{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:41:53Z","timestamp":1760118113492,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,25]],"date-time":"2024-12-25T00:00:00Z","timestamp":1735084800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Banco de Santander","award":["PR108\/20-02"],"award-info":[{"award-number":["PR108\/20-02"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The entropy production in the polarization phenomena occurring in the underlimiting regime, when an electric current circulates through a single cation-exchange membrane system, has been investigated in the 3\u201340 \u00b0C temperature range. From the analysis of the current\u2013voltage curves and considering the electro-membrane system as a unidimensional heterogeneous system, the total entropy generation in the system has been estimated from the contribution of each part of the system. Classical polarization theory and the irreversible thermodynamics approach have been used to determine the total electric potential drop and the entropy generation, respectively, associated with the different transport mechanisms in each part of the system. The results show that part of the electric power input is dissipated as heat due to both electric migration and diffusion ion transports, while another part is converted into chemical energy stored in the saline concentration gradient. Considering the electro-membrane process as an energy conversion process, an efficiency has been defined as the ratio between stored power and electric power input. This efficiency increases as both applied electric current and temperature increase.<\/jats:p>","DOI":"10.3390\/e27010003","type":"journal-article","created":{"date-parts":[[2024,12,25]],"date-time":"2024-12-25T19:27:46Z","timestamp":1735154866000},"page":"3","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Entropy Production in an Electro-Membrane Process at Underlimiting Currents\u2014Influence of Temperature"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1864-8564","authenticated-orcid":false,"given":"Juan Carlos","family":"Maroto","sequence":"first","affiliation":[{"name":"Department of Electronics, Automation, and Communications, Comillas Pontifical University, 28049 Madrid, Spain"},{"name":"Department of Science and Aerospace, Universidad Europea de Madrid, 28670 Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4130-5543","authenticated-orcid":false,"given":"Sagrario","family":"Mu\u00f1oz","sequence":"additional","affiliation":[{"name":"Department of Structure of Matter, Thermal Physics and Electronics, Faculty of Physics, Complutense University of Madrid, 28040 Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0177-8071","authenticated-orcid":false,"given":"Vicenta Mar\u00eda","family":"Barrag\u00e1n","sequence":"additional","affiliation":[{"name":"Department of Structure of Matter, Thermal Physics and Electronics, Faculty of Physics, Complutense University of Madrid, 28040 Madrid, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.desal.2017.12.044","article-title":"Electrodialysis for water desalination: A critical assessment of recent developments on process fundamentals, model and applications","volume":"434","author":"Campione","year":"2018","journal-title":"Desalination"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2741","DOI":"10.1246\/bcsj.42.2741","article-title":"Concentration polarization phenomena in ion-exchange membrane electrodialysis. II. The effect of the condition of the diffusion-boundary layer on the limiting-current density and on the relative transport numbers of ions","volume":"42","author":"Yamane","year":"1969","journal-title":"Bull. Chem. Soc. Jpn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/0022-0728(92)80270-E","article-title":"Polarization phenomena at the interfaces between an electrolyte solution and an ion exchange membrane","volume":"336","author":"Tay","year":"1992","journal-title":"J. Electroanal. Chem."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/S0376-7388(96)00210-4","article-title":"Chronopotentiometric response of an ion-exchange membrane in the underlimiting current-range. Transport phenomena within the diffusion layers","volume":"123","author":"Sistat","year":"1997","journal-title":"J. Membr. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1006\/jcis.1998.5649","article-title":"Current-voltage curves for ion-exchange membranes: A method for determining the liming current density","volume":"205","year":"1998","journal-title":"J. Colloid Interface Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.desal.2005.03.064","article-title":"Influence of the concentration boundary layers on membrane potential in a single membrane system","volume":"184","year":"2005","journal-title":"Desalination"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"13458","DOI":"10.1021\/jp062433f","article-title":"Effect of anion-exchange membrane surface properties on mechanisms of overlimiting mass transfer","volume":"110","author":"Belova","year":"2006","journal-title":"J. Phys. Chem. B"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.cis.2008.01.007","article-title":"Application of chronopotentiometry to determine the thickness of diffusion layer adjacent to an ion-exchange membrane under natural convection","volume":"139","author":"Larchet","year":"2008","journal-title":"Adv. Colloid Interface Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1134\/S1023193512060122","article-title":"Mass transfer in a boundary layer and in an ion exchange membrane: Mechanism of concentration polarization and water dissociation","volume":"48","author":"Tanaka","year":"2012","journal-title":"Russ. J. Electrochem."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.memsci.2013.05.059","article-title":"Chronoamperometric response of ion-exchange membrane systems","volume":"444","author":"Moya","year":"2013","journal-title":"J. Membr. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.desal.2014.01.008","article-title":"Desalination at overlimiting currents: State-of-the-art and perspectives","volume":"342","author":"Nikonenko","year":"2014","journal-title":"Desalination"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.electacta.2016.02.098","article-title":"Accounting for the concentration dependence of electrolyte diffusion coefficient in the Sand and the Peers equation","volume":"195","author":"Mareev","year":"2016","journal-title":"Electrochim. Acta"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"102269","DOI":"10.1016\/j.cis.2020.102269","article-title":"Novel ionic separation mechanisms in electrically driven membrane process","volume":"284","author":"Wenten","year":"2020","journal-title":"Adv. Colloid Interface Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"102439","DOI":"10.1016\/j.cis.2021.102439","article-title":"Investigation of ion-exchange membranes by means of chronopotentiometry: A compressive review on this highly informative and multipurpose technique","volume":"293","author":"Barros","year":"2021","journal-title":"Adv. Colloid Interface Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"119846","DOI":"10.1016\/j.memsci.2021.119846","article-title":"Direct 3D observation and unravelling of electroconvection phenomena during concentration polarization at in exchange membranes","volume":"640","author":"Stockmeier","year":"2021","journal-title":"J. Membr. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Batko, K.M., \u015al\u0119zak-Prochazka, I., \u015al\u0119zak, A., Bajdur, W.M., and \u0160\u010durek, R. (2022). Modelling of the electrical membrane potential for concentration polarization conditions. Entropy, 24.","DOI":"10.3390\/e24010138"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Uzdenova, A. (2023). Ion transport in electromembrane system under the passage of direct current: 1D modelling approach. Membranes, 13.","DOI":"10.3390\/membranes13040421"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1134\/S2517751624010074","article-title":"Modeling of ion transport in a three-layer system with an ion-exchange membrane based on the Nernst Planck and displacement current equations","volume":"6","author":"Uzdenova","year":"2024","journal-title":"Membr. Membr. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"125929","DOI":"10.1016\/j.seppur.2023.125929","article-title":"Modeling non-linear ion transport phenomena in ion-selective membranes: Three simplified models","volume":"333","author":"Nguyen","year":"2024","journal-title":"Sep. Purif. Technol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"120137","DOI":"10.1016\/j.ces.2024.120137","article-title":"Towards optimized cation-exchange membranes for overlimiting current electrodialysis: Correlation between size of resin particles in membranes and mechanism of ion transport through them","volume":"295","author":"Zabolotsky","year":"2024","journal-title":"Chem. Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.desal.2017.03.001","article-title":"Entropy generation analysis of electrodialysis","volume":"414","author":"Chehayeb","year":"2017","journal-title":"Desalination"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"114858","DOI":"10.1016\/j.desal.2020.114858","article-title":"Entropy generation analysis of electrodialysis desalination using multi-component groundwater","volume":"500","author":"Generous","year":"2021","journal-title":"Desalination"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1007\/s11242-016-0807-7","article-title":"Evaluation of S-entropy production in a single-membrane system in concentration polarization conditions","volume":"116","author":"Grzegorczyn","year":"2017","journal-title":"Transp. Porous Med."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kjelstrup, S., Bedeaux, D., Johannessen, E., and Gross, S. (2017). Non-Equilibrium Thermodynamics for Engineers, Word Scientific. [2nd ed.].","DOI":"10.1142\/10286"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kjelstrup, S., and Bedeaux, D. (2008). Non-Equilibrium Thermodynamics of Heterogeneous Systems, Word Scientific.","DOI":"10.1142\/9789812779144"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2673","DOI":"10.1016\/j.ijheatmasstransfer.2006.01.009","article-title":"Fuel cell entropy production with ohmic heating and diffusive polarization","volume":"49","author":"Naterer","year":"2006","journal-title":"Int. J. Heat Mass Tranfer."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"044037","DOI":"10.1103\/PhysRevApplied.11.044037","article-title":"Thermoelectric power of ion exchange membrane cells relevant to reverse electrodialysis plants","volume":"11","author":"Kristiansen","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kujawski, W., Yaroshchuk, A., Zholkovskiy, E., Koter, I., and Koter, S. (2020). Analysis of membrane transport equations for reverse electrodialysis (RED) using irreversible thermodynamics. Int. J. Mol. Sci., 21.","DOI":"10.3390\/ijms21176325"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.compchemeng.2018.06.002","article-title":"Energy efficient design of membrane processes by use of entropy production minimization","volume":"117","author":"Magnanelli","year":"2018","journal-title":"Comp. Chem. Eng."},{"key":"ref_30","unstructured":"Helfferich, F. (1995). Ion Exchange, Dover Publication, INC."},{"key":"ref_31","unstructured":"Levich, V.G. (1962). Physicochemical Hydrodynamics, Prentice-Hall."},{"key":"ref_32","unstructured":"Strathamann, H. (2004). Ion-Exchange Membrane Separation Processes, Elsevier."},{"key":"ref_33","unstructured":"Katchalsky, A., and Curran, P.F. (1967). Thermodynamics in Biophysics, Harvard University Press Cambridge."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Brockris, J.O.M., and Reddy, A.K.N. (1970). Modern Electrochemistry, Plenum Press.","DOI":"10.1007\/978-1-4615-7467-5"},{"key":"ref_35","unstructured":"Lobo, V.M., and Quaresma, J.L. (1989). Handbook of Electrolyte Solutions, Elsevier. Parts A and B."},{"key":"ref_36","unstructured":"Robinson, R.A., and Stokes, R.H. (2003). Electrolyte Solutions, Dover Publication Inc.. [2nd ed.]."},{"key":"ref_37","first-page":"43","article-title":"Comparative study on the proton conductivity of perfluorosulfonic and polybenzimidazole based polymer electrolyte membranes","volume":"48","author":"Avramov","year":"2016","journal-title":"Bulg. Chem. Commun."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"7213","DOI":"10.1016\/j.jpowsour.2010.05.005","article-title":"Effect of cations (Na+, Ca2+, Fe3+) on the conductivity of a Nafion membrane","volume":"195","author":"Hongsirikarn","year":"2010","journal-title":"J. Power Sources"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"A1556","DOI":"10.1149\/1.1517281","article-title":"Ionic conductivity of an extruded Nafion 1100 EW series of membranes","volume":"149","author":"Slade","year":"2002","journal-title":"J. Electrochem. Soc."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1006\/jcis.2002.8407","article-title":"Direct measurement of concentration distribution within the boundary layer of an ion-exchange membrane","volume":"251","author":"Choi","year":"2002","journal-title":"J. Colloid Int. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/S0376-7388(97)00247-0","article-title":"The effect of ion-conducting spacers on mass transfer-numerical analysis of laser interferometry","volume":"139","author":"Shaposhnik","year":"1998","journal-title":"J. Membr. Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/1\/3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:59:41Z","timestamp":1760115581000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/1\/3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,25]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["e27010003"],"URL":"https:\/\/doi.org\/10.3390\/e27010003","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,12,25]]}}}