{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T02:33:41Z","timestamp":1768617221301,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2018,5,3]],"date-time":"2018-05-03T00:00:00Z","timestamp":1525305600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Universiti Kebangsaan Malaysia (UKM)","award":["DIP-2017-010"],"award-info":[{"award-number":["DIP-2017-010"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The problem of entropy generation analysis and natural convection in a nanofluid square cavity with a concentric solid insert and different temperature distributions is studied numerically by the finite difference method. An isothermal heater is placed on the bottom wall while isothermal cold sources are distributed along the top and side walls of the square cavity. The remainder of these walls are kept adiabatic. Water-based nanofluids with Al     2    O     3    nanoparticles are chosen for the investigation. The governing dimensionless parameters of this study are the nanoparticles volume fraction (    0 \u2264 \u03d5 \u2264 0.09    ), Rayleigh number    (  10 3  \u2264 R a \u2264  10 6  )    , thermal conductivity ratio (    0.44 \u2264  K r  \u2264 23.8    ) and length of the inner solid (    0 \u2264 D \u2264 0.7    ). Comparisons with previously experimental and numerical published works verify a very good agreement with the proposed numerical method. Numerical results are presented graphically in the form of streamlines, isotherms and local entropy generation as well as the local and average Nusselt numbers. The obtained results indicate that the thermal conductivity ratio and the inner solid size are excellent control parameters for an optimization of heat transfer and Bejan number within the fully heated and partially cooled square cavity.<\/jats:p>","DOI":"10.3390\/e20050336","type":"journal-article","created":{"date-parts":[[2018,5,3]],"date-time":"2018-05-03T03:20:27Z","timestamp":1525317627000},"page":"336","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Entropy Generation Analysis and Natural Convection in a Nanofluid-Filled Square Cavity with a Concentric Solid Insert and Different Temperature Distributions"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2970-6600","authenticated-orcid":false,"given":"Ammar","family":"Alsabery","sequence":"first","affiliation":[{"name":"Refrigeration &amp; Air-conditioning Technical Engineering Department, College of Technical Engineering, The Islamic University, Najaf 54001, Iraq"},{"name":"Center for Modelling &amp; Data Science, Faculty of Science &amp; Technology, Universiti Kebangsaan Malaysia, UKM Bangi 43600, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhamad","family":"Ishak","sequence":"additional","affiliation":[{"name":"Center for Modelling &amp; Data Science, Faculty of Science &amp; Technology, Universiti Kebangsaan Malaysia, UKM Bangi 43600, Selangor, Malaysia"},{"name":"Department of Engineering, College of Foundation and Diploma Studies, Universiti Tenaga Nasional, Kajang 43000, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8335-3121","authenticated-orcid":false,"given":"Ali","family":"Chamkha","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia"},{"name":"RAK Research and Innovation Center, American University of Ras Al Khaimah, P.O. Box 10021, Ras Al Khaimah 86416, UAE"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4237-7140","authenticated-orcid":false,"given":"Ishak","family":"Hashim","sequence":"additional","affiliation":[{"name":"Center for Modelling &amp; Data Science, Faculty of Science &amp; Technology, Universiti Kebangsaan Malaysia, UKM Bangi 43600, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,5,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3639","DOI":"10.1016\/S0017-9310(03)00156-X","article-title":"Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids","volume":"46","author":"Khanafer","year":"2003","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1016\/j.icheatmasstransfer.2006.02.016","article-title":"Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures","volume":"33","author":"Jou","year":"2006","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1016\/j.ijthermalsci.2007.10.005","article-title":"Study of heat transfer augmentation in a differentially heated square cavity using copper-water nanofluid","volume":"47","author":"Santra","year":"2008","journal-title":"Int. J. Therm. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.ijthermalsci.2010.04.006","article-title":"Natural convection of nanofluids in a shallow cavity heated from below","volume":"50","author":"Alloui","year":"2011","journal-title":"Int. J. Therm. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.icheatmasstransfer.2014.11.001","article-title":"A brief review of natural convection in enclosures under localized heating with and without nanofluids","volume":"60","author":"Yan","year":"2015","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1750030","DOI":"10.1142\/S0218348X1750030X","article-title":"Optimization of the fractal-like architecture of porous fibrous materials related to permeability, diffusivity and thermal conductivity","volume":"25","author":"Xiao","year":"2017","journal-title":"Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1108\/HFF-11-2015-0488","article-title":"Effect of spatial side-wall temperature variation on transient natural convection of a nanofluid in a trapezoidal cavity","volume":"27","author":"Alsabery","year":"2017","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6811","DOI":"10.1166\/jnn.2017.14502","article-title":"Research on relative permeability of nanofibers with capillary pressure effect by means of Fractal-Monte Carlo technique","volume":"17","author":"Xiao","year":"2017","journal-title":"J. Nanosci. Nanotechnol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"645","DOI":"10.2118\/185173-PA","article-title":"The Effects of Perforation Erosion on Practical Hydraulic-Fracturing Applications","volume":"22","author":"Long","year":"2017","journal-title":"SPE J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1840015","DOI":"10.1142\/S0218348X18400157","article-title":"A fractal model for water flow through unsaturated porous rocks","volume":"26","author":"Xiao","year":"2018","journal-title":"Fractals"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1080\/10407789008944791","article-title":"Effect of a centered conducting body on natural convection heat transfer in an enclosure","volume":"18","author":"House","year":"1990","journal-title":"Numer. Heat Transf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1080\/104077899275209","article-title":"Numerical study on transient heat transfer and fluid flow of natural convection in an enclosure with a heat-generating conducting body","volume":"35","author":"Ha","year":"1999","journal-title":"Numer. Heat Transf. Part Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.spmi.2012.05.007","article-title":"Natural convection in a square cavity containing a nanofluid and an adiabatic square block at the center","volume":"52","author":"Mahmoodi","year":"2012","journal-title":"Superlattices Microstruct."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1016\/j.spmi.2016.10.062","article-title":"Transient free convective heat transfer in nanoliquid-saturated porous square cavity with a concentric solid insert and sinusoidal boundary condition","volume":"100","author":"Alsabery","year":"2016","journal-title":"Superlattices Microstruct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/j.ijheatmasstransfer.2017.01.117","article-title":"Mixed convection in alumina-water nanofluid filled lid-driven square cavity with an isothermally heated square blockage inside with magnetic field effect: Introduction","volume":"109","author":"Mehmood","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2124","DOI":"10.1016\/j.ces.2011.02.009","article-title":"Role of entropy generation on thermal management during natural convection in porous square cavities with distributed heat sources","volume":"66","author":"Kaluri","year":"2011","journal-title":"Chem. Eng. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1080\/10407782.2013.784679","article-title":"Heat transfer enhancement and entropy generation in a square enclosure in the presence of adiabatic and isothermal blocks","volume":"64","author":"Mahapatra","year":"2013","journal-title":"Numer. Heat Transf. Part Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/j.ijheatmasstransfer.2013.10.009","article-title":"A numerical study on natural convection and entropy generation in a porous enclosure with heat sources","volume":"69","author":"Lam","year":"2014","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"9","DOI":"10.3390\/e18010009","article-title":"Analysis of entropy generation in natural convection of nanofluid inside a square cavity having hot solid block: Tiwari and Das\u2019 model","volume":"18","author":"Sheremet","year":"2015","journal-title":"Entropy"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.powtec.2016.05.032","article-title":"Heat transfer and entropy generation of natural convection on non-Newtonian nanofluids in a porous cavity","volume":"299","author":"Kefayati","year":"2016","journal-title":"Powder Technol."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Kolsi, L., Mahian, O., \u00d6ztop, H.F., Aich, W., Borjini, M.N., Abu-Hamdeh, N., and Aissia, H.B. (2016). 3D buoyancy-induced flow and entropy generation of nanofluid-filled open cavities having adiabatic diamond shaped obstacles. Entropy, 18.","DOI":"10.3390\/e18060232"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.jtice.2015.09.012","article-title":"Conjugate heat transfer and entropy generation in a cavity filled with a nanofluid-saturated porous media and heated by a triangular solid","volume":"59","author":"Ismael","year":"2016","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.ijmecsci.2017.03.025","article-title":"Natural convection and entropy generation in a nanofluid filled cavity with thick bottom wall: Effects of non-isothermal heating","volume":"126","author":"Bouchoucha","year":"2017","journal-title":"Int. J. Mech. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/j.molliq.2017.03.016","article-title":"Lattice Boltzmann numerical simulation and entropy generation analysis of natural convection of nanofluid in a porous cavity with different linear temperature distributions on side walls","volume":"233","author":"Ghasemi","year":"2017","journal-title":"J. Mol. Liquids"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.euromechflu.2016.11.016","article-title":"Investigation of different nanofluids effect on entropy generation on natural convection in a porous cavity","volume":"62","author":"Ashorynejad","year":"2017","journal-title":"Eur. J. Mech. B\/Fluids"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5065","DOI":"10.1016\/j.energy.2011.06.001","article-title":"Entropy generation due to natural convection in discretely heated porous square cavities","volume":"36","author":"Kaluri","year":"2011","journal-title":"Energy"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1080\/10407782.2011.616778","article-title":"Buoyancy-driven heat transfer in nanofluid-filled trapezoidal enclosure with variable thermal conductivity and viscosity","volume":"60","author":"Roslan","year":"2011","journal-title":"Numer. Heat Transf. Part Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.icheatmasstransfer.2011.11.004","article-title":"Investigation of buoyancy-driven flow and heat transfer in a trapezoidal cavity filled with water-Cu nanofluid","volume":"39","author":"Nasrin","year":"2012","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1492","DOI":"10.1016\/j.ijheatfluidflow.2007.02.004","article-title":"Temperature and particle-size dependent viscosity data for water-based nanofluids-hysteresis phenomenon","volume":"28","author":"Nguyen","year":"2007","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3107","DOI":"10.1063\/1.2093936","article-title":"Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement","volume":"87","author":"Chon","year":"2005","journal-title":"Appl. Phys. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1016\/j.enconman.2010.06.072","article-title":"Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids","volume":"52","author":"Corcione","year":"2011","journal-title":"Energy Convers. Manag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2522","DOI":"10.1016\/j.applthermaleng.2004.11.032","article-title":"Natural convective heat transfer in square enclosures heated from below","volume":"25","author":"Calcagni","year":"2005","journal-title":"Appl. Therm. Eng."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/5\/336\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:03:01Z","timestamp":1760194981000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/5\/336"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,3]]},"references-count":32,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2018,5]]}},"alternative-id":["e20050336"],"URL":"https:\/\/doi.org\/10.3390\/e20050336","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5,3]]}}}