untersuchungen zur Energiespeicherung. [Ph.D. Thesis, Technische Hochschule Aachen]."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"185","DOI":"10.2514\/3.23047","article-title":"Experimental Evaluation of Hydraulic Accumulator Efficiency With and Without Elastomeric Foam","volume":"4","author":"Pourmovahed","year":"1988","journal-title":"J. Propulsion"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1115\/1.2894128","article-title":"An Experimental Thermal Time-Constant Correlation for Hydraulic Accumulators","volume":"112","author":"Pourmovahed","year":"1988","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Jou, D., Casas-V\u00e1zquez, J., and Lebon, G. (2010). Extended Irreversible Thermodynamics, Springer. [4th ed.].","DOI":"10.1007\/978-90-481-3074-0_2"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1515\/jnetdy-2013-0029","article-title":"Heat conduction at micro and nanoscales: A review through the prism of Extended Irreversible Thermodynamics","volume":"39","author":"Lebon","year":"2014","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1515\/jnet-2016-0035","article-title":"Extended Reversible and Irreversible Thermodynamics: A Hamiltonian Approach with Application to Heat Waves","volume":"42","author":"Lebon","year":"2017","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Truesdell, C., and Bharatha, S. (1977). The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Springer Science & Business Media.","DOI":"10.1007\/978-3-642-81077-0"},{"key":"ref_18","unstructured":"Matolcsi, T. (2017). Ordinary Thermodynamics, Society for the Unity of Science and Technology."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/jnet-2017-0005","article-title":"Gradient Dynamics and Entropy Production Maximization","volume":"43","author":"Pavelka","year":"2018","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_20","first-page":"311","article-title":"Endoreversible Thermodynamics","volume":"22","author":"Hoffmann","year":"1997","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2115","DOI":"10.1103\/PhysRevA.21.2115","article-title":"Minimum Entropy Production and the Optimization of Heat Engines","volume":"21","author":"Salamon","year":"1980","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.331592","article-title":"Optimal Staging of Endoreversible Heat Engines","volume":"53","author":"Rubin","year":"1982","journal-title":"J. Appl. Phys."},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1119\/1.16175","article-title":"Observations on Efficiency of Heat Engines Operating at Maximum Power","volume":"58","author":"Gordon","year":"1990","journal-title":"Am. J. Phys."},{"key":"ref_25","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_26","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_27","doi-asserted-by":"crossref","first-page":"114914","DOI":"10.1063\/1.2401313","article-title":"Optimal piston trajectories for adiabatic processes in the presence of friction","volume":"100","author":"Huleihil","year":"2006","journal-title":"J. Appl. Phys."},{"key":"ref_28","first-page":"1","article-title":"Maximum Power, Ecological Function and Efficiency of an Irreversible Carnot Cycle. A Cost and Effectiveness Optimization","volume":"38","year":"2008","journal-title":"Braz. J. Phys."},{"key":"ref_29","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_30","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1515\/jnet-2017-0051","article-title":"General Properties for an Agrowal Thermal Engine","volume":"43","year":"2018","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Sieniutycz, S., and De Vos, A. (2000). Distillation by Thermodynamic Geometry. Thermodynamics of Energy Conversion an Transport, Springer. chapter 12.","DOI":"10.1007\/978-1-4612-1286-7"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1119\/1.16818","article-title":"Generalized Power Versus Efficiency Characteristics of Heat Engines: The Thermoelectric Generator as an Instructive Illustration","volume":"59","author":"Gordon","year":"1991","journal-title":"Am. J. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/0196-8904(94)00035-X","article-title":"Maximum Obtainable Specific Cooling Load of a Refrigerator","volume":"36","author":"Wu","year":"1995","journal-title":"Energy Convers. Manag."},{"key":"ref_34","first-page":"22","article-title":"The Maximum Coefficient of Performance of Thermoelectric Heat Pumps","volume":"17","author":"Chen","year":"1996","journal-title":"Int. J. Power Energy Syst."},{"key":"ref_35","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_36","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_37","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/0379-6787(91)90021-G","article-title":"Is a solar cell an edoreversible engine?","volume":"31","year":"1991","journal-title":"Sol. Cells"},{"key":"ref_38","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_39","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_40","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_41","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_42","unstructured":"Schwalbe, K., Fischer, A., Hoffmann, K.H., and Mehnert, J. (2014, January 15\u201319). Applied endoreversible thermodynamics: Optimization of powertrains. Proceedings of the ECOS 2014\u201427th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, Turku, Finland."},{"key":"ref_43","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_44","doi-asserted-by":"crossref","unstructured":"Masser, R., and Hoffmann, K.H. (2019). Dissipative Endoreversible Engine with Given Efficiency. Entropy, 21.","DOI":"10.3390\/e21111117"},{"key":"ref_45","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_46","unstructured":"Wagner, K. (2014). An Extension to Endoreversible Thermodynamics for Multi-Extensity Fluxes and Chemical Reaction Processes. [Ph.D. Thesis, Technische Universit\u00e4t Chemnitz]."},{"key":"ref_47","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_48","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1007\/s12206-010-0313-8","article-title":"Modeling and simulation of hydrostatic transmission system with energy regeneration using hydraulic accumulator","volume":"24","author":"Ho","year":"2010","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_49","unstructured":"Stephan, P., Kabelac, S., Kind, M., Mewes, D., Schaber, K., and Wetzel, T. (2013). VDI-W\u00e4rmeatlas, Springer."},{"key":"ref_50","unstructured":"Schweizer, A. (2019, February 18). Formelsammlung und Berechnungsprogramme Anlagenbau. Available online: https:\/\/www.schweizer-fn.de\/."},{"key":"ref_51","unstructured":"Addinol (2019, February 18). Sicherheitsdatenblatt ADDINOL Hydraulik\u00f6l HLP 46. Available online: http:\/\/addinol.oilfinder.net\/show_msds.php?id=64764&download=1."},{"key":"ref_52","unstructured":"Bosch Rexroth AG (2019, February 18). Drehzahlvariables Druck- und F\u00f6rderstrom-Regelsystem Sytronix DFEn 5000. Available online: http:\/\/static.mercateo.com\/c0\/986f0946fb5c4a09a9ec8ea992d97080\/pdf\/boschrexroth-rd62241.pdf?v=2."},{"key":"ref_53","unstructured":"Hydac (2019, February 18). Proportional-Druckbegrenzungsventil Schieberausf\u00fchrung, vorgesteuert Einschraubventil UNF - 350 bar. Available online: https:\/\/www.hydac.com\/fileadmin\/pdb\/pdf\/PRO0000000000000000000059913010011.pdf."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/383\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:12:05Z","timestamp":1760173925000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/383"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,26]]},"references-count":53,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["e22040383"],"URL":"https:\/\/doi.org\/10.3390\/e22040383","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,3,26]]}}}