Chapter7 CollegeofNuclearScienceandTechnology NaturalConvectionSystems 1 Chapter7 CollegeofNuclearScienceandTechnology 2 7 1Introduction Natural orfreeconvectionisobservedasaresultofthemotionofthefluidduetodensitychangesarisingfromtheheatingprocess Example Ahotradiatorusedforheatingaroomisoneexampleofapracticaldevicewhichtransfersheatbyfreeconvection Chapter7 CollegeofNuclearScienceandTechnology 3 Themovementofthefluidinfreeconvectionresultsfromthebuoyancyforcesimposedonthefluidwhenitsdensityintheproximityoftheheat transfersurfaceisdecreasedasaresultoftheheatingprocess Definition BodyForces Thebuoyancyforcesthatgiverisetothefree convectioncurrents Chapter7 CollegeofNuclearScienceandTechnology 4 NaturalConvection Inaninfinitespace Inenclosures Thedevelopmentofthermalboundarylayerwillnotbeinfluencedordisturbed Itdoesnotmeaninfiniteingeometry Thedevelopmentofthermalboundarylayerortheflowofthefluidswillbeinfluenced Chapter7 CollegeofNuclearScienceandTechnology 5 7 2Free ConvectionHeat transferOnAVerticalFlatPlate ConsidertheverticalflatplateshowninFigure7 1 Whentheplateisheated afree convectionboundaryisformed Thevelocityprofile Atthewallthevelocityis0becauseoftheno slipcondition itincreasestosomemaximumvalueandthendecreasesto0attheedgeofboundarylayersincethe free streamconditions areatrestinthefree convectionsystem 6 Chapter7 CollegeofNuclearScienceandTechnology Thetemperatureprofileandvelocityprofilealongverticalflatplate Chapter7 CollegeofNuclearScienceandTechnology 7 Theinitialboundary layerdevelopmentislaminar butatsomedistancefromtheleadingedge dependingonthefluidpropertiesandthetemperaturedifferencebetweenwallandenvironment turbulenteddiesareformed andtransitiontoaturbulentboundarylayerbegins Fartheruptheplatetheboundarylayermaybecomefullyturbulent Chapter7 CollegeofNuclearScienceandTechnology 8 Toanalyzetheheat transferproblem wemustobtainthedifferentialequationofmotionfortheboundarylayer ForthispurposewechoosethexcoordinatealongtheplateandycoordinateperpendiculartotheplateasintheanalysesofChapter5 Theonlynewforcewemustconsiderinthederivationistheweightofelementoffluid Equatethesumoftheexternalforcesinthexdirectiontothechangeinmomentumfluxthroughcontrolvolume Chapter7 CollegeofNuclearScienceandTechnology 9 Thepressuregradientinthexdirectionresultsfromthechangeinelevationuptheplate Thus Substitutingthetwoequations Chapter7 CollegeofNuclearScienceandTechnology 10 Thedensitydifferencemaybeexpressedintermsofvolumecoefficientofexpansion Sothat Theequationofmotionforthefree convectionboundarylayer Chapter7 CollegeofNuclearScienceandTechnology 11 Noticethatthesolutionforthevelocityprofiledemandsaknowledgeofthetemperaturedistribution Theenergyequationforthefree convectionsystemisthesameasthatforaforced convectionsystematlowvelocity Thevolumecoefficientofexpansion maybedeterminedfromtablesofpropertiesforthespecificfluid Chapter7 CollegeofNuclearScienceandTechnology 12 Foridealgas itmaybecalculatedfrom Eventhoughthefluidmotionistheresultofdensityvariations thesevariationsarequitesmall andasatisfactorysolutiontotheproblemmaybeobtainedbyassumingincompressibleflow thatis constant Chapter7 CollegeofNuclearScienceandTechnology 13 Forfree convectionsystem theintegralmomentumequationbecomes Weobservethatthefunctionalformboththevelocityandthetemperaturedistributionmustbeknowninordertoarriveatthesolution 7 6 Chapter7 CollegeofNuclearScienceandTechnology 14 Thefollowingconditionsapplyforthetemperaturedistribution Thetemperaturedistribution 7 7 Chapter7 CollegeofNuclearScienceandTechnology 15 Theconditionsforvelocityprofileare AnadditionalconditionmaybeobtainedfromEquation 7 4 bynotingthat Chapter7 CollegeofNuclearScienceandTechnology 16 Asintheintegralanalysisforforced convectionproblems weassumethatthevelocityprofileshavegeometricallysimilarshapesatvariousxdistancesalongtheplate Fortheconvectionproblem wenowassumethatthevelocitymayberepresentedasapolynomialfunctionofymultipliedbysomearbitraryfunctionofx Chapter7 CollegeofNuclearScienceandTechnology 17 Applyingthefourconditionstothevelocityprofilelistedabove wehave Theterminvolvingthetemperaturedifference andmaybeincorporatedintothefunctionsothat 7 8 Chapter7 CollegeofNuclearScienceandTechnology 18 AplotofEquation 7 8 isgiveninFigure7 2 Chapter7 CollegeofNuclearScienceandTechnology 19 SubstitutingEquations 7 7 and 7 8 intoEquation 7 6 andcarryingouttheintegrationsanddifferentiationsyields Theintegralformoftheenergyequationforthefree convectionsystemis 7 9 Chapter7 CollegeofNuclearScienceandTechnology 20 Andwhentheassumedvelocityandtemperaturedistributionsareinsertedintothisequationandtheoperationsareperformed theseresults WeassumethefollowingexponentialfunctionalvariationsforUxand 7 11 Chapter7 CollegeofNuclearScienceandTechnology 21 IntroducingrelationsintoEquations 7 9 and 7 11 gives And Chapter7 CollegeofNuclearScienceandTechnology 22 ThetwoequationsmaybesolvedfortheconstantsC1andC2togive Chapter7 CollegeofNuclearScienceandTechnology 23 Theresultantexpressionforthebound