{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T07:32:14Z","timestamp":1782804734418,"version":"3.54.5"},"reference-count":84,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,4]],"date-time":"2020-02-04T00:00:00Z","timestamp":1580774400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The fractal-type flow-fields for fuel cell (FC) applications are promising, due to their ability to deliver uniformly, with a Peclet number Pe~1, the reactant gases to the catalytic layer. We review fractal designs that have been developed and studied in experimental prototypes and with CFD computations on 1D and 3D flow models for planar, circular, cylindrical and conical FCs. It is shown, that the FC efficiency could be increased by design optimization of the fractal system. The total entropy production (TEP) due to viscous flow was the objective function, and a constant total volume (TV) of the channels was used as constraint in the design optimization. Analytical solutions were used for the TEP, for rectangular channels and a simplified 1D circular tube. Case studies were done varying the equivalent hydraulic diameter (Dh), cross-sectional area (D\u03a3) and hydraulic resistance (DZ). The analytical expressions allowed us to obtain exact solutions to the optimization problem (TEP\u2192min, TV=const). It was shown that the optimal design corresponds to a non-uniform width and length scaling of consecutive channels that classifies the flow field as a quasi-fractal. The depths of the channels were set equal for manufacturing reasons. Recursive formulae for optimal non-uniform width scaling were obtained for 1D circular Dh -, D\u03a3 -, and DZ -based tubes (Cases 1-3). Appropriate scaling of the fractal system providing uniform entropy production along all the channels have also been computed for Dh -, D\u03a3 -, and DZ -based 1D models (Cases 4-6). As a reference case, Murray\u2019s law was used for circular (Case 7) and rectangular (Case 8) channels. It was shown, that Dh-based models always resulted in smaller cross-sectional areas and, thus, overestimated the hydraulic resistance and TEP. The D\u03a3 -based models gave smaller resistances compared to the original rectangular channels and, therefore, underestimated the TEP. The DZ -based models fitted best to the 3D CFD data. All optimal geometries exhibited larger TEP, but smaller TV than those from Murray\u2019s scaling (reference Cases 7,8). Higher TV with Murray\u2019s scaling leads to lower contact area between the flow-field plate with other FC layers and, therefore, to larger electric resistivity or ohmic losses. We conclude that the most appropriate design can be found from multi-criteria optimization, resulting in a Pareto-frontier on the dependencies of TEP vs TV computed for all studied geometries. The proposed approach helps us to determine a restricted number of geometries for more detailed 3D computations and further experimental validations on prototypes.<\/jats:p>","DOI":"10.3390\/e22020176","type":"journal-article","created":{"date-parts":[[2020,2,5]],"date-time":"2020-02-05T03:18:48Z","timestamp":1580872728000},"page":"176","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Fractal-Like Flow-Fields with Minimum Entropy Production for Polymer Electrolyte Membrane Fuel Cells"],"prefix":"10.3390","volume":"22","author":[{"given":"Natalya","family":"Kizilova","sequence":"first","affiliation":[{"name":"Warsaw University of Technology, Institute of Aviation and Applied Mechanics, 00-665 Warsaw, Poland"},{"name":"Department of Applied Mathematics, V.N. Karazin Kharkov National University, 61022 Kharkiv, Ukraine"},{"name":"PoreLab, Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marco","family":"Sauermoser","sequence":"additional","affiliation":[{"name":"PoreLab, Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1235-5709","authenticated-orcid":false,"given":"Signe","family":"Kjelstrup","sequence":"additional","affiliation":[{"name":"PoreLab, Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4928-7378","authenticated-orcid":false,"given":"Bruno G.","family":"Pollet","sequence":"additional","affiliation":[{"name":"Department of Energy and Process Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,4]]},"reference":[{"key":"ref_1","unstructured":"(2019, April 18). Available online: https:\/\/www.shell.com\/."},{"key":"ref_2","unstructured":"Hacker, V., and Mitsushima, S. (2018). Fuel Cells and Hydrogen. From Fundamentals to Applied Research, Elsevier. [1st ed.]."},{"key":"ref_3","unstructured":"(2019, April 18). 2018 Cost Projections of PEM Fuel Cell Systems for Automobiles and Medium-Duty Vehicles. Brian James, Strategic Analysis Inc., Available online: https:\/\/www.energy.gov\/sites\/prod\/files\/2018\/04\/f51\/fcto_webinarslides_2018_costs_pem_fc_autos_trucks_042518.pdf."},{"key":"ref_4","unstructured":"Sauermoser, M., Kizilova, N., Kjelstrup, S., and Pollet, B.G. Flow field patterns for proton exchange membrane (PEM) fuel cells. Front. Phys., (under review)."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/j.ijhydene.2004.09.019","article-title":"Review of bipolar plates in PEM fuel cells: flow field designs","volume":"30","author":"Li","year":"2005","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"10798","DOI":"10.1016\/j.ijhydene.2017.12.149","article-title":"Numerical optimization of channel to land width ratio for PEM fuel cell","volume":"43","author":"Chowdhury","year":"2018","journal-title":"Intern. J. Hydrogen Energy"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.enconman.2016.10.053","article-title":"Flow field bipolar plates in a proton exchange membrane fuel cell: Analysis & Modeling","volume":"133","author":"Kahraman","year":"2017","journal-title":"Energy Convers. Manag."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Shabany, Y. (2009). Heat Transfer: Thermal Management of Electronics, CRC Press.","DOI":"10.1201\/9781439814680"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1109\/EDL.1981.25367","article-title":"High performance heat sinking for VLSI","volume":"2","author":"Tuckerman","year":"1981","journal-title":"IEEE Electron. Dev. Lett."},{"key":"ref_10","unstructured":"Bejan, A. (2000). Shape and Structure, from Engineering to Nature, Cambridge University Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1126\/science.2396104","article-title":"Principles of design of fluid transport systems in zoology","volume":"249","year":"1990","journal-title":"Science"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Losa, G.A., Merlini, D., Nonnenmacher, T.F., and Wiebel, E.R. (2005). Coppens Is the lung an optimal gas exchanger. Fractals in Biology and Medicine, Birkauser.","DOI":"10.1007\/3-7643-7412-8"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Losa, G.A., Merlini, D., Nonnenmacher, T.F., and Wiebel, E.R. (2005). Gas diffusion through the fractal landscape of the lung: How deep does oxygen enter the alveolar system. Fractals in Biology and Medicine, Birkauser.","DOI":"10.1007\/3-7643-7412-8"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"179","DOI":"10.3176\/proc.2008.3.07","article-title":"Long-distance liquid transport in plants","volume":"57","author":"Kizilova","year":"2008","journal-title":"Proc. Estonian Acad. Sci. Ser. Phys. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1007\/978-3-540-24709-8_51","article-title":"Computational approach to optimal transport network construction in biomechanics","volume":"3044","author":"Kizilova","year":"2004","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_16","unstructured":"Roux, W. (1878). Uber die Verzweigungen der Blutgefasse. [PhD Thesis, University of Jena]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1073\/pnas.12.3.207","article-title":"The physiological principle of minimum work. I. The vascular system and the cost of blood volume","volume":"12","author":"Murray","year":"1926","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1085\/jgp.9.6.835","article-title":"The physiological principle of minimum work applied to the angle of branching of arteries","volume":"9","author":"Murray","year":"1926","journal-title":"J. Gen. Physiol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1161\/01.RES.77.5.1017","article-title":"Design principles of vascular beds","volume":"77","author":"Pries","year":"1995","journal-title":"Circ. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/S0022-5193(84)80089-2","article-title":"Cost of departure from optimality in arterial branching","volume":"109","author":"Zamir","year":"1984","journal-title":"J. Theor. Biol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1139\/y86-109","article-title":"Branching characteristics of human coronary arteries","volume":"64","author":"Zamir","year":"1986","journal-title":"Canad. J. Physiol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1152\/jappl.1999.86.2.569","article-title":"Structure-function relationships in the pulmonary arterial tree","volume":"86","author":"Dawson","year":"1999","journal-title":"J. Appl. Physiol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1161\/01.RES.33.2.190","article-title":"Morphometry of the human pulmonary arterial tree","volume":"33","author":"Singhal","year":"1973","journal-title":"Circ. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0034-5687(81)90073-6","article-title":"Design of the mammalian respiratory system","volume":"44","author":"Taylor","year":"1981","journal-title":"Respir Physiol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1023\/A:1001841615640","article-title":"Bionics, biological systems and the principle of optimal design","volume":"46","author":"Popescu","year":"1999","journal-title":"Acta Biotheor."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1126\/science.199.4331.888","article-title":"Tree branch angle: maximizing effective leaf area","volume":"199","author":"Honda","year":"1978","journal-title":"Science"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1006\/jtbi.2001.2252","article-title":"Modeling of Branching Structures of Plants","volume":"209","author":"Zhi","year":"2001","journal-title":"J. Theor. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1038\/nature01444","article-title":"Water transport in plants obeys Murray\u2019s law","volume":"421","author":"McCulloh","year":"2003","journal-title":"Nature"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/S0092-8240(77)80054-2","article-title":"Optimization of diameters and bifurcation angles in lung and vascular tree structures","volume":"39","author":"Uylings","year":"1977","journal-title":"Bull. Math. Biol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"114909","DOI":"10.1063\/1.2396842","article-title":"Heterogeneous porous media as multiscale structures for maximum flow access","volume":"100","author":"Lorente","year":"2006","journal-title":"J. Appl. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"485503","DOI":"10.1088\/0022-3727\/48\/48\/485503","article-title":"From pore network prediction based on the Constructal law to macroscopic properties of porous media","volume":"48","author":"Wattez","year":"2015","journal-title":"J. Phys. D Appl. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Barber, R., Cieslicki, K., and Emerson, D. (2006). Using Murray\u2019s law to design artificial vascular microfluidic networks. Design and Nature III: Comparing Design in Nature with Science and Engineering, WIT Press.","DOI":"10.2495\/DN060241"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1039\/C7EE02161E","article-title":"A lung-inspired approach to scalable and robust fuel cell design","volume":"11","author":"Trogadas","year":"2018","journal-title":"Energy Environ. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.energy.2018.12.143","article-title":"Visualization of liquid water in a lung-inspired flow-field based polymer electrolyte membrane fuel cell via neutron radiography","volume":"170","author":"Cho","year":"2019","journal-title":"Energy"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"7499","DOI":"10.1016\/j.ijheatmasstransfer.2012.07.050","article-title":"A study on the hydraulic and thermal characteristics in fractal tree-like microchannels by numerical and experimental methods","volume":"55","author":"Yu","year":"2012","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Sauermoser, M., Kjelstrup, S., Kizilova, N., Pollet, B.G., and Flekk\u00f8y, E.G. (2020). Seeking minimum entropy production for a tree-like flow-field in a fuel cell. Phys. Chem. Chem. Phys., under review.","DOI":"10.1039\/C9CP05394H"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"073530","DOI":"10.1063\/1.2794379","article-title":"Effect of bifurcation angle in tree-shaped microchannel networks","volume":"102","author":"Wang","year":"2007","journal-title":"J. Appl. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"174302","DOI":"10.1063\/1.4935288","article-title":"Generalizing Murray\u2019s law: An optimization principle for fluidic networks of arbitrary shape and scale","volume":"118","author":"Stephenson","year":"2015","journal-title":"J. Appl. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1080\/10893950290098359","article-title":"Reduced pumping power and wall temperature in microchannel heat sinks with fractal-like branching channel network","volume":"6","author":"Pence","year":"2003","journal-title":"Microscale Thermophys. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2643","DOI":"10.1016\/S0017-9310(02)00013-3","article-title":"Heat transfer and pressure drop in fractal tree-like microchannel nets","volume":"45","author":"Chen","year":"2002","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.icheatmasstransfer.2005.02.001","article-title":"An experimental investigation on the thermal efficiency of fractal tree-like microchannel nets","volume":"32","author":"Chen","year":"2005","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.jpowsour.2003.11.078","article-title":"Investigation of fractal flow-fields in portable proton exchange membrane and direct methanol fuel cells","volume":"131","author":"Oedegaard","year":"2004","journal-title":"J. Power Sources"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1115\/1.1795236","article-title":"Thermal characteristics of microscale fractal-like branching channels","volume":"126","author":"Alharbi","year":"2004","journal-title":"ASME J. Heat Transf."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1063\/1.1757028","article-title":"Tree network channels as fluid distributors constructing double staircase polymer electrolyte fuel cells","volume":"96","author":"Senn","year":"2004","journal-title":"J. Appl. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.jpowsour.2003.12.025","article-title":"Laminar mixing, heat transfer and pressure drop in tree-like microchannel nets and their application for thermal management in polymer electrolyte fuel cells","volume":"130","author":"Senn","year":"2004","journal-title":"J. Power Sources"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1016\/j.enconman.2004.05.008","article-title":"Thermal and hydrodynamic analysis of a fractal microchannel network","volume":"46","author":"Ghodoossi","year":"2005","journal-title":"Energy Convers. Manag."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4558","DOI":"10.1016\/j.ijheatmasstransfer.2006.04.035","article-title":"Constructal tree-shaped parallel flow heat exchangers","volume":"49","author":"Zimparov","year":"2006","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1016\/j.ijthermalsci.2006.01.010","article-title":"Thermal characteristics of tree-shaped microchannel nets for cooling of a rectangular heat sink","volume":"45","author":"Wang","year":"2006","journal-title":"Int. J. Therm. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1115\/1.2159007","article-title":"Numerical analysis of blockage and optimization of heat transfer performance of fractal-like microchannel nets","volume":"128","author":"Wang","year":"2006","journal-title":"J. Electron. Packag."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1115\/1.2229228","article-title":"Laminar heat transfer in constructal microchannel networks with loops","volume":"128","author":"Wang","year":"2006","journal-title":"ASME J. Electron. Packag."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.compfluid.2014.01.034","article-title":"Lattice Boltzmann method for shape optimization of fluid distributor","volume":"94","author":"Wang","year":"2014","journal-title":"Comput. Fluids"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2139","DOI":"10.1016\/j.ijthermalsci.2009.03.018","article-title":"Thermal characteristics of tree-shaped microchannel nets with\/without loops","volume":"48","author":"Xu","year":"2009","journal-title":"Intern. J. Thermal Sci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1080\/01457630701378218","article-title":"Adiabatic flow boiling in fractal-like microchannels","volume":"28","author":"Daniels","year":"2007","journal-title":"Heat Transfer Eng."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"4986","DOI":"10.1016\/j.ijheatmasstransfer.2007.09.006","article-title":"Conjugate heat transfer in fractal-shaped microchannel network heat sink for integrated microelectronic cooling application","volume":"50","author":"Hong","year":"2007","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1016\/j.ijheatmasstransfer.2008.07.048","article-title":"Efficiency of optimized bifurcating treelike and parallel microchannel networks in the cooling of electronics","volume":"52","author":"Escher","year":"2009","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"2018","DOI":"10.1002\/aic.12135","article-title":"Thermal and hydrodynamic characteristics of constructal tree-shaped minichannel heat sink","volume":"56","author":"Chen","year":"2010","journal-title":"AIChE J."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.ijheatmasstransfer.2014.09.016","article-title":"Gas flow in micro tree-shaped hierarchical network","volume":"80","author":"Chen","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.ijheatmasstransfer.2010.09.051","article-title":"Flow boiling in constructal tree-shaped minichannel network","volume":"54","author":"Zhang","year":"2011","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"6366","DOI":"10.1016\/j.jpowsour.2011.03.044","article-title":"Methanol steam reforming in microreactor with constructal tree-shaped network","volume":"196","author":"Chen","year":"2011","journal-title":"J. Power Sources"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1016\/j.applthermaleng.2013.10.042","article-title":"Constructal design and thermal analysis of microchannel heat sinks with multistage bifurcations in single-phase liquid flow","volume":"62","author":"Xie","year":"2014","journal-title":"Appl. Therm. Eng."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.applthermaleng.2014.07.031","article-title":"Laminar thermal performance of microchannel heat sinks with constructal vertical y-shaped bifurcation plates","volume":"73","author":"Li","year":"2014","journal-title":"Appl. Therm. Eng."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"948","DOI":"10.1016\/j.ijheatmasstransfer.2015.07.034","article-title":"Parametric study on thermal performance of microchannel heat sinks with internal vertical y-shaped bifurcations","volume":"90","author":"Xie","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.ijheatmasstransfer.2015.04.006","article-title":"Mass transfer and reaction in methanol steam reforming reactor with fractal tree-like microchannel network","volume":"87","author":"Chen","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.energy.2017.05.139","article-title":"Numerical modeling of a proton exchange membrane fuel cell with tree-like flow field channels based on an entropy generation analysis","volume":"133","year":"2017","journal-title":"Energy"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Li, P., Coopamah, D., and Ki, J.-P. (2008, January 16\u201318). Uniform distribution of species in fuel cells using a multiple flow bifurcation design. Proceedings of the 6th International Fuel Cell Science, Engineering and Technology Conference, Denver, CO, USA.","DOI":"10.1115\/FuelCell2008-65106"},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Damian-Ascencio, C.E., Hernandez-Guerrero, A., Escobar-Vargas, J.A., Rubio-Arana, C., and Elizalde-Blancas, F. (2007, January 11\u201315). Three dimensional numerical prediction of the current density for a constructal theory-based flow field pattern. Proceedings of the ASME 2007 International Mechanical Engineering Congress and Exposition, Seattle, DC, USA.","DOI":"10.1115\/IMECE2007-42449"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.expthermflusci.2008.07.003","article-title":"Experimental investigation of the flow distribution of a 2-dimensional constructal distributor","volume":"33","author":"Fan","year":"2008","journal-title":"Experim. Thermal Fluid Sci"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"12965","DOI":"10.1016\/j.ijhydene.2011.07.017","article-title":"Constructal flow distributor as a bipolar plate for proton exchange membrane fuel cells","volume":"36","author":"Ellis","year":"2011","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"2678","DOI":"10.1016\/j.ijheatmasstransfer.2008.09.042","article-title":"Kockmann, Simulation and experimental investigation of pressure loss and heat transfer in microchannel networks containing bends and T-junctions","volume":"52","author":"Haller","year":"2009","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"1032","DOI":"10.1016\/j.applthermaleng.2015.07.024","article-title":"Conjugate heat transfer in fractal tree-like channels network heat sink for high-speed motorized spindle cooling","volume":"90","author":"Xia","year":"2015","journal-title":"Appl. Thermal Eng."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1016\/j.ijthermalsci.2006.06.006","article-title":"Optimization of tree-shaped fluid networks with size limitations","volume":"46","author":"Gosselin","year":"2007","journal-title":"Int. J. Therm. Sci."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1115\/1.1625684","article-title":"Fluid flow through microscale fractal-like branching channel networks","volume":"125","author":"Alharbi","year":"2003","journal-title":"ASME J. Fluids Eng."},{"key":"ref_73","doi-asserted-by":"crossref","unstructured":"Xu, G., Wang, M., Tao, Z., Ding, S., Wu, H., and Guo, J. (2008, January 11\u201316). Numerical analysis of flow and heat transfer characteristics of Y-fractal-link micro channel networks. Proceedings of the ICHMT International Symposium on Advances in Computational Heat Transfer (CHT-08), Marrakech, Morocco.","DOI":"10.1615\/ICHMT.2008.CHT.480"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"065018","DOI":"10.1088\/0960-1317\/21\/6\/065018","article-title":"Characterization on the performance of a fractal-shaped microchannel network for microelectronic cooling","volume":"21","author":"Hong","year":"2011","journal-title":"J. Micromech. Microeng."},{"key":"ref_75","unstructured":"Pence, D.V. (2000, January 23). Improved thermal efficiency and temperature uniformity using fractal-like branching channel networks. Heat Transfer and Transport Phenomena in Microsystems. Proceedings of the Engineering. Foundation Conference on Heat Transfer and Transport Phenomena in Microscale, Banff, AB, Canada."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"4911","DOI":"10.1016\/S0017-9310(02)00211-9","article-title":"Optimal tree-shaped networks for fluid flow in a disc-shaped body","volume":"45","author":"Wechsatol","year":"2002","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1016\/0017-9310(96)00175-5","article-title":"Constructal-theory network of conducting paths for cooling a heat generating volume","volume":"40","author":"Bejan","year":"1997","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"3105","DOI":"10.1016\/S0017-9310(99)00353-1","article-title":"Convective trees of fluid channels for volumetric cooling","volume":"43","author":"Bejan","year":"2000","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_79","unstructured":"Sauermoser, M., Kizilova, N., and Kjelstrup, S. (2019, January 21\u201324). Minimum entropy production approach to optimal design of the flow field plates in polymer electrolyte fuel cells. Proceedings of the Joint European Thermodynamics Conference (JETC2019), Barcelona, Spain."},{"key":"ref_80","unstructured":"White, F.M. (2003). Fluid Mechanics, McGraw-Hill."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1080\/10407780600669175","article-title":"Optimal temperature distribution in a 3D triple layered skin structure embedded with artery and vein vasculature","volume":"50","author":"Tang","year":"2006","journal-title":"Numer. Heat Transf. A"},{"key":"ref_82","first-page":"11","article-title":"The method of entropy generation minimization","volume":"15","author":"Bejan","year":"1999","journal-title":"Energy and the environment. Environmental science and Technology Library"},{"key":"ref_83","doi-asserted-by":"crossref","unstructured":"Kjelstrup, S., and Bedeaux, D. (2008). Non-equilibrium Thermodynamics of Heterogeneous Systems, World Scientific.","DOI":"10.1142\/9789812779144"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1021\/ie00061a010","article-title":"Equipartition of entropy production. An optimality criterion for transfer and separation processes","volume":"26","author":"Tondeur","year":"1987","journal-title":"Ind. Eng. Chem. Res."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/176\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:54:33Z","timestamp":1760172873000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/176"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,4]]},"references-count":84,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["e22020176"],"URL":"https:\/\/doi.org\/10.3390\/e22020176","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,4]]}}}