Influence of high shear on the effective thermal conduction of spherical micro- and nanoparticle suspensions in view of particle rotation

ContentslistsavailableatScienceDirect

International

Journal

of

Heat

and

Mass

Transfer

journalhomepage:www.elsevier.com/locate/hmt

Influence

of

high

shear

on

the

effective

thermal

conduction

of

spherical

micro-

and

nanoparticle

suspensions

in

view

of

particle

rotation

Q.

Shu,

R.

Kneer,

W.

Rohlfs

∗

Institute of Heat and Mass Transfer, RWTH Aachen University, Augustinerbach 6, Aachen 52056, Germany

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 2 November 2020 Revised 10 March 2021 Accepted 20 March 2021 Available online 12 May 2021 Keywords:

Nanofluids

Microparticle suspensions Shear

Thermal conductivity enhancement Particle rotation

Pipe flow

a

b

s

t

r

a

c

t

Inthepastthreedecades,differentphysicalmodelstodescribethethermalconductivityenhancements ofsuspensionswithmicro-andnanoparticleswithrespecttoparticlesizes,shape,volumefractionetc. wereestablished.However,noneofthesemodelsconsideredtheeffectofshear,althoughaninfluencehas beenobservedexperimentallyinafewstudiesforveryhighshearrates.Possibleunderlyingmechanisms forshearinducedthermalconductivityenhancementhavenotbeenexplainedindetailyet.

Inthisarticle,thefiniteelementmethodisusedtoevaluatehowshearinducedparticlerotationandthe velocityfieldaroundtherotatingparticledrivesthermalconductivityenhancement.Fornanoparticlesit isfoundthatparticlerotationdoesnotenhancetheheattransferinsidetheparticle.However,withinthe Pécletnumber(Pe=γ˙d2_{/a)rangeof0− 300,particlerotationlinearlyenforcestheheattransportupto} 5− 30%,mainly duetoadvectivemotion(eddies)aroundthe particle.Forhighshearrates(Pe>500), theenhancementapproachesaconstantvalue,atwhichtheparticlematerialbecomesirrelevantforthe effectiveconductivityenhancement.Inthiscase,asymmetricadvectivemotionaroundtheparticle domi-natestheheattransfer.NotethatthePécletnumberrangeforwhichanadvectivemotionplaysa signifi-cantroleisrelevantforsuspensionswithmicrometer-sizeparticlesbutnotfornanoparticlesuspensions. In a final step, the influence of the shear-affected thermal conductivity is evaluated for a forced-convectionlaminar pipeflowwith aparabolicvelocityprofile.Utilizing athermal conductivitymodel whichaccountsforthelocalshear,theoverallheattransferenhancementinthechannelflowis deter-mined. It isfoundthat a65%cross-section saturationwithshear-inducedenhancementis hardtobe reached withinthelaminar flowregion. Thus,the shearenforcementsmainly takeeffectclose tothe pipewall.

© 2021TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/ )

1. Introduction

Theexplorationoffluidswithenhancedthermalpropertiescan have massiveapplications in heatingandcooling systemsof var-ious industries[1].One ofthebranchesamongthose attemptsis to mix solid particles intothe fluid to form the suspension. Ini-tially based on theoretical and experimental studies of suspen-sions containing millimeter- or micrometer-sized particles, vari-ous effectivemediumtheory modelsweredeveloped,such asthe Maxwell model[2]consideringthevolumefraction;theHamilton and Crossermodel [3]includinga shape factor fornon-spherical

∗_{Corresponding author.}

E-mail address: rohlfs@wsa.rwth-aachen.de (W. Rohlfs).

particles;andlaterNan’smodel[4]takingparticlesize,shape, vol-umefraction,orientationdistributionandinterfacialthermal resis-tanceinto consideration. Allof thesemodels consider astagnant fluid.

Choiand Eastman[5] were amongthe firstto utilize suspen-sions with particles of nanometer-size, so called Nanofluids, for thermal conductivity enhancement. Following their experimental work, numerous research groups investigated the enhancement ofthermalconductivitybynanoparticles.Theexperimental obser-vations showed inconsistent results which were hardly explain-able by classical theoretical models [6,7]. Different mechanisms were assumed to be responsible for the abnormal increase such asBrownianmotion,liquidlayering,interfacialresistancesbecause of ballistic phonon heat transfer across the interface, clustering

https://doi.org/10.1016/j.ijheatmasstransfer.2021.121251

Nomenclature

Physicalquantities

¯

u averagevelocityoftheHagen-Poiseuilleflowinside thecircularpipem_/s

˙

γ

imposedshearrateonthefluiddomains−1

˙

Q totalheatfluxthroughtheupperorlowerboundary ofthedomainW

λ

thermalconductivityofthematerialW/

(

m· k

)



σ

theaverageshearstressintegratedontheboundary

surfacePa

μ

dynamicviscosityofthefluidPa· s

ω

angular velocity of the rotating spherical particle

s−1

φ

volumefrationontheparticleinthebasefluid →

τ

shearstressonthesurfaceoftheparticlePa

→

n normaldirectionofthesurface

a thermaldiffusivitym2_/s

Br Brinkmannumberindicatingtheratiobetweenheat produced by viscous dissipation and heat trans-portedbyfluid(Br= μ(γ˙l)2

λ(Tupper−Tlower))

cp heatcapacityJ/

(

kg· K

)

d diameteroftheparticlem

k0 theeffectivethermalconductivitywiththefluidand theparticleatrestW/

(

m· k

)

k_∞ theeffectivethermalconductivityofplateauathigh shearratesW/

(

m· k

)

k_eff theeffectivethermalconductivitywiththefluidand theparticleW/

(

m· k

)

l lengthofthedomainm

Nu a dimensionless number indicating the convective heattransfercomparedtothethermalconductivity ofthefluid

Pe Pécletnumberindicatingtheratiobetweenthe ad-vectiveheatfluxandtheconductiveheatflux(Pe=

˙ γd2

a )

R radiusofthecircularpipem

Re shear Reynolds number indicating the inertia of fluidundershearcomparedtoitsviscosity

Sk adimensionlessnumbernamedinthisstudyto in-dicatetheratiobetweentheshearinducedthermal conductivity incrementand its static thermal con-ductivityk0

Tupper temperaturedefined atthe upperboundary ofthe fluiddomainK

Tm the meantemperatureof thefluid inthe fully de-velopedHagen-PoiseuilleflowK

T_lower temperature definedat the lower boundary ofthe fluiddomainK

u0 velocity at the upper and lower boundary in the sheardirectionaccordingtotheshearratem/s Subscripts

adv advection

cond conduction

f fluid

p particle

of nanoparticlesorthermophoresis [1,8].Including some ofthese mechanismsleadstovariousmodelapproachesextendingthe clas-sicmodels:forexample,KooandKleinstreuer[9]includedthe ef-fect of Brownianmotion; Xuan etal.[10] expandedthe Maxwell model considering the Brownian motion and aggregation struc-ture of nanoparticle clusters; Yu and Choi [11,12] developed the

Maxwellaswell astheHamiltonandCrossermodelincludingthe effectofinterfacialnanolayers tothethermal conductivity;Avsec [13] featured the model representing the phonon and electron heat transfer analysisofthermal conductivityfor nanofluids bas-ingonstatisticalnanomechanics;Buongiorno[14]emphasizedthe important mechanismscontributing to the thermalenhancement ofnanofluidsare Browniandiffusion andthermophoresis.Thus, a non-homogeneoustwo-phasemodelwasproposedtodescribethe effectofthenanofluidontheconvectivetransport.

Toresolvetheproblemofinconsistenciesduetoabroadrange of synthesis processes, experimental measurement approaches, andthe often-incompletecharacterizations ofthe nanofluid sam-ples,theInternationalNanodPropertyBenchmarkExercise(INPBE) [15]organizedbetweenSeptember,2007 andJanuary, 2009 mea-surementsby 34 organizationsworldwidewithstandard samples of nanofluids. Based on thislarge set ofexperimental data,they concluded that the classic effective medium theory is applicable forwell-dispersed nanofluids andno anomalous enhancement of thestagnant thermalconductivityexists. Sofar,theenhancement of thermal conductivity of nanofluids is valid with descriptions frommathematicalmodelsverifiedbyexperimentaldata.

Former numerical investigations on nanofluids involved con-vectiveheat transfer fordifferent flow geometries likepipe flow, parallel-plate flow, squeezing flow andjet flow etc., with differ-ent algorithms such as semi-analytical methods, finite difference method,finitevolumemethodandlatticeBoltzmannmethod[16]. Experimentaldataunderlaminarflowconditionsincirculartubes canbefoundbyWen [17],Kim[18]andReaetal.[19]who mea-suredtheentranceregion;furtherHwangetal.[20]extendedthe length anddecreased the diameterof the tubeto enable the fo-cusonthefullydevelopedregionwithuniformheatfluxfromthe wall; andHeyhat et al. [21] performedthe fully developed flow withthe constant wall temperatureboundary. Detailson the pa-rametersadoptedarelistedinTable1.Adirectcomparisonofthe experimental datawithnumericalsimulation resultsusing differ-enteffectivethermalconductivitymodels(seeTable2) isdifficult. Especially because the experimental setup did not always match withtheassumptions madefor thenumericalinvestigation. Nev-ertheless,the comparisonbetweenthesimulationby Ebrahimnia-Bajestanetal.[22]andtheexperimentbyKimetal.[18],alsothe simulationby Haghshenas-Fardetal.[23] andtheexperiment by Zeinali-Herisetal.[24,25]offeredahint,thattherelativeerror be-tweenexperimentaldataandnumericalresultsusingsingle-phase methodcanbeashighas7.5%.Thisindicatesthatpresenteffective thermalconductivitymodelscannotpredicteffectsthattakeplace inamovingliquid,forinstanceshearinducedparticlemigrationor shearinducedparticlerotation.

Studies on shear-dependent thermal conductivities date back to Sohn and Chen [26] in 1981, conducting experiments with micrometer-scale particle suspensions. For this problem, the rel-evant parameter is the Péclet number Pe=

γ

˙d2_{/a, with}

_γ

_˙ de-noting the shear rate, d the particle diameter, and a the fluid thermal diffusivity. Sohn and Chen showed the effect of micro-convection,e.g. theformation of eddiescaused by shear-induced particlerotation, onthe thermalconductivityenhancement. They observed an enhancement only at high Pe numbers (larger than 300). For nanoparticle suspensionswith Pécletnumbers of order one, researches assumed shear-dependent particle rotation being insignificant for the convective thermalenhancement [17]. How-ever,theexperimentalstudiesonnanofluidsunderhighshearrates by Shin and Lee[27] do not support thisstatement. By examin-ing polyethyleneand polypropyleneparticles of 25,100, 180and 300

μ

msize,asignificantenhancementduetoshearwasobserved forparticleslargerthan100

μ

m,withPécletnumbersrangingfrom 20to200.Sunetal.[28] measuredtheheattransferutilizing sili-conoxide(SiO2)nanodswithaverageparticlediametersof10,20,

Table 1

Former experimental investigations on laminar convective heat transfer of nanofluids inside a circular tube.

Author, year Material Particle size Volume fraction Tube size Boundary condition

Reynolds number

Results example: convective increase Diameter Length

Wen, 2004 Water-Al 2 O 3 27 − 56 nm 0%, 0.6%, 1%, 1.6% 4.5mm 0.97m Constant heat flux 500 − 2100 φ= 1 . 6% ,

Re = 1600

x/D = 63 47%

x/D = 173 14% Rea, 2009 Water-Al 2 O 3 50nm 0.6%, 1%, 3%, 6% 4.5mm 1.01m Constant heat flux 400 − 1900 φ= 6% 27%

Water-ZrO 2 0.32%, 0.64%, 1.32% φ= 3 . 5% 3%

Kim, 2009 Water-Al 2 O 3 20 − 50 nm 3.5% 4.57mm 2m Constant heat flux 1460 at entrance 15% at the exit 12%

Water-C 3% at entrance 8%

at the exit 5% Hwang, 2009 Water-Al 2 O 3 30nm 0.01%, 0.02%, 0.03%,

0.1%, 0.2%, 0.3%

1.81mm 2.5m Constant heat flux 5000W/m 2 500 − 800 φ= 0 . 3% 8% Heyhat, 2013 Water-Al 2 O 3 40nm 0.1%, 0.5%, 1%, 1.5%, 2% 5mm 2m Constant wall temperature 330 − 2100 φ= 0 . 1% , Re = 330 3% φ= 2% , Re = 2100 30% Table 2

Former numerical investigation on laminar convective heat transfer of nanofluids inside a circular tube based on single-phase method with different effective thermal conductivity models.

Author, year Material Tube size Boundary condition

Input effective thermal conductivity model Results example: convective increase Diameter Length Maïga, 2005 Water-Al 2 O 3 EG-Al 2 O 3

1mm 1m Uniform wall heat flux 2500 − 5000 W/m 2

Hamilton and Crosser

(1962) φ

= 7 . 5% at the exit

63% Bianco, 2009 Water-Al 2 O 3 10mm 1m Uniform wall heat flux 5000,

7000, 10,000 W/m 2

Hamilton and Crosser

(1962) φ

= 1% at the exit

3.70% Haghshenas-fard, 2010 Water-Al 2 O 3 6mm 1m Uniform wall temperature Yu and Choi (2003) φ= 3%

Pe = 2500 average

13% Ebrahimnia-bajestan, 2011 Water-Al 2 O 3 4.6mm 2m Uniform wall heat flux

2090 W/m 2

Koo and Kleinstreuer

(2004) φ

= 6% at the exit

22%

40,and60nm,withPeintheorderof10−5.Theexperimental re-sults revealeda shear-dependent thermal enhancement up to ca. 10%.BothexperimentalresultsbyShinandLee[27]andSunetal. [28] havein common that the shear-inducedenhancement satu-rates forhighshearrates(400s−1). Thiscould havenot been ob-servedbySohnandChen[26]becauseintheircasetheshearrate waslimitedto25s−1.

Theaimofthispaperistoquantifytheinfluenceofshearand particlerotationonheattransferenhancement.Forthis,numerical simulations usingafiniteelementmethodhavebeenapplied. Ex-amining thesimulationresults,the shear-dependentthermal con-ductivity enhancement and the saturation behavior shall be ex-plained. Further, a shear-dependent correlation for the effective thermal conductivityis applied to determinethe convective heat transferinachannelflow.

2. Methods

2.1. Heattransfersimulationofsingleparticlesinshearflow

In a first attempt, thesuspended particles are assumedto be evenly distributed in the base fluid under shearflow conditions. This includes the absence of any clustering or redistribution ef-fects (e.g. by shear or thermophoresis)and allows to investigate a representative fluid volume with periodic boundary conditions and a singleparticle centrally located inside. This assumption is consideredapplicable intherangeofvolumefractionsbelow20%, whereintheviewofrheology,particle-particleinteractions in di-lute suspension are neglectable [29]. To access the heat transfer ratethroughthisfluidlayerincludingtheparticle,acomputational domainasshowninFig.1isapplied.Fixedvelocityboundary

con-Fig. 1. Boundary conditions set on the computational domain illustrated by a 2D slice.

ditions of −u0/2 at the bottom and u0/2 atthe top impose the shearflow. Forthetemperature, theupperandthelower bound-aryaresettoaconstantvalue.Allotherdomainboundariesareof periodictype.The ratiobetweendomainandparticlesizedefines thevolumefraction.Thethree-dimensionaldomainaswellasthe appliedmeshareshowninFig.2.Computationsareperformed us-ingCOMSOLMultiphysics®.

Fig. 2. Geometry and mesh of the single sphere particle rotating model including 112,946 tetrahedral elements in the bodies and 8560 triangle elements on surfaces. Thefluiddynamicproblemisdefinedbythemassand momen-tum conservation equations for an incompressible fluid and sta-tionarystateconditionswhichread

∇

·V→= 0 , (1)

ρ

V→ ·



∇

V→



= −

∇

P +

μ∇

2 → V . (2)

Therotationspeed oftheparticle

ω

isa directresultfromthe imposedshearrate

γ

˙.Thisisadirectresponsetotheshearforces actingontheparticle.Thepositionoftheparticleisfixed.Thus,a possibletranslatorymotion,thatcanonlybecauseddueto numer-icaltruncationerrorsinthefullysymmetricandlaminarproblem, issuppressed.

Tomodelthisinteraction,twodifferentapproachesareutilized: (1) Thetime-dependentfluid-structure interaction (FSI)solver, which fully couples fluid dynamics and solid mechanics. In this case, thefluid exerts a force on theparticle such that the rotat-ingvelocityconvergesduringthesimulationtoafinalvalue.(2)A steadystatesimulationwithoutfullcouplingoffluiddynamicand solid mechancis,wherea guessedparticlerotationvelocity is im-posed.Inthiscase,aforceontheparticleremainswhichwould ac-celerateordeceleratetheparticle.Iterativelyadaptingtherotation velocity untilthetangentialshearstressontheparticlesurface→

τ

vanisheswillleadtotheequilibriumrotationspeed.

Theshearstressontheparticlesurfaceisevaluatedby →

τ

= →n ·

, (3)

withtheviscousstresstensor

=

μ



∇

V→ +



∇

V→



T



=

⎡

⎣

2

μ

du dx

μ



du dy+ dv dx

μ



du dz+ dw dx

μ



du dy+ dv dx

2

μ

dv dy

μ



dv dz+ dw dy

μ



du dz+ dw dx

μ



dv dz+ dw dy

2

μ

dw dz

⎤

⎦

. (4)

For model validation, the angular velocity

ω

is calculated for the two-dimensional caseofa rotatingcylinderandpresented in Fig. 3 together with literature data. Good agreement is obtained withthevelocity beingsmallerthanhalfoftheshear rate

γ

˙ and beingdependentonReynoldsnumber[30][31].Bothsolution ap-proaches (time-dependentFSI solverandthestatisticsolver) pro-videdthesameresults.

For the particle in the three-dimensional domain and a Reynoldsnumberofrange0.024− 0.335,theangularspeed calcu-lated utilizingbothapproaches takesthevalue of

ω

/

γ

˙ ≈ 0.47326 andisalsoconsistentwiththesecond-orderapproximationbyLin

ω

/

γ

˙ =1/2− 0.1538Re1.5_[32].

Fig. 3. Ratio of angular velocity and shear rate for a rotating two-dimensional cylin- der under shear. Present results computed by two different approaches (green stars) are compared to the results from former studies summerized by Ding and Aidun [31] , among which a linear decreasing tendency at Re > 20 was proposed by Kos- sack and Ding. ( H/r represents the ratio between the hight of the domain and the radius of the cylinder.). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Theeffective viscosityisanother quantitysuitable to compare andvalidate themodelresults.Therelativeviscositycouldbe ob-tained by the average shear stress



σ

on the upper or lower boundary,asvalidatedbyXuetal.[33,34]

μ

eff

μ

f =



σ

/ ˙

γ

μ

f

(5) Table 3 provides data for two different fluids and two different shear rates. All four values are close to the theoretical value of 1.2calculatedwiththe modelofBatchelor [35].Accordingto the theoryfordilutesuspensions,therelativeviscosityisindependent fromrotationalspeedandonlyvarieswithvolumefraction.

Thestationaryheattransferproblemisgovernedbythe conser-vationequation

ρ

cp

→

V ·

(

∇

T

)

=

λ∇

2_{T (6)}

Thisequationissolvedbasedontheconvergedsolutionofthe velocityfield.WiththeBrinkmannumberBr=

μ

(

γ

˙l

)

2_/[

_λ

₍

_T

upper−

T_lower

)

]beingsmallerthan10-3_{,viscousdissipationisneglectable.} Theeffectivethermalconductivityisevaluatedby dividingthe total heatflux perpendicular to the upperorlower surface Q˙ by theimposed temperaturedifference andthedistance l (lengthof thedomain).

keff= ˙ Q

l ·

T. (7)

Theconductivityenhancementdueto particlerotationis com-paredtothethermalconductivitywiththenanofluidandthe par-ticlesatrestk0.Thus,theenhancementrateisdefinedby

k =

keff − k 0

k0 × 100% .

Table 3

Computed relative viscosity of dispersions of volume fraction φ= 7% with different fluid material comparing to the effective medium theory model.

Material μeff

μf at ˙ γ= 500 s

−1 [present study] μeff

μf at ˙ γ= 30 0 0 s

−1 [present study] μeff

μf =1 + 2 . 5 φ+ 5 . 2 φ

2 [Batchelor, 1972]

Water 1.196 1.213 1.200

Silicon oil mixed with kerosene 1.195 1.195 1.200

2.2. Heattransferinfully-developedpipeflowwithsheardependent thermalconductivity

The influence of a shear dependent thermal conductivity on a laminar pipe flow is estimated on an analytical basis. In gen-eral, high shear ratescan lead to shear thinning effects of com-plex dispersions. However, for dilute nanofluids (such as consid-ered here with volume fraction below 7%), the influence of the shear-thinning isneglectable [36]. Assuch, thevelocity profile in thepipeisassumedtobeofparabolicshape(Hagen-Poiseuille re-sult). u

(

r

)

= 2 u¯



1 − r2 R2



(9) andtheresultinglocalshearrateisgivenby

˙

γ

(

r

)

=





∂

u

∂

r





= 4 u¯ R2r, (10)

whereu¯denotestheaveragevelocityandRthepiperadius.With theboundaryconditionofconstantheatflux,thetemperature pro-filecanbeobtainedbyintegratingtheenergyequation

ρ

cp



u

∂

T

∂

z



= k

1 r

∂

∂

r



r

∂

T

∂

r



+

∂

2T

∂

z2



. (11)

With uniform heat flux boundary condition, the temperature increase in flow direction takes the constant value of

∂

T/

∂

z= 2q˙ /

(

R

ρ

cpu¯

)

=const.Thus, thesecond derivate vanishes.Because

the thermalconductivitydependson thelocalshear rate, the ef-fectivethermalconductivitykeff

(

γ

˙

)

isusedtoreplacekleadingto theequation

∂

2_T

∂

r2 + 1 r

∂

T

∂

r = u

(

r

)

keff

(

r

)

2 ˙ q Ru¯. (12)

Todeterminetheeffectiveheattransfercoefficient,

heff = ˙ q

T

(

r = R

)

− T m, (13)

equation(12)isnumericallysolvedfortheunknownwall tempera-tureT

(

R

)

.ThemeantemperatureofthefluidTm,whichisthe ref-erence temperatureforthe heat transfer coefficient, iscalculated bytheintegrationof Tm =  A

ρ

cpu

(

r

)

T

(

r

)

dA

ρ

cpu¯A = 4 R2  R 0 r



1 −r2 R2



T

(

r

)

dr. (14)

3. Resultsanddiscussion

3.1. Advectiveheattransferoftherotatingparticle

Inafirststep,theheattransferenhancementbytheadditional advective heat transport of the rotating particle is analyzed. For this,theheatfluxfromthebottomofthespheretothetopofthe sphere is theoretically addressedwithout considering any contri-bution due to theoutside fluid. Incase ofa non-moving sphere, theheattransportispurelybyconductionsuchthatthetotalheat fluxdensityscaleswith

˙

qp,cond =

λ

p

T

d . (15)

Ifthesphererotateswithagivenspeed,anadvectiveheatflux contributionariseswhichscaleswiththevelocity.

˙

qp,adv =

ρ

pcppu

T. (16)

ThePécletnumbercharacterizestheratiobetweenthetwoheat transfermechanisms Pep = ˙ qp,adv ˙ qp,cond = ud

λ

p/

ρ

pcpp = ud/ap, (17) with a being the thermal diffusivity. For an aluminum-oxide nanoparticlewiththethermaldiffusivitya_Al

2O3=8.23× 10−6m 2_/s, andasize ofd=6× 10−8_{m(nanoparticle) ord=1× 10}−4_m (mi-croparticle), the velocity needs to be in the order of 100m/s or 0.08m/s respectively to have a comparable contribution of con-ductionandconvection.Because ofthisveryhighvelocity, which translatestoashearrateof2.3× 109_s−1 _or820s−1_,the contribu-tionoftheadvectivetransportfromthenanoparticleisneglectable for all shear rates and inside the microparticle for small shear rates.Conductionremainsthedominantmechanism.

3.2. Advectiveheattransferofthesurroundingliquid

Theflowfieldinashearflowwithoutparticlesisstrictly paral-leltothewall.Thus,thereisnocontributiontoheattransferbyan advective velocity component inwall normal direction. However, theflowfieldincludingarotatingparticleshowsavelocity compo-nentin wallnormaldirection(y-direction inFig.4).This velocity contributestotheheattransportyieldingtoanadvectiveheatflux densityof

˙

qf,adv =

ρ

fcpfu

T. (18)

Adirectcomparisonbetweentheheat conductionthroughthe fluidandtheheatadvectionisgivenby

Pef = ˙ qf,adv ˙ qf,cond = ud/af. (19) Using againthe size of the particle asa characteristic length, and the transport properties of water, the wall normal veloc-ity requires to be in the order of 2.5m/s for nanoparticle or 1.5× 10−3_{m/sformicroparticleforanequivalentcontribution.This} value forthe velocitiy can be obtainedforshear rates inthe or-derof4× 107_s−1_{fornanoparticlesor15s}−1 _{formicroparticles.The} commonflow insidea microchannelsuch asthe oneused inthe experiments by Rea and Hwang [19,20] adopts a volume flux of around 10− 20cm3_{/s, which resultsin a maximum shear rateat} wallinthe orderof103_s−1_{,farbelowthethresholdofnanosized} particlesforanequivalenteffectofadvection.Thus,fornanofluids in channel flow the advectiondue tothe particle rotation is ne-glectable.

3.3. Shear-inducedheattransferenhancmentbyparticles

Intheprevioussections,theeffectofparticlerotationandflow field alteration was discussed separately. Here, both effects are combinedbysolving both,theflow andthetemperaturefieldfor differentshearratesinside andoutsideoftherotatingparticle.In linewiththeexperimental investigationofShin andLee[27],the materialpropertiesofsiliconekerosene mixtureandpolyethylene particles withthesize ofd=1× 10−4_{mare used. Fig.5presents}

Fig. 4. The stream line of the velocity field simulated with a rotating particle in the center at Pe = 100 (a); and the resultant temperature field with the isothermal contour lines (b).

Fig. 5. The effective thermal conductivity ratio for polyethylene particles in a sili- con oil and kerosene mixture presented in dimensionless form as function of Pe .

the results forthe effective thermal conductivity ratio forPéclet numbers fromzeroto900. Therein,k0 iscalculatedbasedonthe simulationresultsforanon-movingparticle.Thesimulationresults agree well with the experimental data and show a quasi-linear increase up to a Péclet numberof300. Beyondthat Péclet num-ber,thethermalconductivityratioconvergestoaplateauwiththe valuek_eff/k0≈ 1.36.

Inordertoidentifythereasonfora)thelinearincreaseandb) theplateau,amoredetailedinvestigationoftheflowfieldandthe heat fluxisperformed.Forthis,thedifferentheattransfer contri-butionsthroughthecenter-plainoftheparticleaty=0are evalu-ated.

Fig. 6 shows the three dominant heat transfer contributions. First, heat conduction through theparticle inside (blue line with filled dots); second, the convective heat transport by conduction

Fig. 6. Heat transfer contributions from the heat conduction inside the particle and the convective heat transfer outside the particle (y-axis in blue on the left side). The total convective heat transfer can be further divided into two sections: the flow on the left section transfers heat upwards; while the flow on the right section trans- feres heat downwards (y-axis in orange on the right side). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

andadvectionthroughtheleftsection(x<−0.5D,reddashedline with triangles pointing to the left) and, third, through the right section (x_>0_.5D_, red dashed line withtriangles pointing to the right)(seeFig.7).Thelattertwocontributionsaredividedbecause thewallnormalvelocity

v

pointsinoppositedirection. Whilethe convectiveheattransportthroughthefluidlinearlyincreaseswith theshearrateandtheincreaseinvelocity

v

,theheatconduction throughtheparticledecreases.Thereasonforthedecreasingheat conductionisaquasi-isothermallayerformed aroundtheparticle bythefluidflow,whichisillustratedinFig.4.Thisisothermallayer isaresultofthereducedtemperaturegradientcausedby the

ad-Fig. 7. Two sections to compute the convective heat transfer.

ditionaladvectiveheattransfercontributioninthefluid.Theresult ofthe linearincrease oftheconvective heat transferthrough the leftandrightsectionleadstothesaturationofthetotalheat con-vection,namelytheplateauathighPe.

3.4. Influenceofparticlepropertiesonheattransferenhancement

Becauseitisfoundthattheheattransferbyconductionthrough the particle becomes less important for higher rotation veloci-ties, it’s expectedthat the particle propertiesinthis casedo not influence the overall heat transfer. To confirm this assumption, the effective thermal conductivity at high shear rates is deter-mined for different particle materials, e.g. polyethylene, silicon-oxide, aluminum-oxide, and copper-oxide, whose parameters are listedinTable4.

Fig.8showstheabsolutevalueoftheeffectivethermal conduc-tivity (plot b,d) and the effectivethermal conductivity ratio(a,c) forthedifferentparticlematerials andfluid properties(siliconoil -kerosene mixture (a,b); water(c,d)). As expected, theeffective thermalconductivity increasesfirst withshearrate. However, for

0 100 200 300 400 500 600 700 800 Pe 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 keff /k0

Relative thermal conductivity enhancement in mixture of silicon oil and kerosene

CuO Al₂O₃ SiO₂ Polyethylene (a) 0 100 200 300 400 500 600 700 800 Pe 0.16 0.17 0.18 0.19 0.2 0.21 0.22 k eff [W/(m K)]

Absolute thermal conductivity enhancement in mixture of silicon oil and kerosene

CuO Al₂O₃ SiO₂ Polyethylene (b) 0 100 200 300 400 500 Pe 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 k eff /k 0

Relative thermal conductivity enhancement in water

CuO Al₂O₃ SiO₂ Polyethylene (c) 0 100 200 300 400 500 Pe 0.5 0.6 0.7 0.8 0.9 1 1.1 k eff [W/(m K)]

Absolute thermal conductivity enhancement in water

CuO Al₂O₃ SiO₂ Polyethylene

(d)

Fig. 8. The relative enhancement k eff /k 0 of various particle materials with different initial thermal conductivity k 0 in mixture of silicon oil and kerosene(a) and in water(c); the absolute effective thermal conductivity k eff of suspensions in mixture of silicon oil and kerosene(b) and in water(d) as function of Pe .

Table 4

Different material properties used to validate the irrelervance of thermal conduc- tivity of material on the effective thermal conductivity at high shear rates.

Material Density [kg/m 3 ] Thermal conductivity [W/(m ·K)] Heat capacity [J/(kg ·K)] Fluid Water 997.1 0.613 4179

Silicon oil mixed

with kerosene 915 0.159 1910 Solid CuO 6315 33 531 Al 2 O 3 3970 25 765 SiO 2 2650 1.4 705 Polyethylene 900 0.35 1900 Table 5

Computed effective thermal conduction of dispersions of volume fraction

φ= 7% with different particle material in water comparing to the effective medium theory model.

Material k0 [present study] k0 = λfλλpp+2+2λλff+2−φφ((λλpp−−λλff)) [Maxwell, 1881]

CuO 0.7439 0.7435

Al 2 O 3 0.7415 0.7410

SiO 2 0.6527 0.6524

PE 0.5925 0.5918

highershearrates,allvaluesconvergetothesamevalue indepen-dent from the particle material. Fora less viscous fluid like wa-ter withhigherReynoldsnumberanda decreasingratiobetween particle rotationspeed andshear rate(see Fig.3), there isstill a continuousincrease in effectivethermalconductivitywith higher Pécletnumber.Fora highviscous liquidwithlow Reynolds num-bers, theratiobetweenparticle rotationspeedandshear rate re-mains almost constant, which results ina saturation behavior of theeffectivethermalconductivitywithhigherPécletnumbers.

Considering the effective thermal conductivity ratio, which is veryoftenshowninexperimentalstudies,mightresultina differ-entinterpretation.Becausetheeffectivethermalconductivityratio dependsonk0(fluidandparticleatrest),whichvariesfrom differ-entinserting materialslisted inTable5,therelativethermal con-ductivity increase doesnot converge forthe various particle ma-terials.Comparingthedeterminedvalue ofk0toMaxwell’smodel

[2]forthisquantityshowsasexpectedexcellent agreement.Note thattheHamiltonandCrossermodel[3]andNan’smodel[4]are identicaltoMaxwell’smodelifconsidering onlythesphere with-outinterfacialthermalresistances.

3.5. Influenceoftheshear-dependentthermalconductivityonthe heattransferinfully-developedpipeflows

Based on the 3D simulation results as shown in Fig. 5, an approximation of the shear induced effective thermal conductiv-ity can be drawn with a threshold value Pe_∞=300_, a satura-tion value

(

k/k0

)

∞=1.36, a slope value

∂

k/

∂

Pe=k0

(

(

k/k0

)

∞− 1

)

/Pe∞=1.95× 10−4, leadingto an effective thermalconduction modelconsideringtheshear:

k =



_∂k ∂Pe· P e Pe<Pe∞ k0·



k k0

∞ Pe ≥ Pe∞ (20) Thiscouldbeappliedfurthertoshearrelatedforcedconvection modelstoinvestigateitsmacroinfluences.

Forthe fully-developed Hagen-Poiseuilleflow, theaverage ax-ial velocity u¯ and the radius of the circular tube R are two in-dependentvariables.Indimensionlessform,theReynoldsnumber

Re=2u¯R/

υ

decides the absolute value of the velocity profile. In termsofheattransfer inthethermallyfullydevelopedpipeflow, thedimensionlessheattransfercoefficient(Nusseltnumber)takes thevalueof4.36(withconstantaxialwallheatfluxboundary con-ditionandintheabsence ofthermalenergysources,viscous dis-sipations,flowwork)[37]forconstantfluidproperties.Ifthe ther-malconductivitydependsontheshearrate,an additional dimen-sionless numbercan be introduced,which wedenote by Sk.This numberisdefinedby Sk= 4 ¯ u R·∂ k ∂γ˙ k0 , (21) with

∂

k

∂

˙

γ

=

∂

k

∂

Pe·

∂

Pe

∂

˙

γ

= dp2 kf/

ρ

fcpf ·

∂

k

∂

Pe (22)

andscalesthegradientofthermalconductivitywithrespecttothe shearrate

(

∂

k_/

∂

γ

˙

)

_/k0tothemeanshearrateofthefluidflowu/R. Aphysicaldefinitionofthisdimensionlessnumberis providedin AppendixA.

Fig. 9. The convective heat transfer for the fully-developed Hagen-Poiseuille flow considering the shear-induced effective thermal conductivity enhancements of the fluid (in blue); and the radius proportion of the cross-sectional area which reaches the saturation k ∞ (in orange) in relation to the dimensionless variable Sk .

Based on these two independent dimensionless variables, Re

and Sk, the overall enhancement in convection is presented in Fig.9 (left).Theconvectioncoefficient expressedinNusselt num-berdependsnotsignificantlyontheReynoldsnumber,butlargely onSk.

Becauseofthesaturationbehaviorofthesingleparticle’s ther-mal conductivity enhancement, also the heat transfer coefficient reaches a plateau for high values of Sk. With increasing Sk, the cross-sectional area increases in which the thermal conductivity reachessaturation(seeFig.9,right).ForSk₌1_,morethan65%of theradialcross-sectionalareaissaturatedwiththemaximal ther-malconductivityk_∞/k0=1.36.

The final question that needs to be answered is, whether a valueofSk≈ 1canbeobtainedinrealisticapplications.According to the definitionof ReandSk (21)the two physicalindependent quantitiescanbepresentedas

¯ u=



Re· Sk ·

μ

k0 8

ρ

∂k ∂γ˙ , (23) R=



Re Sk· 2

μ

∂k ∂γ˙

ρ

k0 . (24)

Toreach a valueof Sk=1in alaminarpipe flow usingwater asliquidwithCuOnanoparticlesof100nmdiameter(

∂

k_/

∂

γ

˙ _{≈ 4×} 10−5),thepiperadiusmustbesmallerthan5× 10−5_mtokeepRe within laminar range,whichisnot realistic. Using viscous silicon oil-kerosene mixtureasliquidwithCuOmicroparticles of100

μ

m

instead, (

∂

k/

∂

γ

˙ ≈ 4× 10−6_{), the resulted constrainsis the radius} smaller than 4× 10−3m while the average velocity u¯ in order of 10m_/s,whichisnoteasytorealizeeither.

4. Conclusion

Utilizingthree-dimensionalnumericalsimulationsofa rotating particle inashearflow, theinfluenceofhighshearratesonheat transfer is analyzed. In addition, the effect of shear on the heat transfer in a full-developed Hagen-Poiseuille flow is presented. Withouttakingintoaccounttheeffectofshearonotherfluid prop-erties(e.g.viscosity),thefindingsofthisstudycanbesummarized asfollows

1. For nanoparticles, the contribution of the advective transport inside (due tothe rotary movement)and outsidethe rotating particleis irrelevant,compared tothe conductionthroughthe particle.

2. Formicroparticles, the contributionof theadvective transport insideandoutsidetherotatingparticleplays ratheran impor-tantroleunderhighshear.

3. For highshear rates, an isothermal liquid layer forms around theparticlewhichhindersheattransferthroughtheparticle it-self.

4. Forhighshearratesandhighviscousliquids,theeffective ther-malconductivityexhibitssaturationbehaviorwithPéclet num-ber.

5. Forhighshear ratesandlow viscous liquids(Re>10), the ef-fectivethermalconductivitycontinuouslyincreaseswithPéclet numbercausedby adecreasingratiobetweenparticlerotation speedandshearrate.

6. For high shear rates, the thermal conductivity enhancement becomesindependent fromthe particlematerial properties.A polyethyleneparticleenhancesheattransferinthesame man-nerasacopper-oxideparticle.

7. IntheHagen-Poiseuilleflow,theheattransferenhancement de-pendssignificantlyonthedimensionlessnumberSkbutnoton Reynoldsnumber.

AuthorshipStatement

Manuscripttitle:Influenceofhighshearonthe effective ther-malconduction ofspherical micro- andnanoparticle suspensions inviewofparticlerotation

Allpersonswhomeetauthorshipcriteriaarelistedasauthors, andall authors certifythat they have participated sufficiently in the work to take public responsibility for the content, including participation inthe concept, design, analysis, writing, or revision ofthemanuscript.Furthermore,eachauthorcertifiesthatthis ma-terialorsimilar materialhasnot beenandwillnot be submitted toor publishedinanyother publication before itsappearance in the/nternationalJournalofHeatandMassTransfer.

Authorshipcontributions

Pleaseindicate thespecificcontributionsmadeby eachauthor (list theauthorsinitialsfollowedby their surnames, e.g., Y.L. Che-ung).Thenameofeachauthormustappearatleastonceineach ofthethreecategoriesbelow.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper

Acknowlgedgments

This work is funded by“Deutsche Forschungsgemeinschaft” (DFG) - GRK 1856. Gefördert durch die Deutsche Forschungsge-meinschaft(DFG)-GRK1856.

AppendixA

The dimensionless number given the name Sk is de-duced from the integration of the radial thermal resistance in equation(A.1)and(A.2).

Rthermal= 1 2

π

L  R rm 1 k

(

r

)

rdr, (A.1) withk

(

r

)

=k0+_∂∂γk˙ ·∂ ˙ γ ∂rr=k0+



∂k ∂γ˙ · 4u¯ R2



r withinthelinearzone,

Rthermal = 1 2

π

k0L

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

ln 1 1 +R r· k0· R 4 u¯·∂k ∂γ˙







1/Sk

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦















R rm . (A.2)

Asrm,theradialpositionatwhichthetemperatureisthesame

as the mean temperature (Tr=rm =Tm), is only determined by R (rm

R =



2−



17

6 ≈ 0.5628), so that the only essential dimension-lessvariablehereistheratiobetweenthebyshearratesincreased thermal conductivityand the original staticthermal conductivity

k0,definedhereasSk.

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