of Natural Convection Heat Transfer in Horizontal ...paper.uscip.us/AJHMT/AJHMT.2016.1024.pdf · Columbia International Publishing American Journal of Heat and Mass Transfer (2016)

ColumbiaInternationalPublishing American Journal of Heat and Mass Transfer(2016)Vol.3No.6pp.425‐452doi:10.7726/ajhmt.2016.1024ResearchArticle

______________________________________________________________________________________________________________________________*Correspondingemail:[email protected](M.M.Rahman),Fax:+968241414901DepartmentofMathematicsandStatistics,CollegeofScience,SultanQaboosUniversity,P.O.Box36,P.C.

123,Al‐Khod,Muscat,SultanateofOman2OnleavefromtheDepartmentofMathematics,JagannathUniversity,Dhaka‐1100,Bangladesh

425

InvestigationsofNaturalConvectionHeatTransferinNanofluidsFilledHorizontalSemicircular‐Annulususing

NonhomogeneousDynamicModel

M.J.Uddin1,M.S.Alam1,2,N.Al‐Salti1,andM.M.Rahman1*Received:28July2016;Publishedonline:5November2016©Theauthor(s)2016.Publishedwithopenaccessatwww.uscip.us

AbstractInthispaper,theproblemofunsteadyconvectiveflowofnanofluidinahorizontalsemicircular‐annulususingnonhomogeneous dynamic mathematical model has been investigated. The outer wall of the annulus isconsideredacolderwallandtheinnerismaintainedatthreedifferenttemperatures(constant,quadraticandsinusoidal) while two other walls are thermally insulated. The Galerkin weighted residual finite elementmethodhasbeenemployedtosolvethegoverningpartialdifferentialequationsafterconvertingthemintonon‐dimensionalformusingsuitabletransformationofvariables.Inthenumericalsimulations,thecobalt‐kerosenenanofluid has been taken to gain insight into the flow, thermal fields as well as concentration levels ofnanofluids. Local Nusselt number and temperature gradientmagnitude on the hotter and colder wall aredisplayedaslinegraphs.TheaverageNusseltnumberforcobalt‐kerosenenanofluidisdisplayedaslinegraphsfordifferent flowparameters including amount, shape and size of nanoparticles.Also, the averageNusseltnumber is presented as a bar diagram for 12 types of nanofluids. The result shows that 1‐20 nm sizenanoparticlesareuniformandstableinthesolution.TheaverageNusseltnumberincreasessignificantly,asnanoparticle volume fraction, Rayleigh number increases and nanoparticle diameter decreases. Sinusoidalheatinginnerwallofannulusexhibitshigherheattransferrate.AverageNusseltnumberofcobalt‐kerosenenanofluidismuchhigherthanthatofother11typesofnanofluidswhicharestudiedinthepresentanalysis.Keywords:Nanofluids;Semicircular‐annulus;Dynamicmodel;Heattransfer;Browniandiffusion;Thermophoresis

1. IntroductionNaturalconvectionheattransferinahorizontalannulusbetweenconcentricsemicircleshasbeenthefocusofmanyinvestigationsinrecentagesbecauseofthegrowingneedforabetterunderstandingofthisphenomenonintechnologicalandengineeringapplicationssuchaselectronicpackaging,food

M.J.Uddin,M.S.Alam,N.Al‐Salti,andM.M.Rahman/AmericanJournalofHeatandMassTransfer(2016)Vol.3No.6pp.425‐452

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process, cooling channels of nuclear reactors and solar collector‐receiver. There are severalresearchesonbothnumerical andexperimental investigationofnatural convectionheat transferbetweentwoconcentriccircularcylinders.Poweetal.(1971)studiednumericalsolutionfornaturalconvectionincylindricalannuliandclassifiedtheflowpatternsconferringtosuitablecombinationsoftheRayleighnumberandtheradiusratio.KuehnandGoldstein(1976)conductedanexperimentalandtheoreticalstudyofnaturalconvectioninanannulusbetweenhorizontalconcentriccylinders.Theyobserved that forair, the flowstarted tooscillatenearRayleighnumbersof 510 andsteadylaminar boundary layer regime exists between Rayleigh numbers of 43 10 and 510 . Onyegegbu(1986)studiedtheheattransferwithintheannulargapoftwoinfinitelylongisothermalhorizontalconcentriccylindersusingtheMilne‐Eddingtonapproximation.HehasshowedthatdecliningPlancknumber,increasingthedegreeofnon‐graynessofthefluidorincreasingopticalthickness,increasesthetotalheattransferanddiminishestheinducedbuoyantflowintensityandvelocities.KarkiandPatankar(1989)presentedanumericalstudyforlaminarmixedconvectionintheentranceregionofa horizontal annuluswith a fixed radius. Their result showed that as Grashof number increases,Nusseltnumberandfrictionfactorincreasesinbothdevelopingandfullydevelopedregions.Ghaddar(1992) reported the numerical results of natural convection from a uniformly heated horizontalcylinder placed in a large air‐filled rectangular enclosure and observed that flow and thermalbehaviordependonheatfluxesimposeontheinnercylinderwithintheisothermalenclosure.NoninoandGiudice(1996)conductedfiniteelementanalysisoflaminarmixedconvectionintheentranceregionofhorizontalannularductswithdifferent radiusratios.Cesinietal. (1999)conducted thenumericalandexperimentalanalysisofnaturalconvectionfromahorizontalcylinderenclosedinarectangularcavity.TheirresultshowedthattheNusseltnumberdistributionontheuppercylindersurface is effectively influencedby the buoyancyplumeoriginating from the lower cylinder.TheeffectofsurfaceradiationonconjugatenaturalconvectioninahorizontalannulusdrivenbyinnerheatgeneratingsolidcylinderisinvestigatedbyShaijaandNarasimham(2009).TheirresultshowedthatsurfaceradiationenhancestheoverallNusseltnumber.Mohammedetal.(2010)investigatedexperimentallytheforceandfreeconvectionforthermallydevelopingandfullydevelopedlaminarairflowinsidehorizontalconcentricannuli.Theirinvestigationrevealedthattheheattransferrateisgreaterinthedevelopingflowthanthefullydevelopedflowoverasignificantportionoftheannuli.There are many numerical and experimental investigations on convection heat transfer of theconcentricannulususingairandwaterasworkingfluids.ThenotableworksarefromLunbergetal.(1963),KuehnandGoldstein(1987),LeiandTrupp(1990),MoukalledandAcharya(1996),CharandHsu(1998),Haldar(1998)andYoo(1998).Bytheemergenceofnanoscienceandnanotechnology,nanofluidshavebeendevelopedtoenhancetheheattransferrateinsteadofconventionalheattransferfluidslikewaterandair.Nanofluidsareproducedusingbasefluidslikewater,ethyleneglycol,engineoil,pumpoil,glycerol,etc.and1‐100nm size particles aremade from variousmaterials. Nanofluids have significantly higher thermalconductivityandheattransferratethanconventionalheattransferfluids.ExcellentresultonthermalconductivityofsuspensionsbydispersingnanoparticleshasbeenfoundbyLeeetal.(1999),Dasetal.(2003)andXuanetal.(2003).TremendousresultsonheattransferenhancementofnanofluidshavebeenpublishedbyPakandCho(1998),XuanandLi(2003),WenandDing(2004),Yangetal.(2005)andHerisetal.(2006).AcomprehensivereviewonnanofluidthermalconductivityaswellasaugmentationofheattransfercoefficientanditsapplicationcanbefoundinUddinetal.(2016)andMolana(2016).Differenttypesofnanofluidsareavailablecommerciallyandusedsuccessfullyfortheenhancementofheat transfer.Among themCuO ‐water, 2 3Al O ‐water, ZnO ‐water,Cu-water,


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CNT-waternanofluidsareverycommon.Nowaday,researchonnanofluidscontaining 3 4Fe O ,cobaltandnickelasnanoparticlesisongoing.Thesetypesofnanofluidsshowedpromisingcharacteristicsto enhance heat transfer rate. Sheikholeslami and Ganji (2014) studied ferrohydrodynamic andmagnetohydrodynamic effects on ferrofluid flow in a semicircular annulus. They concluded thatincrementofsolidvolumefractionwiththeincrementsofRayleighnumberplayedpositiveroleforenhancementofheattransfer.The numerical investigations of natural convection heat transfer in different types of concentricannulusfilledwithnanofluidsarethemostrecentattentionbythedifferentinvestigators.Abu‐Nadaet al. (2008) investigatednatural convectionheat transfer enhancement in horizontal concentricannuli filled with nanofluid. They found that at low Rayleigh number, nanofluids containingnanoparticleswithhigherthermalconductivitycausemoreenhancementinheattransfer.Hussainand Hussein (2010) investigated natural convection phenomena in a uniformly heated circularcylinder immersed in square enclosure filledwith air at different vertical locations. Their resultshowedthatsmallRayleighnumbersdoesnothavemuchinfluenceontheflowfieldwhileathighRayleigh numbers have considerable effect on the flow pattern. Zi‐Tao et al. (2012) studied thetransientnaturalconvectionheattransferofaqueousnanofluids inahorizontalannulusbetweentwocoaxialcylinders.Theyfoundthatasnanoparticlevolumefractionisincreased,thetime‐averageNusselt number is gradually lowered at constant Rayleigh number. Soleimani et al. (2012)investigated natural convection heat transfer in a nanofluid filled semi‐annulus enclosure. TheirresultshowedthatthereisanoptimalangleofturninwhichtheaverageNusseltnumberismaximumforeachRayleighnumber.Yangetal.(2013)investigatedconvectiveheattransferofnanofluidsinaconcentric annulus and found that Nusselt number has optimal bulkmean nanoparticle volumefraction for alumina water nanofluid, whereas it only increases monotonously with bulk meannanoparticlevolumefractionfortitania‐waternanofluid.Therearemoretremendousefforthasbeenmadetoanalyzetheconcentricannulususingnanofluids.Thenotablenumerical investigationsofnaturalconvectionheattransferonconcentricannulususingnanofluidsincludeArefmaneshetal.(2012),Sheikholeslamietal.(2014),Seyyedietal.(2015)andBezietal.(2015).Ouraimistofindouttheenhancementofheattransferrateinasemicircular‐annulusfordifferentnanofluidsusingournewlyproposednonhomogeneousdynamicmathematicalmodel(Uddinetal.(2016)).Thereasonofchoosingsemicircular‐annulusshapegeometryisthatithashugepotentialityin the fieldofnuclear, civil,mechanicalandarchitecturalengineering.Specially, thesemicircular‐annulushasacomprehensiveprospectforuseinthecoolingchannelofnuclearreactor,receiverofsolarthermalcollector,aductbywhichheatcanbepassedthroughforthesecurityandsafetyoftheindustry, the design of the electronic and electrical equipment for reducing heat. In engineeringapplications,thegeometrythatoccursismorecomplicatedthananenclosurecontainingconvectivefluids.However,theimportantinsightsofheattransferoftheverycomplexdesigncanbeprovidedbythesimplegeometry.Thegeometricalconfigurationofconcerniswiththeexistenceofphysicalstructureembeddedwithintheenclosure.Nevertheless,comparatively,littleworkhasbeendoneonnatural convectionheat transfer insemicircularannuli.Asperauthors’knowledge, the literaturereview reveled that three different boundary conditions taking into nonhomogeneous dynamicmathematicalmodel inasemicircular‐annulusshapeenclosurefilledwithnanofluidhasnotbeenstudiedyet.Thestudyhasbeencarriedoutnumericallywithanaccuratenumericalprocedure,andthe related results have been shown graphically for cobalt‐kerosene nanofluid in the form ofstreamlines, isotherms, and isoconcentrations. Moreover, the heat transfer rates of 12 types of


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nanofluidshavebeendisplayedinabardiagramforthreedifferentheatsourceswhichisveryrareintheheattransferanalysis.Sincetheenclosureintroducedinthispapercanhaveaninordinateuseinnuclearreactorandmakingsolarcollectors,thenumericalanalysispresentedcanhelptopromotetheuseofrenewableenergy.

Fig.1.Schematicviewofthemodelwithboundaryconditions.

2.ProblemFormulations2.1Physicalmodeling

Aschematicgeometryofthetwo‐dimensionalproblemunderinvestigationisshowninFig.1,wherex and y are the Cartesian coordinates. We have considered a semicircular‐annulus filled with

nanofluids having inner radius ir and outer radius or . The three different thermal boundary

conditions of the inner semicircular curve of the annulus is modeled as ,hT T( )( / 1 / 4)(3 / 4 / )C h CT T T T x L x L and ( )sin(4 / )C h CT T T T x L , where / 4 o iL r r is

thediameterofthesemi‐annulus.Twootherhorizontallinesaremodeledasadiabaticassumingnoheatescapesthroughtheseboundaries.Thetemperatureoftheoutersemicircularboundaryoftheenclosureislowasheatgoesthroughitandmodeledas CT T .Initially,itisassumedthatnanofluid

concentrationiskeptatlowconcentration CC butfor 0t ,itisassumedas hC intheentiredomain

sothat h CC C .Thermophoresis,Browniandiffusionandgravityeffectsareincludedinourstudyintheabsenceofthermalradiationandchemicalreaction.Thebasefluidandthesolidnanoparticles


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arethermallyequilibrium.Cobalt‐kerosenenanofluidasdefaulthasbeenanalyzedintermsofflow,thermalandconcentrations fieldsaswellasheat transferenhancement. Inaddition,12‐ typesofnanofluidshavebeenconsideredforthebestperformersofheattransferenhancementcomparedtothebasefluids.2.2Mathematicalmodeling

2.2.1Conservationequationsfornanofluids

Toconstructconservationequationsofnanofluidsforaccurateresultswehaveconsideredtheslipmechanisms. For the present study, we have also considered the Brownian motion of thenanoparticles, thermophoreticdiffusionandgravitationaleffectas slipmechanisms.Weassumedthatthebasefluidsandthenanoparticlesareinthermallyequilibrium,therearenochemicalreactionbetweensolidsandbasefluids,andnoradiativeheattransfer.Since,thesizeofnanoparticlesisverysmallandtheyareeasilyfluidized,wetreatednanofluidasafluidcontainingnanoparticlesasspeciesofbasefluids.Astheconcentrationisveryimportantinnanofluidstounderstandthedensenessaswell as stable situation of nanoparticles in the base fluids, we have considered the nanofluidsconcentration equation that has been neglected in well‐known one‐component as well as two‐componentmathematicalmodels.Therefore,undertheaforementionedassumptions,thedynamicunsteady conservation equations namely, the continuity, momentum, energy and concentrationequationsfornanofluidshavebeenconstructedasfollows(Uddinetal.2016)):Thenanofluidcontinuityequation:

0u v

x y

(1)

Thenanofluidmomentumequationin x ‐direction:2 2

2 2( )nf nf

u u u p u uu v

t x y x x y

(2)

Thenanofluidmomentumequationin y ‐direction:2 2

*2 2

( ) ( ) ( ) ( ) ( )nf nf nf c nf c

v v v p v vu v g T T g C C

t x y y x y

(3)

Thenanofluidenergyequation:222 2

2 2 + B T

nf

T T T T T D C T C T D T Tu v

t x y x y C x x y y T x y

(4)

Thenanofluidconcentrationequation:2 2 2 2

2 2 2 2T T

B

C C C C C C D T T D C T C Tu v D

t x y x y T x y T x x y y

(5)

where ,u v arethevelocityalong ,x y coordinates,respectively; p isthepressure, g isthegravity,T is the temperature, CT is the reference temperature, C is the concentration and CC is the

reference concentration of nanofluids. Here, nf is the dynamic viscosity of nanofluid, nf is the

density of nanofluid,( )

nfnf

p nfc

is the thermal diffusivity of nanofluid, nf is the thermal

conductivityofnanofluid, ( )p nfc istheheatcapacityofnanofluid, ( )nf isthevolumetricthermal


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expansion of nanofluid and *( )nf is the volumetric mass expansion of nanofluid. The popular

relations of effective viscosity, density, heat capacitance, volumetric thermal expansion, andvolumetricmassexpansionofnanofluidrespectivelyaregivenby

2.5(1 )nf bf (6)

(1 )nf bf p (7)

(1 )( ) ( )p p bf p pnfc c c (8)

(1 )( ) ( )bf pnf (9)

* * *(1 )( ) ( )bf pnf (10)

Themostpopularthermalconductivitymodelsforthistypeofsolid‐liquidmixturearetheMaxwellmodel(1873)andHamiltonandCrosser(1962).Inthesemodels,thethermalconductivityisdefinedasfollows:

2 2 ( )

2 ( )bf p bf p

n f bfbf p bf p

k k f k kk k

k k f k k

+ - -=

+ + - (11)

( 1) ( 1) ( )

( 1) ( )p bf bf p

nf bfp bf bf p

n n

n

k k f k kk k

k k f k k

é ù+ - - - -ê ú= ê ú+ - + -ê úë û (12)

Equation(12)representsthethermalconductivitywithparticlesshapefactor 3 /n ,where isthesphericitydefinedastheratiobetweenthesurfaceareaofthesphereandthesurfaceareaoftherealparticlewithequalvolumes.Thevaluesof areevaluatedexperimentally0.81,0.62,0.52and0.36 for brick, cylinder, platelet and blade shape nanoparticle respectively (see Timofeeva et al.,2009).Hence,Maxwell’s(1873)thermalconductivityequationisdefinedfortheshapefactor 3n ,whichrepresentsthesphericalshapeparticleinthebasefluidandHamiltonandCrosser’s(1962)thermal conductivity of heterogeneous two‐component system includes empirical shape factorwhere 3.7,n 4.9n , 5.7n and 8.6n represents brick, cylinder, platelet and blade shapenanoparticlerespectively.Furthermore,Xuanetal.(2003)consideredtheBrownianmotionofnanoparticleswiththeMaxwellmodelandproposedamodifiedformulaforthedynamiceffectivethermalconductivitymodelwhichisasfollows:

2 2 ( ) ( ) 2

2 ( ) 2 3bf p bf p p p B C

nf bfbf p bf p nf p

c k T

d

k k f k k r fk k

k k f k k pm+ - -

= ++ + -

(13)

where, Bk is the Boltzmann constant, pd is the diameter of a nanoparticle. In equation (13), the

secondtermontherighthandsideisduetotheBrownianmotionofnanoparticles.Therefore,consideringshapeofthenanoparticlestheequation(13)canbewrittenwithHamiltonandCrosser’s(1962)thermalconductivityequationas

( 1) ( 1) ( ) ( ) 2

( 1) ( ) 2 3bf p bf p p p B C

nf bfbf p bf p nf p

n n c k T

n d

k k f k k r fk k

k k f k k pm- + - - -

= +- + + -

(14)

Thismodel(equation(14))hasbeenusedinthepresentstudyinordertodefinetheeffectivethermalconductivityofnanofluids.TheBrowniandiffusioncoefficientfornanoparticlesisdefinedby,


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3B C

Bnf p

k TD

d (15)

Thethermophoreticvelocityequationfornanofluidsisdefinedas(seeUddinetal.,2015)

0.126 (ln )nf bf nfT

bf nf

V Tk b mk r

=

(16)

where, bfb isthethermalexpansionofthebulkfluidandthesymbol isacorrectionfactordependsonthesizeandshapeoftheparticlesas 0.0002 0.1537pd .Foraparticlesizeof100nm,the

valuesof isfound0.1337andfortheparticlesizeof1nmthevalueof is0.1535.Thethermaldiffusioncoefficient, TD canbeincurredfromthethermophoreticvelocityequationas(Uddinetal.,2015)

0.126 nf bf nfT

bf nf

Dk b mk r

=

(17)

Thus, for water and hexane based nanofluids with polystyrene latex nanoparticles at roomtemperature, thermal diffusion coefficient, TD can be obtained from equation (17) as

12 2 -1 -13.639 10 m s KTD .ThisresultisveryconsistentwiththeexperimentalresultofShiunduetal.(2003)andIacopinietal.(2006).2.2.2BoundaryconditionsTheappropriateinitialandboundaryconditionsforthepresentproblemalongwiththeabovestatedmodelareasfollows:For 0t ,entiredomain: 0u v , CT T , CC C , 0p (18a)For 0t ,Ontheinnersemicircularwall:Case‐1: 0u v , hT T , hC C (18b)Case‐2: 0u v , ( )( / 1 / 4)(3 / 4 / )C h CT T T T x L x L , hC C (18c)Case‐3: 0u v , ( )sin(4 / )C h CT T T T x L , hC C (18d)Ontheoutersemicircularwall: 0u v , CT T , hC C (18e)

Onthetwohorizontalinsulatedwalls: 0u v , 0T

n

, hC C (18f)

2.2.3Non‐dimensionalgoverningequationsTodescribeseveraltransportmechanismsinnanofluids,itismeaningfultomaketheconservationequations non‐dimensional. The main benefits of non‐dimensionalization are (i) one can easilyunderstandthecontrollingflowparametersofthesystem,(ii)getridofdimensionalconstraintand(iii) make a generalization of the size and shape of the geometry. For this purpose, the aboveequationscanbeconvertedtonon‐dimensionalforms,usingthefollowingdimensionlessquantities:

bf

uLU

,

bf

vLV

,

xX

L ,

yY

L , CT T

T

,

2

2bf bf

pL

,

2

bf t

L

, CC C

C

(19)


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where, bf , L , h CT T T , h CC C C and CT and CC arethethermaldiffusivityofthebasefluid,

referencelengthofthegeometry,thetemperaturedifference,thenominalconcentrationdifference,referencetemperatureandreferenceconcentrationwithinthenanofluid,respectively.Introducingtheabovetransformationsintotheconservationequations(1)‐(5),assumingconstantpropertiesofnanofluids and recognizing that ( / ) 1CT T and ( / ) 1,CC C the governing equations (1)‐(5)canbewrittenindimensionlessformsas:Continuityequation:

0U V

X Y

(20)

Momentumequationin X ‐direction:2 2

2 2Pr( )bf nf

nf bf nf

U U U U UU V

X Y X X Y

(21)

MomentumequationinY ‐direction:

2 2

2 2

( )Pr( ) Pr Prbf nf nf

T Cnf bf nf bf nf

V V V V VU V Ra Ra

X Y Y X Y

(22)

Energyequation:2 22 2

2 2

1 Prnf TBT

bf

NU V

x y X Y Le X X Y Y Sc X Y

(23)

Concentrationequation:2 2 2 2

2 2 2 2

Pr Pr + TBTC TBTU V N N

X Y Sc X Y Sc X Y X X Y Y

(24)

The dimensionless parameters introduced in the above equations (20)‐(24) are as follows:

3bf

Tbf bf

L g TRa

is the local thermal Rayleigh number,

*3 ( )nfC

bf bf nf

L g CRa

is the local solutal

Rayleighnumber, Pr bf

bf

isthePrandtlnumber,( )

bf C

p bf B

CLe

c D C

isthemodifiedLewisnumber,

TTBT

B C

D TN

D T

is the dynamic thermo‐diffusion parameter, T C

TBTCB C

D T CN

D C T

is the dynamic

diffusionparameterand bf

bf B

ScD

istheSchmidtnumber.

2.2.4Non‐dimensionalboundaryconditionsTheinitialandboundaryconditionsinthedimensionlessformforthepresentproblemcanbewrittenas:For 0 ,entiredomain: 0, 0, 0U V , P 0 (25a)For 0 ,


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Ontheinnercircularwall:Case‐1: 0, 1, 1U V (25b)

Case‐2:1 3

0, ( )( ), 1, 0.25 0.754 4

U V X X X (25c)

Case‐3: 0, sin(4 ), 1, 0.25 0.75U V X X (25d)Ontheoutercircularwall: 0,U V 1 (25e)

Onthetwohorizontalinsulatedwalls: 0, 0,U Vn

1 (25f)

3.ThermophysicalPropertiesThethermophysicalpropertiesofnanofluidsdirectlydependonthethermophysicalpropertiesofnanoparticlesandbase fluids.Thermophysicalpropertiesofbase fluidsandnanoparticlescanbesimplydefinedasthecharacteristicsoffluidssystemaswellasthematerialpropertiesthesevarieswiththestatevariablestemperatureandpressurewithoutalteringtheirchemicalidentity.Forourcurrent nanofluid research, we use specific heat capacity, density, thermal conductivity andvolumetricthermalexpansioncoefficientsofbasefluidsandparticles,aswellastheviscosityofthebasefluidsatroomtemperature.ThesethermophysicalpropertiesareshowninTable1(OztopandAbu‐Nada,2008;RahmanandAziz,2012;Mutuku,2014).

4.NusseltNumberandTemperatureGradientMagnitude

Thelocalnusseltnumberontheheatsourcesurfacecanbeexpressedas

nfL

bf

NuY

(26)

TheaverageNusseltnumberisevaluatedbyintegrating LNu alongtheheatsource

40

4nfave

bf

Nu dXY

27)

Thetemperaturegradientmagnitudeoftheflowofnanofluidcanbedefinedas

2 2

TGMX Y

(28)


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Table1Thermophysicalpropertiesofbasefluidsandmaterialsatroomtemperature.

Items

pc -1 -1(Jkg K )

3(kg m )

-1 -1(Wm K )

1 1(kgm )s

1(K )

Water 2(H O) 4179 997.1 0.613 0.001003 521 10

Kerosene 2090 780 0.149 0.00164 599 10

2 3Al O 765 3970 40 50.85 10

2TiO 686.2 4250 8.9538 50.9 10

CuO 531.8 6320 76.5 51.8 10

ZnO 495.04 5610 29 54.7 10

3 4Fe O 670 5180 80.4 520.6 10

Cobalt 420 8900 100 51.3 10

5.ComputationalProceduresTheGalerkinweightedresidualschemeoffiniteelementmethod(FEM)hasbeenappliedtosolvethegoverning dimensionless equations (20)‐(24) together with the boundary conditions (25). Thedetailswiththebenefitsof thisnumericalmethodarewellanalyzedbyZienkiewiczetal. (2005),Codina(1998)andAlKalbanietal.(2016).Inthismethod,thesolutiondomainisdiscretizedintofiniteelementmeshes,whicharecomposedofnon‐uniformtriangularelements.Sixnodetriangularelementsareusedinthisworkforthedevelopmentofthefiniteelementequations.Allsixnodesareassociated with velocities, temperature as well as isoconcentration; only the corner nodes areassociatedwithpressure.Thismeansthatalowerorderpolynomialischosenforpressureandwhichis satisfied through the continuity equation. Then the nonlinear governing partial differentialequations (i. e. conservation of mass, momentum, energy and concentration equations) aretransferredintoasystemofintegralequationsbyapplyingGalerkinweightedresidualmethod.Theintegration involved in each term of these equations is performed by using Gauss's quadraturemethod. The nonlinear algebraic equations so obtained aremodified by imposition of boundaryconditions.Tosolve thesetof theglobalnonlinearalgebraicequations in the formofmatrix, theNewton‐Raphsoniterationtechniquehasbeenadaptedthroughpartialdifferentialequationsolverwith MATLAB interface. The convergence criterion of the numerical solutions along with errorestimationhasbeensetto 1 510m m ,where isthegeneraldependentvariable ( , , , )U V

andm isthenumberofiteration.


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Fig.2.Grid sensitivity test for cobalt‐kerosene nanofluidwith 510TRa , 10 nmpd , =0.05 at

1 .5.1GridindependencytestTo test and assess grid independenceof the solution for cobalt‐kerosenenanofluid, an extensivemesh testing technique has been conducted for case‐3 with the selected values of 510TRa ,

310CRa , 0.05 , 10 nmpd , 3n (sphericalshapenanoparticle)at 1 .Inthepresentwork,weinspectsixdifferentnon‐uniformgridsystemswiththefollowingnumberofelementswithintheresolutionfield:1276,2012,3064,9080,24020and33372.ThenumericaldesigniscarriedoutforhighlyprecisekeyintheaverageNusseltnumber ( )aveNu attheheatedinnersemicircularwallfortheaforesaidelementstodevelopanunderstandingofthegridfinenessasshowninFig.2.Ascanbeseen that the average Nusselt number enhances, as the number of triangular elements of thegeometryincreasesuptoacertainstageandafterthatitbecomesplateau.Theseindicatethataftergeneratingcertainnumberofelementsinthemesh,alltheresultsareindependentofmeshsizes.Inthe present study, the scale of aveNu for 24020 elements shows a very little differencewith theresultsobtainedfortheelementsof33372.Hence,fromthenumericalexperiment,24020elementsarefoundtomeettherequirementofthegridindependencystudy.5.2TheflowparametersTheparametersoftheflowarecalculatedbytheirdefinitionwiththethermophysicalpropertiesofthenanoparticlesandbase fluids.Thevaluesof thePrandtlnumberarecalculated forwaterandkerosene as 6.8377 and 23.004 respectively. Let us consider 10KT , 0.01C , 300K,CT

1,CC 10 nmpd , 3n and 0.05 .Withthesevalues,thephysicalparametersenteredintothe


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equations (20)‐(24) for cobalt‐kerosene nanofluid can be calculated as follows: 53.8793 10Le ,89239Sc , 112.3561 10 ,BD 114.0352 10 ,TD 5.7089TBTCN and 0.057089TBTN .Theother

parameters such as the Rayleigh number ( )TRa , the solutal Rayleigh number ( )CRa are varied toanalyzetheflowandthermalcharacteristicsofaspecificnanofluid.Asthethermophysicalpropertiesofnanofluidsdependonthethermophysicalpropertiesofnanoparticlesandbasefluids,thevaluesoftheabovestatedparameterswillbedifferentforeachnanofluid.ItisimportanttonotethatastheBrownian and thermophoretic diffusions strongly depend on the diameters of nanoparticles, theBrowniandiffusioncoefficient ( )BD andthethermaldiffusioncoefficient ( )TD changeswiththesizeofthenanoparticles.Also,thevaluesoftheparametersespecially ,Le ,Sc TBTCN and TBTN willchangeaccordingtothesizeoftheparticles,nanoparticlesvolumefractionandthermophysicalpropertiesofnanofluids.

7.ResultsandDiscussionInthissection,theobtainednumericalresultsfornaturalconvectiveheattransferofnanofluidsinasemicircular‐annulusshapedenclosurewiththreedifferentthermalboundaryconditions(case‐1:uniform,case‐2:parabolicandcase‐3:sinusoidaldistributionoftemperature)attheinnercircularwall are discussed. The physical phenomenon is investigated for a wide range of controllingparameters.Inthenumericalsimulation,wehaveconsideredcobalt‐kerosenenanofluidasdefaultforthedescriptionoftheflow,thermalandconcentrationfieldsaswellasaverageNusseltnumberfordifferentmodelparametersoftheproblem.Themostimportantfeaturesofnanofluidsaretheeffect of nanoparticle in terms of amount, size and shape, are also investigated and displayedgraphically.Theratioof thebuoyancydrivenparameters TRa and CRa arekept fixed to 210 .Theresultsare analyzedvaryingnanoparticlevolume fraction ( 0.0, 0.05, 0.10 ),nanoparticle size

( 1, 20, 30 nm)pd andRayleighnumbers ( 510TRa , to 710TRa )onstreamlines, isothermsandisoconcentrations.TheaverageNusseltnumberandtemperaturegradientmagnitudeversus X ,ontheinnercircularhotwallandoutercolderwallforthreedifferentcasesaredisplayedaslinegraphs.Finally,theaverageNusseltnumberof12typesofnanofluidsfordifferentsolidvolumefractionforthreedifferentthermalboundarycaseshasbeendisplayedinabardiagram.Abriefalbeitsufficientexplanationfollowseachresult.Isothermsarehelpfultodetecttheeffectivenessofheattransferinafluidandalsotellusaboutthedictating mode of heat transfer whether it is strong convection or weak convection. Figure 3representstheeffectofsolidvolumefractiononisothermsforthreedifferentboundaryconditionsontheinnersemicircularwallwith 3n , 10 nmpd , 710TRa at 1 .Ascanbeseenthattherearemushroom like isotherm distributions from the inner wall to the upper curved wall and asnanoparticlevolumefractionincreases,thedisseminationofisothermslittlebitcompressestotheinnerhotwallthroughoutthecases.Itisalsointerestingtonotethatisothermsaresymmetricallydistributed relative to the annulus for both case‐1 and case‐2 whereas, two mushroom likeasymmetricisothermsspreadintheannulusforcase‐3.However,thepatternofisothermsbycase‐1andcase‐3aremoredispersive,full‐bodiedanddistortedinthemiddleoftheannulusthanthatofcase‐2.Thisindicates,convectionmodeofheattransferishigherandstrongerforcase‐1andcase‐3thanthatofcase‐2.Forarelativecomparisonofisothermsbetweenusingcase‐1andcase‐3,case‐3givesmoredistortedisothermsthancase‐1whichisaclearindicationthathigherheattransferrate


437

occursforcase‐3.Ascase‐3hassignificantpotentialitytoriseaugmentationofheattransferrate,thiscondition is worthwhile to understand the convection, flow field and concentration levels ofnanofluidintheannulus.

0.0

0.05

0.1

Case‐1 Case‐2 Case‐3

Fig.3.Effectof onisothermsforselectedvaluesof 10 nmpd , 710TRa , 3n at 1 for

threedifferentthermalboundaryconditions.Theeffectofnanoparticlediameteronisoconcentrationsforselectedvaluesof TRa with 0.05 ,

3n at 1 hasbeendisplayed inFig.4.Ascanbeseen that isoconcentrationsofCo ‐kerosenenanofluid represented by three loops in the annulus due to the sinusoidal boundary conditions.Nanoparticle diameter determines the concentration levels of nanofluid in the annulus. Asnanoparticle diameter increases, the isoconcentrations in the loops becomeweaker which is anindicationoflowerheattransferrateforhighervaluesofparticlediameter.Furthermore,densityofisoconcentrationsdecreaseswhenparticlediameter increases. Itcanalsobeseenthatasparticledimeterexceeds20nm,isoconcentrationsclustersonthewallsandnotuniformintheannulus.Thestrength and density of isoconcentrations in the middle of the annulus slightly increases for

20 nmpd whereas, remarkable enhancement can be found for particle diameter 1‐20 nm, asbuoyancydrivenparameterincreases.ThesemaybeduetothehigherBrowniandiffusionforlowerdiameterofparticle.Itisagoodindicationfromthisfigurethatnanoparticlesinthebasefluidbecomeuniformandstableinthesemicircularannulusforparticlediameterwithintherangeof1‐20nm.


438

1

20

30

pd

(nm)

510TRa 610TRa 710TRa

Fig.4.Effectof pd onisoconcentrationsforselectedvaluesof TRa with 0.05 , 3n at 1 .

1

20

30

pd

(nm)


Fig.5.Effectof pd onisoconcentrationsforselectedvaluesof TRa with 0.05 , 3n at 10 .


439

Figure5illustratesthecontinuationeffectofthenanoparticlesdiameteronisoconcentrationforthetime domain, 10 (steady state). It is clearly seen that strength of isoconcentration increasessignificantlythanitwasinthetime, 1 (unsteadystate).Thedensityofisoconcentrationdecreasesas the nanoparticles diameter and buoyancy driven parameter, Rayleigh number increases. Thesimilarresultsarefoundforthenanoparticlesdiameter, 20 nmpd ,asitwasinthetimedomain,

1 .Itcanbefoundfromthefigurethatfortheparticlediameter, 30 nmpd ,theisoconcentrationsbecomesignificantlyweakercomparedtootherisoconcentrationsfornanoparticlesdiameter(1‐20nm). Sowe can summarize thatnanofluidholding1‐20nmsizeparticles areuniformand stablesolutioninthesemicircularannulus.Figure6depicts the influenceofaddingnanoparticles to the fluid inside theannulusbyshowingstreamlinemapsagainst TRa with 10 nmpd , 3n at 1 .Itisseenthattherearethreerotatingvorticesinsidetheannuluswheretwovorticesnearthecorneroftheannulusrotatingclockwiseandotheronerotatingcounterclockwiseinthemiddle.Thispatternistheresultofthermalboundarycondition at the inner wall. As buoyancy driven parameter, TRa increases, distributions ofstreamlinesarefoundintheentireannulus,thatis,afullbodiedflowoccurs.Thestrengthofthefluidcurrentsincreaseswith TRa ;ashigher TRa resultsinhigherconvectiveforce.Anotherinterestingobservation is, as nanoparticle volume fraction increases, vortices inside the annulus slightlycompressesanddensityofstreamlinesincreases.Thesephenomenacanbedescribedbythefactthatnanoparticlesincreasethetotalmassofthefluidwhichisreflectedashigherinertiaofthefluid.Thishigherinertiaiscausingthefluidtoslowdownabit.

0.0

0.05

0.1

510TRa 610TRa

710TRa

Fig.6.Effectof onstreamlinesforselectedvaluesof TRa forcase‐1with 10 nmpd , 3n

at 1 .


440

Figure 7 shows the isothermpattern for thepresent problemunderdifferent values of TRa and

nanoparticlevolume fraction, with 10nmpd , 3n andcase‐1at 1 .For thebase fluid; theisotherms are more dispersive and almost parallel to each other in the entire annulus. Asnanoparticlevolumefractionincreases,isothermsarestronger,rigidandnotparallelinmostoftheregion of the annulus. This is due to heat diffusion by nanoparticles. Furthermore, the isothermpatternseemsto indicate thatat lowvalueof TRa , convection isweaker insidethecavityas twoloopsofisothermsareseenalmostparalleltoeachotherneartheheatedinnerwall.Asthevalueof

TRa increases,theisothermsbecomemoreandmoredistortedandformingtwodistortedmushroomlike isothermpatterns in the twosidesof theannulus.Thisparticularpattern suggests thatheatenergyisflowingstronglyintothefluidfromtheinnerheatedcurvedwallandconvectionmodeofheattransferdominates.

0.0

0.05

0.1


Fig.7.Effectof onisothermsforselectedvaluesof TRa forcase‐1with 10 nmpd , 3n at

1 .Theeffectofsolidvolumefractiononisoconcentrationsforselectedvaluesof TRa forcase‐1with

10 nmpd , 3n at 1 has beenportrayed in Fig. 8. It is observed that isoconcentrations aremoreorlessuniformforallcasesintheentiresemicircularannulus;however,isoconcentrationsofnanofluidareslightlymorenoticeablethanbasefluid.Asthenanoparticlevolumefractionincreasesthelevelofisoconcentrationsalsoincreases.Thisisduetothemoreinteractionsamongparticlesinfluid.Thestrengthoftheisoconcentrationsincreaseswhen TRa increases.Itisinterestingtoseethattheeffectofsolidvolumefractionsonthepatternofisoconcentrationsinannulusisalmostsimilarto thatof streamlines.Thiskindofoutcome is very rare inother typesof enclosure.This clearlyindicates that annulus shape enclosure plays very important role to obtain full‐bodied flow anduniformisoconcentrationsofnanofluid.


441

0.0

0.05

0.1


Fig.8. Effect of on isoconcentrations for selected values of TRa with 10 nmpd , 3n , at

10 .Figures9(a)‐9(c)andFigures10(a)‐10(c)respectively,showvariationoflocalNusseltnumberalongtheinnerheatedcurvewallandupperrelativelycolderwallfordifferentsolidvolumefractionusing(a)case‐1,(b)case‐2,and(c)case‐3thermalconditionswith 10 nmpd , 710TRa , 3n at 1 .ItisobservedfromthefiguresthatlocalNusseltnumberonbothheatedandrelativelycolderwallisremarkably higher for nanofluid than that of base fluid. It is also found that if there is morenanoparticles,therewillbemoreheattransferrate.FromFig.9(a),itisfoundthatdistributionoflocalNusseltnumberattheheatedwallfornanofluidisalmostaparabolaopeningupwardswhereas,itisU‐shapedforbasefluidwithaxisofsymmetryat 0.5X .Inthiscase,thehighestvalueoflocalNusseltnumber is foundat thecornersof theannulus.However, thedistributionof localNusseltnumberatuppercolderwallfornanofluidisacompleteparabolaopeningdownwardandforbasefluid,itisalmostastraightlinewithalittlebumpinthemiddle(Fig.10(a)).Inthiscase,themaximumvalueoflocalNusseltnumberisfoundatthemiddleoftheuppercurvedwall.Thesedistinguishablephenomenaareduetonanoparticlesinthesolutionwherenanoparticlescarrysignificantamountofheatfrombottomwalltouppercolderwall.Fig.9(b)representstheMshapedistributionoflocalNusselt numberwith twopeaks at the two sides of the heatedwall, though there is a quadraticthermal boundary condition. Nevertheless, at the outer cold wall, 10(b) also shows parabolicdistributionoflocalNusseltnumber.Interestingresultsarefoundforcase‐3.FromFig.9(c)andFig.10(c), it isdemonstratedthatlocalNusseltnumberisnoticeablyhigherfornanofluidthanthatofbasefluidcomparetoothertwocases.LocalNusseltnumberismaximumfornanofluidonthehotterwall at 0.55X andon theouterwall at 0.8X ;whichclearly indicates thatheat isdiffusedbynanoparticleintheannulus.ThedistributionoflocalNusseltisstronglywavyfornanofluidthanthatofbasefluid.Theseclearlyindicatethatheattransferrateishigherforsinusoidalthermalboundaryconditionthanothertwocases.


442

(a) (a)

(b) (b)

(c) (c)

Fig.9.EffectoflocalNusseltnumberontheinnerhot round wall for (a) case‐1, (b) case‐2 and (c)case‐3for 10 nmpd , 710TRa , 3n at 1 .

Fig.10.EffectoflocalNusseltnumberontheoutercoldroundwall for (a)case‐1, (b)case‐2and(c)case‐3for 10 nmpd , 710TRa , 3n at 1 .


443

(a) (a)

(b) (b)

(c)

(c)

Fig. 11. Effect of temperature gradientmagnitude on the inner hot roundwall for (a)case‐1,(b)case‐2and(c)case‐3for 10 nmpd ,

710TRa , 3n at 1 .

Fig. 12. Effect of Temperature gradientmagnitudeon theouter cold roundwall for (a)case‐1,(b)case‐2and(c)case‐3for 10 nmpd ,

710TRa , 3n at 1 .


444

(a). (b).

(c)

Fig.13.TheaverageNusseltnumberversus(a)nanoparticlediameterfor 710TRa , 3n (b) Rayleigh number with 10 nmpd , 3n and (c) shapes of nanoparticle for

10 nmpd , 710TRa usingcase‐3at 1 .Thevariationof temperaturegradientmagnitude( TGM )at the innerhotandoutercoldwall forsolidvolumefractionusing(a)case‐1,(b)case‐2and(c)case‐3thermalboundaryconditionswith

10 nmpd , 710TRa , 3n at 1 has been displayed in Fig. 11(a)‐11(c) and Fig. 12(a)‐12(c)respectively.Itisobservedthatthetemperaturegradientmagnitudeatthehotterorcolderwallishigherforbasefluidthanthatofnanofluidthroughoutthecases.Asnanoparticlevolumefractionincreases,temperaturegradientmagnitudeonthehotterandrelativelycolderwalldecreaseswhichiscompletelyareverseeffectoflocalNusseltnumber.Thisisduetothethermophysicalpropertiesofnanofluidandbasefluid.So,itcanbeconcludedthatthevariationoflocalNusseltnumberwithrespecttothesolidvolumefractionisinverselyproportionaltothevariationoftemperaturegradientmagnitude alone at the hotter or colderwall. For case‐1, as can be seen that the distribution oftemperaturegradientmagnitudeisanecklaceshapeforbasefluidandaparabolashapefornanofluid


445

openingupwardathottersurface,whereas,atthecoldersurface,itisalsoadownwardparabola.ItisseenfromFig.11(b)thatatthehotterwall,temperaturegradientmagnitudeisanMshapeforbasefluidandassolidvolumefractionincreases,itturnstobeaUshape.Fig.12(b)representsa shapetemperaturegradientdistributionattheoutercolderwall.Usingcase‐3thermalboundarycondition,itisfoundfromFig.11(c)andFig.12(c)thatatcertainpoints,temperaturegradientmagnitudeofbasefluidisequaltothatofnanofluidwhichiscompletelydifferentfromothercases.Thevariationoftemperaturegradientmagnitudeofbasefluidandnanofluidiscomparativelyclosetoeachotherinthiscase.

Fig. 14. Average Nusselt number enhancement in different water and kerosene based

nanofluidsfor 5% withtheselectedvaluesof 710TRa , 10nmpd , 3n at 1

.Figure13displaystheaverageNusseltnumber( aveNu )fordifferentflowparametersenteredintotheproblemsusingcase‐3thermalboundarycondition.ItclearlyindicatesfromFig.13thataverageNusselt number significantly increases for the increase of the solid volume fraction. Fig.13 (a)representstheaverageNusseltnumberversusthediameterofthenanoparticlefordifferentsolidvolume fraction. This figure exhibits that average Nusselt number decreases significantly, asdiameteroftheparticleincreases.Fortheparticlediameter1to20nm,aswiftdecreaseinaverageNusseltnumberhasbeenobservedand for 20 nmpd , it isdecreasedsteadily fordifferentsolidvolumefraction.Moreover,thedeclinerateofaverageNusseltnumberincreasessignificantlywiththeincreaseofnanoparticlediameterassolidvolumefractionincreases.Fig.13(b)depictsaverageNusseltnumberversusdifferentvaluesoftheRayleighnumberforsolidvolumefraction.Average


446

Nusselt number increases significantly for the increasing values of the solid volume fraction fordifferentRayleighnumbercomparedtothebasefluid.Also,asbuoyancydrivenparameterrises,theaverageNusseltnumberrises.AverageNusseltnumberisalmostremainsteadyfrom 410TRa to

510TRa and after that it is gradually increased up to 610TRa . A dramatic enhancement of

averageNusseltnumberhasbeenfoundafter 610TRa .Fromthesechangeitmaybeconcludedthat

convectionandconductionmodeofheattransfercanbefoundfrom 510TRa to 610TRa andafterthatconvectionmodeofheattransferdominantsintheannulus.Only spherical shape of nanoparticles has been considered for displaying the afore‐mentionedresults. To justify the shape matters on the heat transfer enhancement of nanofluid, we haveconsideredvariousshapesofnanoparticlesinthepresentstudy.TheaverageNusseltnumberwithrespecttonanoparticlevolumefractionfordifferentshapessuchassphere,brick,cylinder,plateletandbladeofthecobalt(Co)nanoparticlesaredisplayedinFig.13(c).Fromfigure,itisclearthattheaverageNusseltnumbersaresignificantlyhigherforthebladeshapenanoparticlesthanthesphericalones.Thisisduetothelesssphericityofthebladeshapeparticles;thatmeanshighertotalsurfaceareaofthebladeshapesolid‐liquidinterfacecomparedtothetotalsurfaceareaofthesphericalshapenanoparticle‐liquidinterfaceforthesameamountofnanoparticlesvolumefraction.Fromthisfigure,asequenceofhigherperformerto lowerperformerofheattransferwithrespecttoshapesofthenanoparticleisblade,platelet,cylinder,brickandsphererespectively.Formorewide‐ranginganalysis,theaverageNusseltnumberenhancementcomparedtobasefluidontheheatedwallforthreedifferentthermalboundarycasesofseveralwater‐basedandkerosene‐basednanofluidshavebeencalculatedanddisplayedinFig.14fortheselectedvaluesof, 710TRa ,

0.05 , 10nmpd , 3n at 1 .Atafirstglimpse,itcanbeseenthatkerosene‐basednanofluidshavehigherheattransferratethanwaterbasednanofluids.Itisalsodepictedthattheenhancementrate of average Nusselt number for cobalt‐kerosene nanofluid is higher than other 11 types ofnanofluidswhich are studied in the present analysis. Also, ZnO ‐water nanofluid has lower heattransferratethanothers.Cobalt‐waternanofluidhashighestheattransferrateamongwaterbasednanofluids.Forwaterbasednanofluids,case‐2andcase‐3thermalboundaryconditionshaveequalamountofpotentialitytoenhanceheattransferrate,whereascase‐1andcase‐3thermalboundaryconditions play a significant role for kerosene based nanofluids. Specifically, for kerosene basednanofluids,higherheattransferratesarefoundusingcase‐3thermalboundarycondition.

7.ConclusionsInheattransferstudies,semicircular‐annulusshapeenclosureisrelativelynew.Besides,kerosenebased nanofluids containing cobalt and 3 4Fe O are gaining notice due to their improved physicalproperties, heat transfer rates and relatively lower cost. In the present study, the analysis ofconvective unsteady flow of nanofluids in semicircular‐annulus shape enclosure has beeninvestigated numerically using nonhomogeneous dynamic model for the understanding of heattransfermechanisms.Allthenumericalresultsarediscussedfromthephysicalpointofviewandthemajorfindingsofthestudycanbelistedasfollows:


447

Convectionandconductionmodesofheattransferoccurswhen TRa rangesfrom 510 to 610 andafterthatconvectionmodeofheattransferdominantsintheannulus.

1‐20nmsizenanoparticlesareuniformandstableinthesolution. Semicircular‐annulusshapeenclosureplaysaveryimportantroletoobtainafull‐bodiedflow

anduniformisoconcentrationsofnanofluid. The non‐uniform sinusoidal heating of the inner curved wall exhibits a higher local Nusselt

numberattherightregimeoftheannulus. Variation of local Nusselt number with respect to the solid volume fraction is inversely

proportionaltothevariationoftemperaturegradientmagnitude. AverageNusseltnumberforabasefluidismuchlowerthanananofluid.AverageNusseltnumber

decreasessignificantly,asdiameteroftheparticleincreases.Forparticlediameter1to20nm,aswiftdecreaseinaverageNusseltnumberhasbeenobservedandfor 20 nmpd ,itisdecreasedsteadilyfordifferentsolidvolumefraction.Moreover,averageNusseltnumberishigherforbladeshapeofnanoparticlesthanothershapes.

Kerosenebasednanofluidshavehigherheattransferratethanthatofwaterbasednanofluids. Heattransferrateofcobalt‐kerosenenanofluidishigherthanother11typesofnanofluidswhich

arestudiedinthepresentanalysis.Quadraticandsinusoidalheatingofinnerwallexhibitsequalamount of heat transfer enhancement rate for water based nanofluids which is significantlyhigherwheninnerwallheatsuniformly.Ontheotherhand,sinusoidalanduniformheatinginnercurvedboundarygivessubstantiallyhigherheattransferenhancementrateforkerosenebasednanofluids

Nomenclature

C concentrationofnanofluid -3(mol m )

CC referenceconcentration 3(mol m )

( )p pc nanoparticlesspecificheat -1 -1(J kg K )

BD Browniandiffusioncoefficient 2 -1(m s )

pd diameterofnanoparticle (nm)

TD thermaldiffusioncoefficient 2 -1(m s ) g accelerationduetogravity -2(ms )

h heattransfercoefficientofnanofluid -2 -1(Wm K )

Bk Boltzmannconstant -1(JK )

L characteristiclength (m)

Le Lewisnumber

n empiricalnanoparticleshapefactor

TBTCN dynamicdiffusionparameter


448

TBTN dynamicthermo‐diffusionparameter

aveNu averageNusseltnumber

LNu localNusseltnumberp dimensionalmodifiedpressure (Pa)

P dimensionlessmodifiedpressure

Pr Prandtlnumber

TRa thermalRayleighnumber

CRa solutalRayleighnumber

Sc Schmidtnumber

t dimensionaltime (s)

T nanofluidtemperature (K)

CT referencetemperature (K)

,U V dimensionlessnanofluidvelocity( , )u v dimensionalnanofluidvelocity -1(ms )

TV thermophoreticvelocity -1(ms ) ,X Y dimensionlesscoordinates

Greeksymbols

bf thermaldiffusivityofbasefluid 2 -1(m s )

nf thermaldiffusivityofnanofluid 2 -1(m s )

bf thermalexpansioncoefficientforbasefluid -1(K )

nf thermalexpansioncoefficientofnanofluid -1(K ) *nf massexpansioncoefficientofnanofluid -1(mol)

nanoparticlesvolumefraction dimensionlessconcentration sphericityofthenanoparticleCo cobaltnanoparticle

bf viscosityofbasefluid 1 1(kg m )s

nf viscosityofnanofluid 1 1(kgm )s

bf kinematicviscosityofbasefluid 2 -1(m s )

bf basefluidthermalconductivity -1 -1(Wm K )

nf nanofluidthermalconductivity -1 -1(Wm K )

p nanoparticlethermalconductivity -1 -1(Wm K )

C filmconcentrationdrop -3(molm )

T filmtemperaturedrop (K)

dimensionlesstemperature


449

bf densityofbasefluid -3(kgm )

nf densityofnanofluid -3(kgm )

correctionfactor dimensionlesstime

AcknowledgmentsM. M. Rahman is grateful to The Research Council (TRC) of Oman for funding under the OpenResearch Grant Program: ORG/SQU/CBS/14/007 and College of Science for the GrantIG/SCI/DOMS/16/15.M.J.UddinisgratefultoTRCforfundingthroughaDoctoralSponsorship.

ConflictofInterestNone

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