{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T13:29:46Z","timestamp":1762176586332,"version":"3.41.2"},"reference-count":16,"publisher":"ASME International","issue":"9","content-domain":{"domain":["asmedigitalcollection.asme.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2012,9,1]]},"abstract":"<jats:p>In this work, we propose an approximate methodology to estimate the Nusselt number and friction factor in fully developed non-Newtonian laminar flow in circular pipes for a constant wall heat flux. The methodology was tested using several constitutive equations, including generalized Newtonian fluids and viscoelastic models such as the simplified Phan-Thien\u2013Tanner (sPTT), Herschel\u2013Bulkley, Bingham, Casson, and Carreau\u2013Yasuda constitutive equations. The error of the approximate methodology was found to be smaller than 3.2%, except for the fluids with yield stress for which the maximum error increased to about 8% for the cases analyzed, which cover a wide range of shear viscosity curves.<\/jats:p>","DOI":"10.1115\/1.4006288","type":"journal-article","created":{"date-parts":[[2012,7,9]],"date-time":"2012-07-09T22:35:51Z","timestamp":1341873351000},"update-policy":"https:\/\/doi.org\/10.1115\/crossmarkpolicy-asme","source":"Crossref","is-referenced-by-count":24,"title":["A Simplified Method for Calculating Heat Transfer Coefficients and Friction Factors in Laminar Pipe Flow of Non-Newtonian Fluids"],"prefix":"10.1115","volume":"134","author":[{"given":"D. A.","family":"Cruz","sequence":"first","affiliation":[{"name":"Departamento de Engenharia Mec\u00e2nica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]},{"given":"P. M.","family":"Coelho","sequence":"additional","affiliation":[{"name":"CEFT, Departamento de Engenharia Mec\u00e2nica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]},{"given":"M. A.","family":"Alves","sequence":"additional","affiliation":[{"name":"CEFT, Departamento de Engenharia Qu\u00edmica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]}],"member":"33","published-online":{"date-parts":[[2012,7,9]]},"reference":[{"key":"2019100516542355500_c1","first-page":"1","article-title":"Ueber die viskositat und elastizitat von solen","volume":"145","author":"Rabinowitsch","journal-title":"Z Physik Chem. Ser. A"},{"key":"2019100516542355500_c2","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1122\/1.2116364","article-title":"Explicit Formulas for Slip and Fluidity","volume":"2","author":"Mooney","journal-title":"J. Rheol."},{"key":"2019100516542355500_c3","doi-asserted-by":"publisher","first-page":"522","DOI":"10.1061\/(ASCE)0733-9429(1998)124:5(522)","article-title":"Pressure Loss Equations for Laminar and Turbulent Non-Newtonian Pipe Flow","volume":"124","author":"Chilton","journal-title":"J. Hydraul. Eng."},{"key":"2019100516542355500_c4","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1063\/1.1745165","article-title":"Simple Derivations of Some Important Relationships in Capillary Flow","volume":"4","author":"Schofield","journal-title":"Physics"},{"issue":"4","key":"2019100516542355500_c5","doi-asserted-by":"publisher","first-page":"434","DOI":"10.1002\/aic.690010409","article-title":"Flow of Non-Newtonian Fluids\u2014Correlation of the Laminar, Transition and Turbulent-Flow Regions","volume":"1","author":"Metzner","journal-title":"AIChE J."},{"key":"2019100516542355500_c6","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/S0377-0257(01)00159-8","article-title":"Study of Steady Pipe and Channel Flows of a Single-Mode Phan-Thien-Tanner Fluid","volume":"101","author":"Alves","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"2019100516542355500_c7","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1002\/cite.330280808","article-title":"W\u00e4rme\u00fcbergang an nicht-Newtonsche fl\u00fcssigkeiten bei laminarer rohrstr\u00f6mung","volume":"28","author":"Grigull","journal-title":"Chem.-Ing.-Tech."},{"key":"2019100516542355500_c8","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/S0017-9310(96)00094-4","article-title":"Fully Developed Laminar Forced Convection in Circular Ducts for Power-Law Fluids With Viscous Dissipation","volume":"40","author":"Barletta","journal-title":"Int. J. Heat Mass Transfer"},{"key":"2019100516542355500_c9","unstructured":"Cruz, D. A.\n          , 2010, \u201cC\u00e1lculo simplificado do coeficiente de atrito e do n\u00famero de Nusselt em escoamentos laminares de fluidos n\u00e3o-Newtonianos em condutas circulares,\u201d Final Project Report (in Portuguese)"},{"key":"2019100516542355500_c10","first-page":"10.12","volume-title":"Non-Newtonian Fluids. Handbook of Heat Transfer","author":"Hartnett","edition":"3rd ed."},{"key":"2019100516542355500_c11","doi-asserted-by":"publisher","first-page":"202","DOI":"10.1016\/j.jnnfm.2008.07.001","article-title":"A Generalized Brinkman Number for Non-Newtonian Duct Flows","volume":"156","author":"Coelho","journal-title":"J. Non-Newtonian Fluid Mech."},{"article-title":"Non-Newtonian Heat Transfer","volume-title":"Encyclopedia of Life Support Systems (EOLSS)","author":"Pinho","key":"2019100516542355500_c12"},{"key":"2019100516542355500_c13","doi-asserted-by":"crossref","first-page":"954","DOI":"10.1152\/jappl.1965.20.5.954","article-title":"Pressure-Flow Relations of Human Blood in Hollow Fibers at Low Flow Rates","volume":"20","author":"Merrill","journal-title":"J. Appl. Physiol."},{"key":"2019100516542355500_c14","doi-asserted-by":"publisher","first-page":"556","DOI":"10.1115\/1.2826016","article-title":"Laminar Convective Heat Transfer of a Bingham Plastic in a Circular Pipe With Uniform Wall Heat Flux: The Graetz Problem Extended","volume":"121","author":"Min","journal-title":"ASME Trans. J Heat Transfer"},{"key":"2019100516542355500_c15","doi-asserted-by":"crossref","first-page":"361","DOI":"10.3233\/BIR-1975-12606","article-title":"Heat Transfer to Blood Flowing in a Tube","volume":"12","author":"Victor","journal-title":"Biorheology"},{"key":"2019100516542355500_c16","first-page":"162","article-title":"The Carreau-Yasuda Fluids: A Skin Friction Equation for Turbulent Flow in Pipes and Kolmogorov Dissipative Scales","author":"Andrade","journal-title":"J. Braz. Soc. Mech. Sci. Eng."}],"container-title":["Journal of Heat Transfer"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/asmedigitalcollection.asme.org\/heattransfer\/article-pdf\/doi\/10.1115\/1.4006288\/5918361\/091703_1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/asmedigitalcollection.asme.org\/heattransfer\/article-pdf\/doi\/10.1115\/1.4006288\/5918361\/091703_1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,5]],"date-time":"2019-10-05T20:54:31Z","timestamp":1570308871000},"score":1,"resource":{"primary":{"URL":"https:\/\/asmedigitalcollection.asme.org\/heattransfer\/article\/doi\/10.1115\/1.4006288\/468170\/A-Simplified-Method-for-Calculating-Heat-Transfer"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,7,9]]},"references-count":16,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2012,9,1]]}},"URL":"https:\/\/doi.org\/10.1115\/1.4006288","relation":{},"ISSN":["0022-1481","1528-8943"],"issn-type":[{"type":"print","value":"0022-1481"},{"type":"electronic","value":"1528-8943"}],"subject":[],"published":{"date-parts":[[2012,7,9]]},"article-number":"091703"}}