Heat Chap14134.docx

Heat Chap14134.docx

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HeatChap14134

14-134Acircularpanfilledwithwateriscoolednaturally.Therateofevaporationofwater,therateofheattransferbynaturalconvection,andtherateofheatsupplytothewaterneededtomaintainitstemperatureconstantaretobedetermined.

Assumptions1Thelowmassfluxmodelandthustheanalogybetweenheatandmasstransferisapplicablesincethemassfractionofvaporintheairislow（about2percentforsaturatedairat25C）.2ThecriticalReynoldsnumberforflowoveraflatplateis500,000.3Radiationheattransferisnegligible.4Bothairandwatervaporareidealgases.

PropertiesTheair-watervapormixtureisassumedtobedilute,andthuswecanusedryairpropertiesforthemixtureattheaveragetemperatureof

=（15+20）/2=17.5C=290.5K.Thepropertiesofdryairat290.5Kand1atmare,fromTableA-15,

Themassdiffusivityofwatervaporinairattheaveragetemperatureof290.5Kis,fromEq.14-15,

Thesaturationpressureofwaterat20Cis

Propertiesofwaterat15Care

（TableA-9）.Thespecificheatofwaterattheaveragetemperatureof（15+20）/2=17.5CisCp=4.184kJ/kg.C.ThegasconstantsofdryairandwaterareRair=0.287kPa.m3/kg.KandRwater=0.4615kPa.m3/kg.K（TableA-1）.

Analysis（a）Theairatthewatersurfaceissaturated,andthusthevaporpressureatthesurfaceissimplythesaturationpressureofwateratthesurfacetemperature（1.7051kPaat15C）.Thevaporpressureofairfarfromthewatersurfaceisdeterminedfrom

Treatingthewatervaporandtheairasidealgasesandnotingthatthetotalatmosphericpressureisthesumofthevaporanddryairpressures,thedensitiesofthewatervapor,dryair,andtheirmixtureatthewater-airinterfaceandfarfromthesurfacearedeterminedtobe

Atthesurface:

and

Awayfromthesurface:

Notethat

andthusthiscorrespondstohotsurfacefacingdown.Theareaofthetopsurfaceofthewater

anditsperimeteris

.Therefore,thecharacteristiclengthis

Thenusingdensities（insteadoftemperatures）sincethemixtureisnothomogeneous,theGrashoffnumberisdeterminedtobe

Recognizingthatthisisanaturalconvectionproblemwithcoldhorizontalsurfacefacingup,theNusseltnumberandtheconvectionheattransfercoefficientsaredeterminedtobe（Eq.14-13）

and

Thentherateofheattransferfromtheairtothewaterbyforcedconvectionbecomes

（towater）

（b）Utilizingtheanalogybetweenheatandmassconvection,themasstransfercoefficientisdeterminedthesamewaybyreplacingPrbySc.TheSchmidtnumberisdeterminedfromitsdefinitiontobe

Therefore,theSherwoodnumberinthiscaseisdeterminedfromTable14-13tobe

UsingthedefinitionofSherwoodnumber,themasstransfercoefficientisdeterminedtobe

Thentheevaporationrateandtherateofheattransferbyevaporationbecome

and

（c）Thenetrateofheattransfertothewaterneededtomaintainitstemperatureconstantat15Cis

DiscussionNotethatifnoheatissuppliedtothewater（byaresistanceheater,forexample）,thetemperatureofthewaterinthepanwoulddropuntiltheheatgainbyconvectionequalstheheatlossbyevaporation.

14-135Airisblownoveracircularpanfilledwithwater.Therateofevaporationofwater,therateofheattransferbyconvection,andtherateofenergysupplytothewatertomaintainitstemperatureconstantaretobedetermined.

Assumptions1Thelowmassfluxmodelandthustheanalogybetweenheatandmasstransferisapplicablesincethemassfractionofvaporintheairislow（about2percentforsaturatedairat25C）.2ThecriticalReynoldsnumberforflowoveraflatplateis500,000.3Radiationheattransferisnegligible.4Bothairandwatervaporareidealgases.

PropertiesTheair-watervapormixtureisassumedtobedilute,andthuswecanusedryairpropertiesforthemixtureattheaveragetemperatureof

=（15+20）/2=17.5C=290.5K.Thepropertiesofdryairat290.5Kand1atmare,fromTableA-15,

Themassdiffusivityofwatervaporinairattheaveragetemperatureof290.5Kis,fromEq.14-15,

Thesaturationpressureofwaterat20Cis

Propertiesofwaterat15Care

（TableA-9）.Also,thegasconstantsofwaterisRwater=0.4615kPa.m3/kg.K（TableA-1）.

Analysis（a）Takingtheradiusofthepanr0=0.15mtobethecharacteristiclength,theReynoldsnumberforflowoverthepanis

whichislessthan500,000,andthustheflowislaminarovertheentiresurface.TheNusseltnumberandtheheattransfercoefficientare

Thentherateofheattransferfromtheairtothewaterbyforcedconvectionbecomes

（towater）

（b）Utilizingtheanalogybetweenheatandmassconvection,themasstransfercoefficientisdeterminedthesamewaybyreplacingPrbySc.TheSchmidtnumberisdeterminedfromitsdefinitiontobe

Therefore,theSherwoodnumberinthiscaseisdeterminedfromTable14-13tobe

UsingthedefinitionofSherwoodnumber,themasstransfercoefficientisdeterminedtobe

Theairatthewatersurfaceissaturated,andthusthevaporpressureatthesurfaceissimplythesaturationpressureofwateratthesurfacetemperature（1.7051kPaat15C）.Thevaporpressureofairfarfromthewatersurfaceisdeterminedfrom

Treatingthewatervaporandtheairasidealgases,thevapordensitiesatthewater-airinterfaceandfarfromthesurfacearedeterminedtobe

Atthesurface:

Awayfromthesurface:

Thentheevaporationrateandtherateofheattransferbyevaporationbecome

and

（c）Thenetrateofheattransfertothewaterneededtomaintainitstemperatureconstantat15Cis

DiscussionNotethatifnoheatissuppliedtothewater（byaresistanceheater,forexample）,thetemperatureofthewaterinthepanwoulddropuntiltheheatgainbyconvectionequalstheheatlossbyevaporation.

Also,therateofevaporationcouldbedeterminedalmostasaccuratelyusingmassfractionsofvaporinsteadofvaporfractionsandtheaverageairdensityfromtherelation

.

14-136Asphericalnaphthaleneballishangedinacloset.Thetimeittakesforthenaphthalenetosublimatecompletelyistobedetermined.

Assumptions1Theconcentrationofnaphthaleneintheairisverysmall,andthelowmassfluxconditionsexistsothattheChilton-Colburnanalogybetweenheatandmasstransferisapplicable（willbeverified）.2Bothairandnaphthalenevaporareidealgases.3Thenaphthaleneandthesurroundingairareatthesametemperature.4Theradiationeffectsarenegligible.

PropertiesThemolarmassofnaphthaleneis128.2kg/kmol.Becauseoflowmassfluxconditions,wecanusedryairpropertiesforthemixtureatthespecifiedtemperatureof298Kand1atm,atwhich

and

（TableA-15）.

AnalysisTheincomingairisfreeofnaphthalene,andthusthemassfractionofnaphthaleneatfreestreamconditionsiszero,wA,=0.Notingthatthevaporpressureofnaphthaleneatthesurfaceis11Pa,themassfractionofnaphthaleneontheairsideofthesurfaceis

Normallywewouldexpectnaturalconvectioncurrentstodeveloparoundthenaphthaleneballbecausetheamountofnaphthalenenearthesurfaceismuchlarger,anddeterminetheNusseltnumber（anditscounterpartinmasstransfer,theSherwoodnumber）fromEq.14-16,

Butthemassfractionvaluedeterminedaboveindicatesthattheamountofnaphthaleneintheairissolowthatitwillnotcauseanysignificantdifferenceinthedensityofair.Withnodensitygradient,therewillbenonaturalconvectionandthustheRayleighnumbercanbetakentobezero.ThentheNusseltnumberrelationabovewillreducetoNu=2oritsequivalentSh=2.ThenusingthedefinitionofSherwoodnumber,themasstransfercoefficientcanbeexpressedas

Themassofnaphthaleneballcanbeexpressedas

.Therateofchangeofthemassofnaphthaleneisequaltotherateofmasstransferfromnaphthalenetotheair,andisexpressedas

Simplifyingandrearranging,

Integratingfrom

attimet=0toD=0（completesublimation）attimet=tgives

Substituting,thetimeittakesforthenaphthalenetosublimatecompletelyisdeterminedtobe

14-137EAswimmerextendshiswetarmsintothewindyairoutside.Therateatwhichwaterevaporatesfrombotharmsandthecorrespondingrateofheattransferbyevaporationaretobedetermined.

Assumptions1Thelowmassfluxmodelandthustheanalogybetweenheatandmasstransferisapplicablesincethemassfractionofvaporintheairislow（about2percentforsaturatedairat60F）.2Thearmcanbemodeledasalongcylinder.

PropertiesBecauseoflowmassfluxconditions,wecanusedryairpropertiesforthemixtureattheaveragetemperatureof（40+80）/2=60Fand1atm,forwhich=0.159×10-3ft2/s,and=0.077lbm/ft3（TableA-15E）.Thesaturationpressureofwaterat40ºFis0.1217psia.Also,at80ºF,thesaturationpressureis0.5073psiaandtheheatofvaporizationis1048Btu/lbm（TableA-9E）.ThemolarmassofwaterisR=0.5956psia.ft3/lbm.R（TableA-1E）.Themassdiffusivityofwatervaporinairat60ºF=520R=288.9KisdeterminedfromEq.14-15tobe

AnalysisTheReynoldsnumberforflowoveracylinderis

TheSchmidtnumberinthiscaseis

Thenutilizingtheanalogybetweenheatandmassconvection,theSherwoodnumberisdeterminedfromEq.10-32byreplacingPrnumberbytheSchmidtnumbertobe

UsingthedefinitionofSherwoodnumber,themasstransfercoefficientisdeterminedtobe

Theairatthewatersurfaceissaturated,andthusthevaporpressureatthesurfaceissimplythesaturationpressureofwateratthesurfacetemperature（0.5073p