{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T05:54:01Z","timestamp":1777614841408,"version":"3.51.4"},"reference-count":58,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,9,7]],"date-time":"2018-09-07T00:00:00Z","timestamp":1536278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Chinese National Natural Science Foundation","award":["51579244"],"award-info":[{"award-number":["51579244"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A heat conduction model in a radial-pattern disc by considering non-uniform heat generation (NUHG) is established in this paper. A series of high conductivity channels (HCCs) are attached on the rim of the disc and extended to its center. Constructal optimizations of the discs with constant and variable cross-sectional HCCs are carried out, respectively, and their maximum temperature differences (MTDs) are minimized based on analytical method and finite element method. Besides, the influences of the NUHG coefficient, HCC number and width coefficient on the optimal results are studied. The results indicate that the deviation of the optimal constructs obtained from the analytical method and finite element method are comparatively slight. When the NUHG coefficient is equal to 10, the minimum MTD of the disc with 25 constant cross-sectional HCCs is specifically reduced by 48.8% compared to that with 10 HCCs. As a result, the heat conduction performance (HCP) of the disc can be efficiently improved by properly increasing the number of HCCs. The minimum MTD of the disc with variable cross-sectional HCC is decreased by 15.0% when the width coefficient is changed from 1 to 4. Therefore, the geometry of variable cross-sectional HCC can be applied in the constructal design of the disc to a better heat transfer performance. The constructal results obtained by investigating the non-uniform heat generating case in this paper can contribute to the design of practical electronic device to a better heat transfer performance.<\/jats:p>","DOI":"10.3390\/e20090685","type":"journal-article","created":{"date-parts":[[2018,9,7]],"date-time":"2018-09-07T11:47:41Z","timestamp":1536320861000},"page":"685","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Constructal Optimization for Cooling a Non-Uniform Heat Generating Radial-Pattern Disc by Conduction"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8137-4180","authenticated-orcid":false,"given":"Jiang","family":"You","sequence":"first","affiliation":[{"name":"Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, China"},{"name":"Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, China"},{"name":"College of Power Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huijun","family":"Feng","sequence":"additional","affiliation":[{"name":"Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, China"},{"name":"Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, China"},{"name":"College of Power Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lingen","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, China"},{"name":"Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, China"},{"name":"College of Power Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7045-648X","authenticated-orcid":false,"given":"Zhihui","family":"Xie","sequence":"additional","affiliation":[{"name":"Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, China"},{"name":"Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, China"},{"name":"College of Power Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.icheatmasstransfer.2016.04.016","article-title":"Generating optimal topologies for heat conduction by heat flow paths identification","volume":"75","author":"Li","year":"2016","journal-title":"Int. Commun. Heat Mass Transfer"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1016\/j.applthermaleng.2016.10.134","article-title":"A review about the engineering design of optimal heat transfer systems using topology optimization","volume":"112","author":"Dbouk","year":"2017","journal-title":"Appl. Therm. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1016\/j.ijheatmasstransfer.2016.08.077","article-title":"Design explorations of heat conductive pathways","volume":"104","author":"Manuel","year":"2017","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_4","unstructured":"Bejan, A. (2000). Shape and Structure, from Engineering to Nature, Cambridge University Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"041301","DOI":"10.1063\/1.2221896","article-title":"Constructal theory of generation of configuration in nature and engineering","volume":"100","author":"Bejan","year":"2006","journal-title":"J. Appl. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bejan, A., and Lorente, S. (2008). Design with Constructal Theory, Wiley.","DOI":"10.1002\/9780470432709"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1115\/1.2426357","article-title":"Constructal design of cooling channel in heat transfer system by utilizing optimality of branch systems in nature","volume":"129","author":"Ding","year":"2007","journal-title":"J. Heat Transfer"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lorenzini, G., Moretti, S., and Conti, A. (2011). Fin Shape Thermal Optimization Using Bejan\u2019s Constructal Theory, Morgan & Claypool Publishers.","DOI":"10.1007\/978-3-031-79333-2"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1299\/jtst.7.677","article-title":"Radiation effect on constructal design analysis of a T-Y-shaped assembly of fins","volume":"7","author":"Hajmohammadi","year":"2012","journal-title":"J. Therm. Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1007\/s11431-011-4701-9","article-title":"Progress in study on constructal theory and its application","volume":"55","author":"Chen","year":"2012","journal-title":"Sci. China Technol. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"151301","DOI":"10.1063\/1.4798429","article-title":"Constructal law of design and evolution: Physics, biology, technology, and society","volume":"113","author":"Bejan","year":"2013","journal-title":"J. Appl. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.enconman.2012.09.031","article-title":"Valuable reconsideration in the constructal design of cavities","volume":"66","author":"Hajmohammadi","year":"2013","journal-title":"Energy Convers. Mgmt."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/j.applthermaleng.2013.08.009","article-title":"Heat transfer improvement due to the imposition of non-uniform wall heating for in-tube laminar forced convection","volume":"61","author":"Hajmohammadi","year":"2013","journal-title":"Appl. Therm. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1144","DOI":"10.1016\/j.ijheatmasstransfer.2014.06.046","article-title":"Optimization of shell-and-tube heat exchangers using a general design approach motivated by constructal theory","volume":"77","author":"Yang","year":"2014","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"061003","DOI":"10.1115\/1.4029850","article-title":"Constructal law: optimization as design evolution","volume":"137","author":"Bejan","year":"2015","journal-title":"Trans. ASME J. Heat Transfer"},{"key":"ref_16","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 Transfer"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1080\/10407782.2015.1023148","article-title":"Constructal parallel-flow and counter-flow microchannel heat sinks with bifurcations","volume":"68","author":"Zhang","year":"2015","journal-title":"Numer. Heat Transfer A-Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1080\/10407782.2014.937286","article-title":"Numerical analysis of constructal water-cooled microchannel heat sinks with multiple bifurcations in the entrance region","volume":"67","author":"Zhang","year":"2015","journal-title":"Numer. Heat Transfer A-Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chen, L.G., Feng, H.J., and Xie, Z.H. (2016). Generalized thermodynamic optimization for iron and steel production processes: Theoretical exploration and application cases. Entropy, 18.","DOI":"10.20944\/preprints201609.0089.v1"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"074901","DOI":"10.1063\/1.4941554","article-title":"Complexity, organization, evolution, and constructal law","volume":"119","author":"Bejan","year":"2016","journal-title":"J. Appl. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.ijthermalsci.2017.07.010","article-title":"Introducing a \u03c8-shaped cavity for cooling a heat generating medium","volume":"121","author":"Hajmohammadi","year":"2017","journal-title":"Int. J. Therm. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.ijheatmasstransfer.2017.08.011","article-title":"Multi-disciplinary, multi-objective and multi-scale constructal optimizations for heat and mass transfer processes performed in Naval University of Engineering, a review","volume":"115","author":"Feng","year":"2017","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chen, L.G., Xiao, Q.H., and Feng, H.J. (2018). Constructal optimizations for heat and mass transfers based on the entransy dissipation extremum principle, performed at the Naval University of Engineering; a review. Entropy, 20.","DOI":"10.3390\/e20010074"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Miguel, A.F., and Rocha, L.A.O. (2018). Tree-Shaped Fluid Flow and Heat Transfer, Springer.","DOI":"10.1007\/978-3-319-73260-2"},{"key":"ref_25","unstructured":"Chen, L.G., Feng, H.J., and Xie, Z.H. (2017). Progress of constructal theory in China over the past decade. Int. J. Heat Mass Transfer, 55."},{"key":"ref_26","first-page":"309","article-title":"Constructal law, twenty years after","volume":"18","author":"Bejan","year":"2018","journal-title":"Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1016\/0017-9310(96)00175-5","article-title":"Constructal-theory network of conducting path for cooling a heat generating volume","volume":"40","author":"Bejan","year":"1997","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1136","DOI":"10.1063\/1.370855","article-title":"Constructal-theory tree networks of \u201cconstant\u201d thermal resistance","volume":"86","author":"Neagu","year":"1999","journal-title":"J. Appl. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"7107","DOI":"10.1063\/1.371799","article-title":"Three-dimensional tree constructs of \u201cconstant\u201d thermal resistance","volume":"86","author":"Neagu","year":"1999","journal-title":"J. Appl. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4922","DOI":"10.1063\/1.1562008","article-title":"Exact solution for cooling of electronics using constructal theory","volume":"93","author":"Ghodoossi","year":"2003","journal-title":"J. Appl. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.enconman.2006.05.009","article-title":"On the \u201carea to point\u201d flow problem based on constructal theory","volume":"48","author":"Wu","year":"2007","journal-title":"Energy Convers. Mgmt."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1111","DOI":"10.1016\/j.apenergy.2008.06.010","article-title":"The volume-point constructal optimization for discrete variable cross-section conducting path","volume":"86","author":"Wei","year":"2009","journal-title":"Appl. Energy"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1134\/S1810232817040051","article-title":"Geometric optimization of T-shaped constructs coupled with a heat generating basement: A numerical approach motivated by Bejan\u2019s constructal theory","volume":"26","author":"Lorenzini","year":"2017","journal-title":"J. Eng. Thermophys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.ijheatmasstransfer.2012.11.040","article-title":"Constructal design of X-shaped conductive pathways for cooling a heat-generating body","volume":"58","author":"Lorenzini","year":"2013","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.ijthermalsci.2013.04.021","article-title":"Constructal design of non-uniform X-shaped conductive pathways for cooling","volume":"71","author":"Lorenzini","year":"2013","journal-title":"Int. J. Therm. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.ijthermalsci.2013.10.015","article-title":"Phi and Psi shaped conductive routes for improved cooling in a heat generating piece","volume":"77","author":"Hajmohammadi","year":"2014","journal-title":"Int. J. Therm. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"061001","DOI":"10.1115\/1.4029847","article-title":"Evolution in the design of V-shaped highly conductive pathways embedded in a heat-generating piece","volume":"137","author":"Hajmohammadi","year":"2015","journal-title":"Trans. ASME J. Heat Transfer"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.ijheatmasstransfer.2015.07.105","article-title":"Constructal design for \u201c+\u201d shaped high conductive pathways over a square body","volume":"91","author":"Feng","year":"2015","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1016\/j.ijheatmasstransfer.2015.10.015","article-title":"Constructal design of I-shaped high conductive pathway for cooling a heat-generating medium considering the thermal contact resistance","volume":"93","author":"Lorenzini","year":"2016","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_40","first-page":"2","article-title":"Constructal design of non-uniform T-shaped conductive pathways for cooling heat generating bodies","volume":"26","author":"Beckel","year":"2016","journal-title":"Vetor Rio Grande"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1884","DOI":"10.1016\/j.ijheatmasstransfer.2017.01.008","article-title":"Geometrical evaluation of T-shaped high conductive pathway with thermal contact resistance for cooling of heat-generating medium","volume":"108","author":"Lorenzini","year":"2017","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1016\/S0196-8904(03)00190-0","article-title":"Conductive cooling of triangular shaped electronics using constructal theory","volume":"45","author":"Ghodoossi","year":"2004","journal-title":"Energy Convers. Manag."},{"key":"ref_43","first-page":"47","article-title":"Global constructal optimization design for triangular assembly at micro and nanoscales","volume":"63","author":"Feng","year":"2012","journal-title":"CIESC J."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1093\/ijlct\/cts071","article-title":"Constructal re-optimization of heat conduction with the triangular elemental area","volume":"9","author":"Chen","year":"2014","journal-title":"Int. J. Low-Carbon Technol."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.joei.2015.01.016","article-title":"\u201cVolume-point\u201d heat conduction constructal optimization based on minimization of maximum thermal resistance with triangular element at micro and nanoscales","volume":"89","author":"Feng","year":"2016","journal-title":"J. Energy Inst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4257","DOI":"10.1016\/j.ijheatmasstransfer.2004.04.024","article-title":"Disc cooled with high-conductivity inserts that extend inward from the perimeter","volume":"47","author":"Silva","year":"2004","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_47","first-page":"043005","article-title":"Using incomplete variable cross-section highly conductive inserts for cooling a disc","volume":"3","author":"Sharifi","year":"2012","journal-title":"Front. Heat Mass Transfer"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1643","DOI":"10.1016\/S0017-9310(01)00269-1","article-title":"Constructal design for cooling a disc-shaped area by conduction","volume":"45","author":"Rocha","year":"2002","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2626","DOI":"10.1016\/j.ijheatmasstransfer.2006.01.017","article-title":"Conduction tree networks with loops for cooling a heat generating volume","volume":"49","author":"Rocha","year":"2006","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1016\/j.ijthermalsci.2011.01.016","article-title":"Constructal optimization for \u201cdisc-to-point\u201d heat conduction without the premise of optimized last-order construct","volume":"50","author":"Xiao","year":"2011","journal-title":"Int. J. Therm. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1016\/j.applthermaleng.2016.09.138","article-title":"Constructal design of high-emissivity radiation inserts embedded in a disc-shaped heat generation body","volume":"112","author":"Bahadormanesh","year":"2016","journal-title":"Appl. Therm. Eng."},{"key":"ref_52","first-page":"15","article-title":"Cooling a solid disc with uniform heat generation using inserts of high thermal conductivity within the constructal design platform","volume":"12","author":"Sharfi","year":"2016","journal-title":"Int. J. Therm. Environ. Eng."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1016\/j.ijheatmasstransfer.2013.08.051","article-title":"Constructal optimization for \u201cdisc-point\u201d heat conduction at micro and nanoscales","volume":"67","author":"Chen","year":"2013","journal-title":"Int. J. Heat Mass Transfer"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"031401","DOI":"10.1115\/1.4007903","article-title":"Micro and nano-scale conductive tree-structures for cooling a disc-shaped electronic piece","volume":"135","author":"Daneshi","year":"2013","journal-title":"Trans. ASME J. Heat Transfer"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"024902","DOI":"10.1063\/1.4926620","article-title":"The natural emergence of asymmetric tree-shaped pathways for cooling of a non-uniformly heated domain","volume":"118","author":"Cetkin","year":"2015","journal-title":"J. Appl. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1140\/epjp\/i2016-16274-4","article-title":"Constructal design for a rectangular body with nonuniform heat generation","volume":"131","author":"Feng","year":"2016","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1007\/s11431-016-0135-x","article-title":"Constructal entransy dissipation rate minimization of a rectangular body with nonuniform heat generation","volume":"59","author":"Feng","year":"2016","journal-title":"Sci. China Technol. Sci."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"896","DOI":"10.1016\/j.ijheatmasstransfer.2017.10.032","article-title":"Heat conduction constructal optimization for nonuniform heat generating area based on triangular element","volume":"117","author":"You","year":"2018","journal-title":"Int. J. Heat Mass Transfer"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/9\/685\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:19:26Z","timestamp":1760195966000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/9\/685"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,7]]},"references-count":58,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2018,9]]}},"alternative-id":["e20090685"],"URL":"https:\/\/doi.org\/10.3390\/e20090685","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,9,7]]}}}