{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T02:16:46Z","timestamp":1768789006806,"version":"3.49.0"},"reference-count":55,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2018,6,16]],"date-time":"2018-06-16T00:00:00Z","timestamp":1529107200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Major research instrumentation program of the Deutsche Forschungsgemeinschaft (DFG)","award":["INST 187\/630-1 FUGG"],"award-info":[{"award-number":["INST 187\/630-1 FUGG"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The calculation of the entropy production rate within an operational high temperature solid oxide fuel cell (SOFC) is necessary to design and improve heating and cooling strategies. However, due to a lack of information, most of the studies are limited to empirical relations, which are not in line with the more general approach given by non-equilibrium thermodynamics (NET). The SOFC 1D-model presented in this study is based on non-equilibrium thermodynamics and we parameterize it with experimental data and data from molecular dynamics (MD). The validation of the model shows that it can effectively describe the behavior of a SOFC at 1300\u00a0K. Moreover, we show that the highest entropy production is present in the electrolyte and the catalyst layers, and that the Peltier heat transfer is considerable for the calculation of the heat flux in the electrolyte and cannot be neglected. To our knowledge, this is the first validated model of a SOFC based on non-equilibrium thermodynamics and this study can be extended to analyze SOFCs with other solid oxide electrolytes, with perovskites electrolytes or even other electrochemical systems like solid oxide electrolysis cells (SOECs).<\/jats:p>","DOI":"10.3390\/e20060469","type":"journal-article","created":{"date-parts":[[2018,6,18]],"date-time":"2018-06-18T10:57:11Z","timestamp":1529319431000},"page":"469","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Impact of Multi-Causal Transport Mechanisms in an Electrolyte Supported Planar SOFC with (ZrO2)x\u22121(Y2O3)x Electrolyte"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4858-5314","authenticated-orcid":false,"given":"Gerardo","family":"Valadez Huerta","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Thermodynamik, Gottfried Wilhelm Leibniz Universit\u00e4t Hannover, Callinstra\u00dfe 36, D-30167 Hannover, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vincent","family":"Flasbart","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Thermodynamik, Gottfried Wilhelm Leibniz Universit\u00e4t Hannover, Callinstra\u00dfe 36, D-30167 Hannover, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tobias","family":"Marquardt","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Thermodynamik, Gottfried Wilhelm Leibniz Universit\u00e4t Hannover, Callinstra\u00dfe 36, D-30167 Hannover, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7616-1148","authenticated-orcid":false,"given":"Pablo","family":"Radici","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Thermodynamik, Gottfried Wilhelm Leibniz Universit\u00e4t Hannover, Callinstra\u00dfe 36, D-30167 Hannover, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stephan","family":"Kabelac","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Thermodynamik, Gottfried Wilhelm Leibniz Universit\u00e4t Hannover, Callinstra\u00dfe 36, D-30167 Hannover, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.rser.2017.01.148","article-title":"Recent advances and challenges of fuel cell based power system architectures and control\u2014A review","volume":"73","author":"Das","year":"2017","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/S1364-0321(02)00014-X","article-title":"Solid oxide fuel cells (SOFCs): A review of an environmentally clean and efficient source of energy","volume":"6","author":"Traversa","year":"2002","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/j.energy.2017.05.044","article-title":"Solid oxide fuel cells powered by biomass gasification for high efficiency power generation","volume":"131","author":"Thomsen","year":"2017","journal-title":"Energy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1093\/nsr\/nww099","article-title":"Prospects of fuel cell technologies","volume":"4","author":"Wang","year":"2017","journal-title":"Natl. Sci. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1149\/06801.0039ecst","article-title":"The Promise and Challenges of Intermediate Temperature Fuel Cells","volume":"68","author":"Litzelman","year":"2015","journal-title":"ECS Trans."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1016\/j.rser.2017.09.071","article-title":"Advances in reforming and partial oxidation of hydrocarbons for hydrogen production and fuel cell applications","volume":"82","author":"Sengodan","year":"2018","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Valadez Huerta, G., \u00c1lvarez Jord\u00e1n, J., Dragon, M., Leites, K., and Kabelac, S. (2018). Exergy analysis of the diesel pre-reforming solid oxide fuel cell system with anode off-gas recycling in the SchIBZ project. Part I: Modeling and validation. Int. J. Hydrogen Energy.","DOI":"10.1016\/j.ijhydene.2018.04.216"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1149\/07801.0171ecst","article-title":"Diesel Based SOFC Demonstrator for Maritime Applications","volume":"78","author":"Nehter","year":"2017","journal-title":"ECS Trans."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2745","DOI":"10.1016\/j.ceramint.2017.11.009","article-title":"Structure, densification and electrical properties of Gd3+ and Cu2+ co-doped ceria solid electrolytes for SOFC applications: Effects of Gd2O3","volume":"44","author":"Santos","year":"2018","journal-title":"Ceram. Int."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Irshad, M., Siraj, K., Raza, R., Ali, A., Tiwari, P., Zhu, B., Rafique, A., Ali, A., Ullah, M.K., and Usman, A. (2016). A Brief Description of High Temperature Solid Oxide Fuel Cell\u2019s Operation, Materials, Design, Fabrication Technologies and Performance. Appl. Sci., 6.","DOI":"10.3390\/app6030075"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1038\/35104620","article-title":"Materials for fuel-cell technologies","volume":"414","author":"Steele","year":"2001","journal-title":"Nature"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2831","DOI":"10.1016\/j.renene.2009.04.010","article-title":"SOFC cogeneration system for building applications, part 1: Development of SOFC system-level model and the parametric study","volume":"34","author":"Lee","year":"2009","journal-title":"Renew. Energy"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1016\/j.ijhydene.2017.11.026","article-title":"Safe heating-up of a full scale SOFC system using 3D multiphysics modelling optimization","volume":"43","author":"Peksen","year":"2018","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1149\/07542.0015ecst","article-title":"Recent Developments in 3D Multiphysics Modelling of Whole Fuel Cell Systems for Assisting Commercialization and Improved Reliability","volume":"75","author":"Peksen","year":"2017","journal-title":"ECS Trans."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"235","DOI":"10.4186\/ej.2017.21.3.235","article-title":"Transient Modeling of Tubular-Designed IIR-SOFC Fueled by Methane, Methanol, and Ethanol","volume":"21","author":"Dokmaingam","year":"2017","journal-title":"Eng. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1016\/j.jpowsour.2016.11.070","article-title":"Dynamic model of a micro-tubular solid oxide fuel cell stack including an integrated cooling system","volume":"342","author":"Hering","year":"2017","journal-title":"J. Power Sources"},{"key":"ref_17","first-page":"728","article-title":"Numerical modeling of ceria-based SOFCs with bi-layer electrolyte free from internal short circuit: Comparison of two cell configurations","volume":"138","author":"Wang","year":"2017","journal-title":"Electrochim. Acta"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.jpowsour.2004.10.011","article-title":"Performance analysis of a tubular solid oxide fuel cell\/micro gas turbine hybrid power system based on a quasi-two dimensional model","volume":"142","author":"Song","year":"2005","journal-title":"J. Power Sources"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.ijhydene.2004.04.018","article-title":"SOFC mathematic model for systems simulations\u2014Part 2: Definition of an analytical model","volume":"30","author":"Bove","year":"2005","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/S0378-4371(97)00296-3","article-title":"Jumps in electric potential and in temperature at the electrode surfaces of the solid oxide fuel cell","volume":"244","author":"Kjelstrup","year":"1997","journal-title":"Physica A"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Kjelstrup, S., and Bedeaux, D. (2017). Non-Equilibrium Thermodynamics of Heterogeneous Systems, World Scientific.","DOI":"10.1142\/10286"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1103\/PhysRev.37.405","article-title":"Reciprocal Relations in Irreversible Processes. I","volume":"37","author":"Onsager","year":"1931","journal-title":"Phys. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1103\/PhysRev.38.2265","article-title":"Reciprocal Relations in Irreversible Processes. II","volume":"38","author":"Onsager","year":"1931","journal-title":"Phys. Rev."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.cej.2004.02.005","article-title":"Electrochemical model of the integrated planar solid oxide fuel cell (IP-SOFC)","volume":"102","author":"Costamagna","year":"2004","journal-title":"Chem. Eng. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1893","DOI":"10.1016\/j.rser.2010.12.011","article-title":"Mathematical modeling of solid oxide fuel cell: A review","volume":"15","author":"Hajimolana","year":"2011","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1016\/j.applthermaleng.2017.07.126","article-title":"Heat generation in lithium-ion batteries with different nominal capacities and chemistries","volume":"125","author":"Nazari","year":"2017","journal-title":"Appl. Therm. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/S0378-7753(02)00724-3","article-title":"Performance comparison of Fick\u2019s, dusty gas and Stefan-Maxwell models to predict the concentration overpotential of a SOFC anode","volume":"122","author":"Suwanwarangkul","year":"2003","journal-title":"J. Power Sources"},{"key":"ref_28","unstructured":"Taylor, R., and Krishna, R. (1993). Multicomponent Mass Transfer, John Wiley & Sons, Inc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1515\/jnet-2017-0054","article-title":"Ergodicity, Maximum Entropy Production, and Steepest Entropy Ascent in the Proofs of Onsager\u2019s Reciprocal Relations","volume":"43","author":"Benfenati","year":"2018","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1103\/PhysRevE.62.845","article-title":"Steady adiabatic state: Its thermodynamic, entropy production, energy dissipation, and violation of Onsager relations","volume":"62","author":"Alahverdyan","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1016\/j.energy.2009.03.007","article-title":"Entropy generation analysis in a monolithic-type solid oxide fuel cell (SOFC)","volume":"34","author":"Sciacovelli","year":"2009","journal-title":"Energy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"012601","DOI":"10.1115\/1.4001063","article-title":"Entropy Generation Minimization in a Tubular Solid Oxide Fuel Cell","volume":"132","author":"Sciacovelli","year":"2010","journal-title":"J. Energy Resour. Technol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1044","DOI":"10.1016\/j.ijheatmasstransfer.2017.12.111","article-title":"Energy and entropy study of a SOFC using biogas from different sources considering internal reforming of methane","volume":"120","year":"2018","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"577","DOI":"10.2298\/TSCI151221127R","article-title":"Entropy Generation Analysis of a Solid Oxide Fuel Cell by Computational Fluid Dynamics. Influence of Electrochemical Model and Its Parameters","volume":"22","year":"2018","journal-title":"Therm. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1016\/j.energy.2017.06.064","article-title":"Thermodynamic assessment of SOFC-ICGT hybrid cycle: Energy analysis and entropy generation minimization","volume":"134","author":"Choudhary","year":"2017","journal-title":"Energy"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"9968","DOI":"10.1016\/j.ijhydene.2016.02.089","article-title":"A review on micro-level modeling of solid oxide fuel cells","volume":"41","author":"Timurkutluk","year":"2016","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.pecs.2014.12.001","article-title":"Numerical thermomechanical modelling of solid oxide fuel cells","volume":"48","author":"Peksen","year":"2015","journal-title":"Prog. Energy Combust. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/jnet-2016-0025","article-title":"Local Entropy Production Rates in a Polymer Electrolyte Membrane Fuel Cell","volume":"42","author":"Siemer","year":"2017","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2374","DOI":"10.1149\/1.2085979","article-title":"The Transported Entropy of Oxygen Ion in Yttria-Stabilized Zirconia","volume":"138","author":"Kjelstrup","year":"1991","journal-title":"J. Electrochem. Soc."},{"key":"ref_40","first-page":"108","article-title":"A phenomenological study of yttria stabilized zirconia at 1300 K with the Green-Kubo formulation and equilibrium molecular dynamics","volume":"485","author":"Kelle","year":"2017","journal-title":"Chem. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"F790","DOI":"10.1149\/2.0641707jes","article-title":"Revisiting the Temperature Dependent Ionic Conductivity of Yttria Stabilized Zirconia (YSZ)","volume":"164","author":"Ahamer","year":"2017","journal-title":"J. Electrochem. Soc."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.jpowsour.2013.01.140","article-title":"Sintering, phase composition and ionic conductivity of zirconia-scandia-ceria","volume":"233","author":"Grosso","year":"2013","journal-title":"J. Power Sources"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"10247","DOI":"10.1021\/ie301009v","article-title":"Fick Diffusion Coefficients in Ternary Liquid Systems from Equilibrium Molecular Dynamics Simulations","volume":"51","author":"Liu","year":"2012","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Gao, C.Y., and Limmer, D.T. (2017). Transport Coefficients from Large Deviation Functions. Entropy, 19.","DOI":"10.3390\/e19110571"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1002\/ctpp.201700102","article-title":"On the Kubo-Greenwood model for electron conductivity","volume":"58","author":"Dufty","year":"2018","journal-title":"Contrib. Plasma Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1021\/acs.jpcc.7b08710","article-title":"Approach to the Coulomb Contribution of Thermodynamic Properties from the Mean Electrostatic Potential of the Ions in (ZrO2)1-x(Y2O3)x","volume":"122","author":"Siemen","year":"2018","journal-title":"J. Phys. Chem. C"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/s10010-005-0016-y","article-title":"Thermodynamische Stoffdaten f\u00fcr Biogase","volume":"70","author":"Kabelac","year":"2005","journal-title":"Forschung im Ingenieurwesen"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1955","DOI":"10.1021\/acs.jced.7b01094","article-title":"A Diffusivity Study of (Sc2O3)0.1(CeO2)0.01(ZrO2)0.89 between 1100 and 1500 K at Zero Pressure with Molecular Dynamics","volume":"63","author":"Reus","year":"2018","journal-title":"J. Chem. Eng. Data"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.1149\/1.3205700","article-title":"Bulk and Grain Boundary Conductivities as Function of Temperature and Oxygen Partial Pressure of Scandia-Stabilized Zirconia Co-Doped with Yttria and Ceria","volume":"25","author":"Preis","year":"2009","journal-title":"ECS Trans."},{"key":"ref_50","unstructured":"Atkins, P., and de Paula, J. (2006). Physical Chemistry, Oxford University Press."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1149\/1.3570081","article-title":"Exchange Current Density of Solid Oxide Fuel Cell Electrode","volume":"35","author":"Yonekura","year":"2011","journal-title":"ECS Trans."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1016\/j.ssi.2004.09.059","article-title":"Modeling of cation diffusion in oxygen ion conductors using molecular dynamics","volume":"175","author":"Kilo","year":"2004","journal-title":"Solid State Ion."},{"key":"ref_53","first-page":"1","article-title":"The MATLAB ODE Suite. SIAM","volume":"18","author":"Shampine","year":"1997","journal-title":"J. Sci. Comput."},{"key":"ref_54","first-page":"25","article-title":"Validation of Solid Oxide Fuel Cell Thermodynamic Models for System-level Integration","volume":"11","author":"Eveloy","year":"2016","journal-title":"Int. J. Therm. Environ. Eng."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"011017","DOI":"10.1115\/1.2971132","article-title":"Impact of the Temperature Profile on Thermal Stress in a Tubular Solid Oxide Fuel Cell","volume":"6","author":"Fischer","year":"2009","journal-title":"J. Fuel Cell Sci. Technol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/469\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:09:02Z","timestamp":1760195342000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/469"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,16]]},"references-count":55,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2018,6]]}},"alternative-id":["e20060469"],"URL":"https:\/\/doi.org\/10.3390\/e20060469","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,16]]}}}