Thermophoresis and its effect on particle impaction on a cylinder for low and moderate Reynolds numbers

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International Journal of Heat and Mass Transfer

journalhomepage:www.elsevier.com/locate/hmt

Thermophoresis and its effect on particle impaction on a cylinder for low and moderate Reynolds numbers

Nils Erland L. Haugen

^a^,^b^,^∗

, Jonas Krüger

^a

, Jørgen R. Aarnes

^c

, Ewa Karchniwy

^c^,^d

, Adam Klimanek

^d

aSINTEF Energy Research, Trondheim N-7465, Norway

bNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, Stockholm SE-10691, Sweden

cDepartment of Energy and Process Engineering, Norwegian University of Science and Technology, Kolbjørn Hejes vei 1B, Trondheim NO-7491, Norway

dInstitute of Thermal Technology, Silesian University of Technology, Konarskiego 22, Gliwice 44-100, Poland

a rt i c l e i nf o

Article history:

Received 11 March 2021 Revised 14 September 2021 Accepted 20 September 2021 Available online 30 September 2021 Keywords:

Particle deposition Thermophoresis Overset grids

a b s t r a c t

The effectof thermophoresis onthe impaction ofparticleson acylinder is investigated fordifferent particlesizes,particleconductivities,temperaturegradientsandforReynoldsnumbersbetween100and 1600.ThisisthefirstsuchstudyperformedusingDirectNumericalSimulations(DNS),wherealltemporal andspatialscalesofthefluidareresolved.SimulationsareperformedusingthePencilCode,ahigh-order finitedifferencecodewithanoverset-gridmethodpreciselysimulatingtheflowaroundthecylinder.

The ratio ofparticlesimpacting the cylinder tothe number ofparticles insertedupstream ofthe cylinderisusedtocalculateanimpactionefficiency.Itisfoundthatboththeparticleconductivityand thetemperaturegradienthaveaclosetolinearinfluenceontheparticleimpactionefficiency forsmall particles.HigherReynoldsnumbersresultinhigherimpactionefficiencyformiddle-sizedparticles,while theimpactionefficiencyissmallerforsmallerparticles.Ingeneral,itisfoundthatthermophoresisonly hasan effectonthe smallparticles, whileforlarger particlestheimpaction is dominatedbyinertial impaction.

Analgebraicmodelispresentedthatpredictstheeffectofthethermophoreticforceonparticleim- pactiononacylinder.Themodelisdevelopedbasedonfundamentalprinciplesandvalidatedagainstthe DNSresults,whicharefaithfullyreproduced.Assuch,thismodelcanbeusedtounderstandthemech- anismsbehindparticledeposition dueto thethermophoreticforce, and,moreimportantly,to identify meansbywhichthedeposition ratecanbereduced.Thisisrelevantforexampleinordertominimise foulingonsuper-heatertubebundlesinthermalpowerplants.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Particle impaction onsurfaces canbe found ina multitude of industrial systems, such as ﬁlters and heat exchangers. The impaction and deposition of material on thesesurfaces can significantly alter their performance, necessitating decreased mainte- nanceintervalsoranincreasedrateofreplacementofcomponents.

In order to improve the design of surfaces exposed to particle laden ﬂows,athoroughunderstandingoftheunderlyingeffectsis needed.

Inthiswork,wewillfocusonhowparticlesaretransportedto the solidsurface. Foraparticletodepositon thesurface,itmust ﬁrstbe transportedto thesurface,beforeithastosticktoit.The

∗Corresponding author at: SINTEF Energy Research, N-7465 Trondheim, Norway.

E-mail address: [email protected] (N.E.L. Haugen).

lattermechanismisoutsidethescopeofthisstudy.Inthefollow- ing,allparticlesimpactingonthesurfacewillthereforebecounted towardsparticledeposition.

Thetransportofmaterialtothesurfaceisgovernedbytheim- pactionefficiency.The impactionefficiencyisdefinedastheratio ofparticlesthatactuallycomeincontactwiththecylindertopar- ticlesthat would comein contactwiththe cylinderif they were unaffected bythe changeinfluid velocity duetothe presenceof thecylinder.Foranoverviewoftheprogressandchallengesinthe field ofparticle impactionincoal andbiomass-fired systems,the readerisreferredtothereviewofKleinhansetal.[1].

Duetoitssimplicity,acylinderplacedinaparticleladenﬂowis awidespreadtestcaseusedtostudytheimpactionofparticleson solidsurfaces orheat exchangertubes. Asketchofsuch acaseis showninFig.1,wheretheparticles(showningreen)areinserted from a plane (red) that has the same size asthe projected area

https://doi.org/10.1016/j.ijheatmasstransfer.2021.121996

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Fig. 1. Sketch of particle deposition analysis case.

ofthecylinder. Theinitialvelocity oftheparticlesisequaltothe flow velocity atthe insertion plane. Ifthe particles followed the flowfromlefttoright,withoutanychangeinvelocity,allparticles would hit the cylinder, leadingto an impaction efficiency (

η

⁾ ^of

unity.Forallrealisticcases

η

<1,asthefluidisflowingaroundthe cylinder andparticles aredragged along withit. Aparticle’sabil- itytofollowthefluidisexpressedastheparticle’sStokesnumber, St,which is theratio ofthe particleresponse time and thefluid timescale(detailsinSection2.2).Ingeneral,particleswithStokes numbers aboveunity donotfollowthe flowvery well,whilethe oppositeistrueforparticleswithsmallStokesnumbers.

IsraelandRosner[2]developedacorrelationbasedonpotential flowtheory,whichallowsforcalculationofimpactionefficiencyon an isothermal cylinder.This correlation is a well established tool forpredictingtheisothermalimpactionefficiencyoflargeparticles inalaminarflow,butitishighlyinaccurateforsmallparticles[3], anditisundefinedforStokesnumbersbelow0.125.

Themassaccumulationrateonacylinderisdeterminedbythe capture efficiency – the product of the impaction efficiency and thestickingefficiency.Thestickingefficiencyisthefractionofthe impacting particles that stick to thesurfacerather thanrebound.

If either the particles or the cylinder surfaceis at leastpartially melted,thestickingeﬃciencyisclosetounity.Ontheotherhand, for cold and clean surfaces, particles will mostlikely bounce off thesurface,andthestickingeﬃciencyisclosetozero.

TheexperimentalstudybyKasperetal.[4],investigatedtheef- fect ofmassaccumulationonthecaptureefficiencyandproposed an empirical powerlawforit.Moreover, theauthors presenteda new fitfunction forthecapture efficiency,whichis boundedbe- tween 0 and 1. The particle Stokes numbers in said study were between 0.3and 3.Haugen and Kragset [3]investigated the impaction efficiencyusing Direct NumericalSimulation withan immersed boundary method and found a steep drop in impaction efficiency below a certain Stokes number, as particles become smaller and follow the flow better. Extending this work, Aarnes etal.studied thesamecaseusingoversetgrids, obtaining results that are deemed more accurate with significantly less computa- tionaleffort[5,6].

The effectofthermophoresis onthecaptureefficiencyisstud- ied byseveralgroups,bothexperimentally andnumerically.Beck- mann etal.[7]measured depositionof flyashfroma pulverised coal jet flame and simulated the deposition rateusing Reynolds Averaged Navier Stokes (RANS) based CFD withthe k−

^model.

They found that thermophoresis increases the capture efficiency for smaller particles, and that the relative increase is higherthe smaller the particles in question are. Experimental data from a pilot-scale furnace was compared to numerical results by Yang etal.[8],where theinfluenceofdeposition growthon theparti- cleimpactionandstickingefficiencywasstudied.Inthiswork,the RANS approachofAnsysFluentwasusedforthenumericalsimu- lations. Theyreportedthat thehighersurfacetemperaturedueto

depositgrowthresultsin areducedeffectof thermophoresisand anincreasedstickingeﬃciency.Atlatertimes,therateofshedding ofmaterialfromthesurfaceanddepositionofmaterialonthesur- face fromtheﬂow balance out, sono netchange ofthe massof thedepositisobserved[9].

In the work of Kleinhans et al. [10] the effect of the thermophoretic force was studied both experimentally and numer- ically. In the experimental part, the deposition of material on cooled and un-cooled probes that are inserted into the particle ladenﬂowabovetheburnersectionofacombinedheatandpower (CHP)plantwasstudied.Large-Eddy simulations (LES)were used to studythe inﬂuence ofthe sticking modeland thermophoresis on deposition ratepredictions.The authors presenta modelthat cantakeintoaccountdifferentstickingmechanismsbywhichlarge and small particles of different composition deposit. It was re- portedthatthermophoresisaccountsforthreequartersoftheob- serveddepositionrate.GarcíaPrezetal.[11]usedunsteadyRANS simulationstostudytheeffectofthermophoresisonparticledepo- sitionandfoundthatthethermophoreticforcewasthedominating depositionmechanismforverysmallparticles.

The effect that the thermophoretic force has on particle impaction on a cylinder in a cross-ﬂow is controlled entirely by processes occurring in the boundary layer around the cylinder.

As such, it is essential that the boundary layeris completely re- solvedinorderforsimulations oftheprocesstohaveanypredic- tive abilities. RANS simulations, andto some extent LES, involve crude approximations and cannot guarantee that the high accu- racy required to properly resolve the boundary ﬂow is achieved tosuchextentthattrajectoriesofsmallparticles affectedbyther- mophoreticforcesareaccuratelycomputed.DirectNumericalSim- ulations(DNS)isahighlytime-consumingmethodforﬂowsimula- tions,whereallspatialandtemporalscalesareresolved,applicable forcaseswhereveryhighaccuracyisparamount.

Totheknowledge ofthecurrentauthors,DNShavenot previ- ouslybeenusedtoperformaparameterstudyoftheeffectofthe thermophoreticforce ontheparticledeposition rateonacylinder inacrossflow. Thismotivatestheauthorsofthecurrentstudyto investigatetheinfluenceofdifferentflowconditions,suchasflow Reynoldsnumber,temperaturegradientandparticleattributeson the effect of thermophoresis. Furthermore, no generic analytical modeldescribing theeffectthat the thermophoreticforce hason particleimpactiononacylinderinacrossflowexist.Suchamodel will be developed in the following section. The model will later be validated against the DNS results. The novelty of the current studyis thereforetwofold:1) the first everaccurate DNSofpar- ticleimpaction undera wide rangeof conditionsand2) an ana- lyticalmodel,based onfundamentalprinciples, that predicts and explainsthe effect that the thermophoreticforce has on theim- pactionrate.

2. Theory 2.1. Fluidequations

The governing ﬂuid equations are the compressible equations forcontinuity, momentum andenergy. Pressureis takeninto ac- count through the ideal gas lawand the Mach number is≈0.1, lowenoughtoconsidertheﬂowasessentiallyincompressible.The continuityequationisgivenby

∂ρ ∂

^t ⁺^u^·

∇ρ

⁼⁻

ρ∇

^·^u^, ⁽¹⁾

with

ρ

^,^t ândû^being^density,^time ând^velocity, respectively.The equationgoverningtheconservationofmomentumis

ρ ∂ ∂

^u^t ⁺

ρ

^u·

∇

^u=−

∇

^p+

∇

·

(

²

ρν

^S

)

, (2)

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where pisthepressure,

ν

^the^kinematic^viscosity,^and S=1

2

∇

^u⁺

( ∇

^u

)

^T

−1

3I

∇

^·^u ⁽³⁾

istherateofstrain tensorwhereIistheidentitymatrix.Theen- ergyequationissolvedfortemperatureby

∂

^T

∂

^t ⁺^u^·

∇

^T⁼

ρ

^k^c^fv

∇

²^T⁺²

ν

^S²

c_v −

( γ

⁻¹

)

^T

∇

^·^u^, ⁽⁴⁾

where

γ

=cp/c_v=5/3, c_v andcp are theheat capacities at con- stantvolumeandpressure,respectively,andk_f isthethermalcon- ductivity.Theidealgaslawisusedtotiepressureanddensityto- gether:

p=

ρ

^r^u^T^, ⁽⁵⁾

whereru=cp−c_visthespeciﬁc gasconstant.Tosimplifythein- vestigation, the kinematic viscosity,

ν

^,^is ^assumed^to ^be ^constant

sincethetemperaturevariationsintheﬂuidarerelativelysmall.In this study, the Pencil Code is used to solve the governing equations. Since the code is an explicit compressible DNS code, the time step, dt,is limitedby the speed of sound through the CFL number, i.e., dt=C_CF_L×min(mesh)/(^cs+max(^u))^, ^where mesh

is themeshspacing,CCFL=0.8istheCFL number,andthespeed ofsoundisgivenbycs=

γ

^r^u^T=

cp(

γ

−1)^T^.Ône^can^therefore usecpasafreeparameterinordertoartificiallylowerthespeedof soundtoobtainlargertime steps.Thisisavalidapproachaslong astheMachnumberiskeptlowerthan 0.1andtheviscousheat- ingofthefluidisnegligible.Tomaintainaconstantthermaldiffu- sivity(D_thermal=k_f/(

ρ

^c^p)⁾ôf^the^gas^phase,ând^henceâ^constant Prandtl number(Pr=

ν

/D_thermal),theconductivity(k_f)ischanged proportionallytothespeciﬁcheatcapacityoftheﬂuid(cp).

2.2. Particleequations

The particles considered hereare spherical andhavelow Biot numbers, makingthem spatiallyisothermal. Numericallythey are treatedaspointparticlesthat areinfluenced bythefluid,butare too dilute to have any significant back-reaction on the fluid. In other words,they areactedonbytheflow buthaveno effecton it.This assumptionisapplicable fordiluteflows,whichis thefo- cusofthecurrentwork.TheparticlesizeisdescribedbyitsStokes number

St=

τ

^St

τ

f

, (6)

where

τ

St=^S₁₈^ρ^d_ν²^p istheparticleStokestimeand

τ

f=^Du istheﬂow timescale.Here,S_ρ=^ρ_ρ^p isthedensityratiobetweenparticleand ﬂuid,dpistheparticlediameterandDisthediameterofthecylin- der. Twoforcesareactingonthe particle:thedragforce andthe thermophoretic force (gravity is neglected forthe small particles studiedhere).Thedragforceisgivenby:

FD=mp

τ

^p

⁽

^u⁻

v

p

)

^, ⁽⁷⁾

where

τ

p,m_p and

v

p are the particle’s response time, mass and velocity, respectively. Using the Stokes time with the Schiller- Naumanncorrectionterm[12]toaccountforlowtomoderatepar- ticleReynoldsnumbers,yieldstheparticleresponsetime

τ

^p⁼

τ

^St

f , (8)

where

f =1+0.15Re⁰_p^.⁶⁸⁷ (9)

andRep=dp

| v

p−u

|

/

ν

^is^the^particle^Reynolds^number.

Thethermophoreticforcepushesparticlesfromregionsofhigh temperature to regions oflow temperature. As such, itis similar

Fig. 2. Comparison of different thermophoretic force terms.

totheSoreteffectforgases.Itwasﬁrstobservedin1870by Tyn- dall[13],andithaslaterbecomewidelystudiedbothexperimen- tallyandtheoretically.Atheoreticalanalysisofthethermophoretic forcecanbefoundintheworksofZheng[14].Young[15]givesan overviewoverthedifferentregimesofthermophoresis, whichare determinedbytheparticlesKnudsen numberKn=

λ

/dp,where

λ

isthemeanfreepathofthegas.Forthepresentstudy,allparticles arein thecontinuum regime(Kn1), hencethethermophoretic forceiscalculatedby

Fth=

μ

²^rp

∇

^T

ρ

^T ^, ⁽¹⁰⁾

whererp=dp/2isparticleradius,

μ

=

ρν

îs^dynamic^viscosityând îs^thethermophoreticforceterm.Theexpressionforîs^taken fromEpstein[16]:

=−12

π

^K^tc

2+

^, ⁽¹¹⁾

wherethe conductivityratio betweenthe particleand thegas is given by =kp/k_f, while the temperaturecreep coeﬃcient, Ktc, usedinthisworkhasavalueof1.1, whichisinthemiddleofthe rangereported by Sharipov [17].This rathersimple model for simpliﬁestheanalysis,whilestillprovidingagreeableresultswhen comparedwiththewidelyusedapproachproposedbyTalbotetal.

[18].FromFig.2weseethatthelargestrelativedifferencebetween thesimplifiedând^theôneôbtained^whenûsing^theâpproachôf Talbotislessthanafactoroftwo.

2.3. Theory

Duetotheirshortresponsetimes,verysmallparticleswillfol- lowtheﬂuidalmostperfectly,essentiallybehavingliketracerpar- ticles. For isothermal situations, Haugen and Kragset [3] showed thatasmallfractionoftheseparticleswillneverthelessimpacton thecylindersurfaceduetotheirsmallbutﬁniteradii.

For the non-isothermal case, where the temperature of the cylinder is lower than that of the surrounding gas, the thermophoreticforce willinduce a relativevelocity betweenthepar- ticlesandtheﬂuidthattransporttheparticlesinthedirectionto- wardsthecylinder.Theeffectofthisisthatalargerfractionofthe particles impacton thecylindersurface. Inthe followingwe will trytoquantifythiseffect.

Forlaminarflows,afluidstreamlinethatstartsfarupstreamof thecylinderwithadisplacement^x^from^the^central^line^(the^line paralleltothemeanflow,goingthroughthecentreofthecylinder) willmoveintheboundarylayerofthecylinderwitharadialdis- placementfromthecylindersurfaceof^rf=^x^(see ^Fig.³^).^Far upstreamofthecylinder,themassflowratebetweenastreamline andthecenterlineofthecylinderisgivenby

˙

mu=Hu0

ρ

⁰

x, (12)

where^x ^is^the ^distance^between^the^streamline ^and^the^centre lineandHistheheightofthecylinder.Withintheboundarylayer

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Fig. 3. Sketch of a particle track of a particle with a small Stokes number under the inﬂuence of thermophoresis.

ofthecylinder, however,the massﬂowratebetweenthestream- lineandthecylindersurfaceisgivenby

˙ m_b=

rf

0

ρ

^H^u_θ^drcyl=

ρ

^H^u⁰^Re¹^/²

2BD

(

^rf

)

², (13)

where

ρ

îs^theâverage^fluid^density^within^the^boundary^layer.^By

followingHaugen andKragset [3],we haveused thefactthatthe tangentialﬂuidvelocitywithintheboundarylayerisgivenby u_θ

(

^rcyl

)

=u0Re¹^/²

BD r_cyl, (14)

whereu₀isthefarﬁeldﬂuidvelocity,Bisaconstantoftheorder ofunityandr_cylisthenormaldistancefromthecylindersurface.

The impactioneﬃciencyfora stationarynon-turbulent ﬂowis givenby

η

⁼²

^xmax/D, (15)

where ^xmax isthemaximum ^xînside ^which^the ^particles^can start out in orderto impact on the surface. Since streamlines of laminar flowsdo notcross eachother, itisclearthat fora given type of particles, all particles inside ^x^max ^will împact ôn ^the cylindersurfacewhilenoneoftheparticlesoutsideof^xmaxwill impact. From now on, a particle startingout atthe limitingdis- tanceof^x^maxâway^from^thecentre-linewillbereferredtoasthe lastimpactingparticle.ForReynoldsnumbersabove≈48,theflow willbecomeunsteadyandvonKrmneddieswilloccurinthewake ofthecylinder.Theeffectofthisunsteadinessonthefrontsideim- pactionisminor.Backsideimpactionmay,however,bestronglyaf- fectedbythevonKrmneddies.Sincethefocusofthecurrentwork istostudyfrontsideimpaction,thedefinitionoftheimpactionef- ficiencyasgivenbyEq.(15)willalsobeusedforunsteadyflows.

The above equationscan be solved to findthe impactioneffi- ciencyforsmallparticlesbycalculatingthedistance,disp,aparticle willmove in theradial directiondueto the thermophoretic force duringthe timeitisinthefrontside boundarylayerofthe cylinder, andsettingthis distanceequal to^rf. Inthe following, the focuswilltherefore beon findingdisp. Here,smallparticles are defined asparticles that are sosmall that the maincause of impactionisthethermophoreticforce.Thisistypicallythecasefor St0.1.

By combining Eq. (7) and Eq. (10), while setting the radial component of the gas phase velocity to zero, we ﬁnd the ther- mophoreticvelocityoftheparticles(intheradialdirection)as

v

th=

ν

6

π

^f

∇

^T

T . (16)

ThecorrectiontermtotheStokestime,asgivenby f (seeEq.(9)), is always close to unity ifthe thermophoreticforce is the main driver of the particle velocity relative to the surrounding ﬂuid.

We therefore set f=1 for the remainder of this analysis. From Prandtl’sconceptofthinboundarylayers,weknowthatthethick-

nessofthevelocityboundarylayercanbeapproximatedby

δ

vel= D

Re¹^/²B. (17)

ThethermalboundarylayerthicknessisthengivenbySchlichting [19]

δ

thermal=

δ

velPr⁻¹^/³. (18)

Hence, the average thermal gradient in the boundary layer be- comes

∇

^T^≈

δ

thermal

^T

=

^T

D BRe¹^/²Pr¹^/³. (19) Theeffectofthethermophoreticforce onthepositionofthepar- ticlecannowbeconsideredastheradialdisplacementofthepar- ticle(disp) fromtheposition(xstream) itwouldhavewithoutthe inﬂuenceofthethermophoreticforce (see Fig.3).Thisradial dis- placementisgivenby

disp=

v

th

τ

th, (20) where

τ

th= D

u_θ

(

disp

)

⁽²¹⁾

isthetimetheparticlestays withinthefrontsideboundarylayer andu_θ(disp)îs^the^tangential^velocityôf^the^flowⁱⁿ^the^boundary layera distancedisp away fromthesurfaceof thecylinder (see Fig. 3). From Eqs. (14) and (21) we can now find the time that thelastimpactingparticlestayswithinthecylinderboundarylayer beforeithitstheboundaryas

τ

th= BD² u0Re¹^/²

disp

. (22)

Combining Eq.(22)withEq.(16),Eqs. (19)and(20), andsolving fordisp,thenyields

²disp=

^B²^Pr¹^/³^D²

6

π

^Re

^T^T^, ⁽²³⁾

whereweusethatRe=u₀D/

ν

^.^Since^the^ﬂuid^is^not^turbulent^(i.e.,

streamlinesdonotcrosseachother),we knowfrommassconser- vation thatthemassflux betweenthestreamlineandthe central lineupstream ofthe cylinder (m˙_u) isequal to the mass flux be- tweenthestreamlineandthecylindersurface(m˙_b).Havingfound disp fromEq.(23),wethereforeproceedbysettingthetwomass fluxesdefinedinEqs.(12)and(13)equaltoeach otherandsolve for^x^max^to^find

^xmax=

ρ ρ

⁰

Re¹^/²

2BD

²disp. (24)

Intheabove wehaveusedthefactthat^x=^x^max ^when^rf= disp.FromEq.(15),wenowﬁndthatforanon-negligibletemper- ature difference, the captureeﬃciency for smallStokes numbers (St0.1)isgivenby

η

⁼²

_D^x^max ⁼

^B^Pr¹^/³

6

π

^Re¹^/²

T T

ρ ρ

⁰ ⁼

2KtcBPr¹^/³ Re¹^/²

(

²+

)

^T T

ρ ρ

⁰^. ⁽²⁵⁾

Forlaminar flows, Haugen and Kragset [3]found that B is inde- pendentofReynolds numberbutvarieswithangularposition on thecylindersurface.Inparticular,theyfound1/Btobeverysmall atthe frontstagnation point whilethe minimumvalue of Bwas foundtobe0.45ataposition60degreesfurtherdownstream.For the remainder of this paper we chose B=1.6, which is a value that yields good model predictions. In order to obtain the last partofEq.(25),wehaveused thesimplified versionof thethermophoreticforceterm(^),^butâny^version^can^beûsed^here.

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The above approach, yielding an impaction eﬃciency due to thermophoreticforcesforsmallStokesnumbers,isstrictlyapplica- bleonlywhenthedistancetheparticletravelswithinthebound- arylayerislessthanafraction

α

^of^the^thickness^of^the^boundary

layeritself,i.e.when

disp

δ

thermal

=B²

2KtcPr

^T

(

²+

)

^T ^<

α

^. ⁽²⁶⁾

We shalllater seethat the criticalvalue of

α

^is ^somewhere ^be-

tween0.5and1.

3. Numericalmethods

The simulationsforthe presentwork areperformedusingthe Pencil Code, an open source, highly parallelizable code for compressible ﬂows with a wide range of implemented methods to modeldifferentphysicaleffects[20–22].

The effect of thermophoresis on the impaction efficiency is studied by releasing a large number of particles upstream of a cylinderinan establishedquasi-steadyflowfield.Everytime step, newparticlesareinserted atrandompositions ontheparticlein- sertion plane (shownasa red lineinFig. 1) withavelocity that isequaltotheinletfluidvelocity.Thedomainistwo-dimensional and has a width of6D anda length of 12D, where D is the diameter of the cylinder. The flow enters the domain on the left and leaves the domain through the outlet on the right. Navier- Stokes characteristicboundary conditions are applied atboth inlet andoutlet to ensure that they are non-reflective foracoustic waves [23].All other boundariesare periodic. A cylindricalover- set gridis placedaround thecylinderto accuratelyrepresentthe cylinder atlow computational cost. Thiscylindrical grid commu- nicates with the Cartesian background grid via its outer points.

Summation-by-parts is usedfor derivativeson the surfaceofthe cylinder,andaPadé ﬁlterisusedtomitigatehighfrequencyoscil- lations onthecylindricalgrid.Fordetailsconcerningthecylindri- cal oversetgrid,includingaccuracyassessmentandvalidation,the readerisreferredtoAarnesetal.[5],6],24].

The backgroundgrid uses288and576cellsforthewidthand length, respectively, except for the cases with Reynolds number of1600, wheretheresolutionisdoubled.The cylindricalgrid has 144 cells in the radial and 480 cells in the tangential direction for the caseswith Reynolds numbersup to 400,and doublethe amountforthecasewithaReynoldsnumberof1600.Theresolu- tion increase is done toensure that the boundarylayer -scaling as1/Re¹^/² -isaccuratelyresolvedatRe=1600.Gridstretchingin the radialdirectionoftheoversetgrid isusedtoensure approxi- matelymatchingcellsizesontheoutergridpointsofthecylindri- cal cells, where the backgroundand the oversetgrids communi- cate. Thecodeusesasixth-orderﬁnitedifferenceschemeforspa- tial discretisationanda third-orderRunge-Kuttascheme fortem- poraldiscretisation.Sincethecellsizeclosetothecylindersurface ismuchsmallerthanthegeneralcellsizeofthebackgroundgrid, thetimestepofthebackgroundgridcanbeamultipleofthetime stepofthecylindricalgrid,withthecylindricalﬂowbeingupdated moreoften.Fordetailsoftheparticletrackingscheme,thereader isreferredtoHaugenandKragset[3]orAarnesetal.[6].

4. Simulations

The inﬂowtemperatureis873Kandthedensityis0.4kg/m³, correspondingtothedensityofairatthistemperatureandapres- sureof1bar.Theparticlematerialdensityis400kg/m³,yieldinga densityratioS_ρ of1000basedontheﬂuiddensityunderinletcon- ditions.Thesevaluesarechosenbasedontheirrelevanceforparti- cledepositiononsuper-heatertubesinthermalpowerplants.The results are neverthelessgeneric since they aregiven asfunctions

Table 1

Range of parameters studied.

Parameter Values

Reynolds number [-] 100, 400, 1600 Conductivity ratio [-] 1, 12, 144 T [K] 0, 1, 3, 10, 173, 400

ofnon-dimensional numbers.The cylinder temperatureis imple- mentedasaDirichletboundaryconditionandsettoaﬁxedvalue.

The inﬂow velocity u₀ is set so that the ﬂow Reynolds number Re=u₀D/

ν

^is¹⁰⁰^for^the^reference^case.^For^the^cases^studying^the

Reynoldsnumbereffect,the viscosityischanged toobtain differ- entReynoldsnumbers.TostudythesamerangeofStokesnumbers, theparticlesizeisadjusted accordingly,andthethermaldiffusiv- ityisdecreasedtoachieve a constantPrandtlnumber.The differ- entaspectsofthethermophoreticeffectareanalysed bychanging one criticalparameteratthe time, whileholdingthe otherscon- stant.Thesecriticalparametersare:1)Reynoldsnumber,2)Prandtl number,3)temperaturedifferencebetweenﬂuidandcylinderand 4)conductivityratio.Thelatterisgivenby

=kp

kf = kp

Dthermalcp

ρ

⁼

k_{p f}

Dthermal, (27)

wherek_{p f}=k_p/(^cp

ρ

)^.^Since^we^keep^the^thermaldiffusivityofthe ﬂuid constant whenchanging the conductivityratio,the conduc- tivityratioisessentiallychangedbychangingk_{p f}.

ForeachoftheparameterslistedinTable1,theimpactioneﬃ- cienciesofparticlesintheStokesnumberrangebetween0.01and 10areobtainedfromtheDNSsimulations.

Foreachparticlesize(Stokesnumber),acertainnumberofpar- ticleshavetoimpactthesurfacetogetsufficientstatisticsinorder to estimate an accurate impactionefficiency. Sincethe impaction efficiencydecreases significantly withStokesnumber, wehave to releasemoresmallthanlargeparticles.Therefore,15,000particles arereleasedforeach particlesizeforSt>1,200,000-400,000 for particleswith0.1<St<1,and2millionforeachparticlesizefor particleswithSt<0.1.Theparticlesareinsertedoverseveralvor- texsheddingtimestomitigatetheeffecttheinstantaneousvortex sheddingcouldhaveontheresults.

Allsimulations canbedescribedby then=10uniqueandin- dependentvariables that arelisted inTable 2.Fromthe table we seethatall variablesinvolveatotal ofk=4differentunits(m, s, kgandK).

FromtheBuckingham-Pitheorem,wethusknowthatthesim- ulations can be described by exactly p=n−k=6 different dimensionless numbers.Thesedimensionless numbersare listed in Table3.

Inthiswork,westudytheeffectofvariationsinallofthesedi- mensionlessnumbers,exceptforS_ρ,whichisalwayskeptconstant atS_ρ=1000.ChangingS_ρmeansthatanotherdimensionlessnum- ber,D/d_p=

ReS_ρ/(¹⁸^St)^,^will^change.^Theêffectôf^thisîsâ^shift intheleveloftheinterceptionmode.I.e.,ashiftwilloccurinthe mode by which very small particles intercept the cylinder when onlypureimpactionisaccountedfor.Thisisimportanttoaccount forwhencomparingsimulationresultswithdifferentvaluesofS_ρ, whichforexampleisdoneinFig.11bofKleinhansetal.[25].The parametervariationsinoursimulationsarelistedinTable4.

5. Results

Inthis section we willstudythe effectof thethermophoretic forceontheimpactioneﬃciency.Inparticular,wewilllookathow the impaction eﬃciency is affected by changes in the Reynolds number,temperaturedifferenceandconductivityratio.

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Table 2

Independent variables describing the simulations.

Variable value unit Description

k p f (1.43, 17.2 or 206) ×10 ⁻³ m ²/s Norm. therm. diff. of particles

ρp 400 kg/m ³ Material density of particles

d p varies m Diameter of particles

ρ 0.4 (at inlet) kg/m ³ Material density of ﬂuid

D thermal 1 . 43 ×10 ⁻³ m ²/s Thermal diffusivity of ﬂuid

u 1 (at inlet) m/s Velocity of ﬂuid

ν (6.25, 25 or 100) ×10 ⁻⁵ m ²/s Viscosity of ﬂuid

D 0.1 m Diameter of cylinder

T f 873 K Far-ﬁeld temperature of ﬂuid

T c 700–873 K Temperature of cylinder

Table 3

Dimensionless numbers describing the simulations.

Dimensionless number Description = k p f/D thermal Conductivity ratio Re = Du/ν Reynolds number

= T f/T c Temperature ratio between far-ﬁeld and cylinder S ρ= ρp/ρ Density ratio between particle and ﬂuid St = S ρd ²pu/ (18 νD ) Stokes number

Pr = ν/D thermal Prandtl number

Table 4

Overview of the simulated cases. Particles with Stokes number ranging from 0.01 to 10 are inserted in all cases.

Sim. Re Pr T

’Base case’ 100 0.7 - -

0 100 0.7 173 12

C1 100 0.7 173 1

C144 100 0.7 173 144

dT3 100 0.7 3 12

dT10 100 0.7 10 12

dT400 100 0.7 400 12

R400 400 0.7 173 12

R1600 1600 0.7 173 12

RPv400 400 0.175 173 12

RPv1600 1600 0.043 173 12

Fig. 4. Comparison of ηfor data from Aarnes et al. [5] with data obtained from an isothermal case with and without thermophoretic force.

Aarnesetal.[5]usedDNSto findtheefficiencybywhichpar- ticlesembeddedinan isothermalcrossflow impactonacylinder.

Intheirstudytheyusedanoversetgrid,buttheydidnotconsider the thermophoreticforce nordidthey solve theenergyequation.

Inthefollowing,wewillusetheirresultsasareferencecase,from now oncalledthe“Base case”. Fig.4compares theimpactionef- ﬁciencyof the“Base case” withwhatisfound forthesame conditions inthecurrentwork.From theﬁgure,we seethat theim-

Fig. 5. Front side impaction efficiency as a function of Stokes number. For the non- isothermal cases ( T = 173 K), results with thermophoretic factors as given by Ep- stein and Talbot are compared. The red dotted line corresponds to the impaction efficiency predicted by the model for small Stokes numbers as given in Eq. (25) . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

paction efficiencyshowsaslight decreasewithdecreasing Stokes numberforStokes numbersabove 1,followedby asteep dropof impactionefficiencyinthe Stokesnumberrangebetween0.1and 1.For even smallerStokes numbers, the impactionefficiency de- creaseslinearly withdecreasing Stokesnumber. As expected, our simulations of isothermal cases both with (blue line) and without(orangeline)thethermophoreticforce(Eq.(10))includedyield the same impaction efficiency profile as the ’Base case’. A case with Brownian forces on the particles has also been performed, andaweakeffectofBrownianforcesforthesmallestStokesnum- bersisvisible.Thiseffectis quiteweak,and, aswe shallsee,the thermophoreticforcewillhaveamuchstrongereffectontheim- pactionefficiencyevenforverysmalltemperaturegradients.

In Fig. 5, the “Base case” (isothermal) is compared to non- isothermalcaseswiththermophoresis.Simulationswithatemper- aturedifferencebetweentheinletgasandthecylindersurfaceof ^T=173 Kand a particle conductivityratio of 12 wasused for thethermophoreticcases. Itwasshownin §2.2that the models ofEpsteinandTalbotgavecomparablevaluesof^.^The^two ^solid linesinFig.5showthecorrespondingdifferenceinimpactioneffi- ciency.Forthesmallerparticles,theimpactionefficiencypredicted bythemodelofTalbotetal.isonlyabout10%higherthantheone predictedbyEpstein,a differencethatdisappears forlarger particles.Itisclearfromourresultsthattheparticleimpactionisunaf- fectedbythethermophoreticforceforlargeStokesnumbers,while itisdominatedbythethermophoreticforceforsmallStokesnum- bers.Instarkcontrasttowhatisobservedfortheisothermalcase, theimpactionefficiencybecomesindependentoftheparticlesize forsmallparticles.Thetheoreticalpredictionoftheimpactioneffi- ciencyforsmallStokesnumbers,aspresentedinEq.(25),isrepre-

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Fig. 6. Front side impaction eﬃciency over Stokes number for different conduc- tivity ratios. This corresponds to simulations ’Base case’, 0, C1 and C144 as listed in Table 4 . The dotted lines correspond to the impaction eﬃciency predicted by Eq. (25) for small Stokes numbers.

Fig. 7. For the simulations marked in the grey area, the model from Eq. (25) is applicable. From lightest to darkest the grey areas correspond to α= 1 . 0 , 0.5 and 0.25. For the cases that are in the white area, however, Eq. (26) yields no applicabil- ity. The simulations with different Reynolds number and constant Prandtl number (“R400” and “R1600”) are positioned at the same place as simulation “0”.

sentedbythereddottedhorizontallineintheﬁgureandonecan seethatitﬁtswellwiththenumericalresultsforSt<0.1.

The effectofdifferentconductivityratios ontheimpactionef- ficiency is shownin Fig. 6.We see that the impaction efficiency for small Stokes numbers is higher forlower values of the conductivity ratio.Thisis becausea smallconductivityratioyields a largethermophoreticforceterm(^),^which^resultsⁱⁿâ^large^thermophoreticforce, leadingtohighimpactionefficiency.This effect can also be seen directly fromEq.(25),where the impactionefficiency for small Stokes numbers scales linearly with ^. ^When thesimplifiedexpressionforîsûsed,â^linear^dependenceôn

meansthattheimpactioneﬃciencyisinverselyproportionalwith (²+)^.

Theapplicabilityofourmodel,whichisgivenasafunctionof

α

ⁱⁿÊq.⁽²⁶⁾^,îs^visualisedⁱⁿ^Fig.⁷^for^different^valuesôf

α

^.^From

theseresultsitis apparent thatthe modelisnot strictlyapplica- ble for the smallest conductivity ratio (case C1) if

α

0.5. This is probablythe reasonwhy themodelled impaction eﬃciencyat smallStokes numbersfor=1(asrepresentedbythehorizontal bluedottedlinein Fig.6)deviatessomewhat fromthesimulated results(bluesolidline).Itshouldbenotedthataconductivityratio of1isquiteimprobable.Zhangetal.[26]giveavalueforthecon- ductivity ratio ofsmall char particles of ≈9. The conductivity ratiomay,however,be differentforother solids.Inthe following weuse=12asabaselineforoursimulations.

Bycomparing how well our model reproduces the simulation results, it seems reasonable to assume that

α

∼0.5−1.0. Based onEq.(26),wecanthen ﬁndthat forcharparticles(≈10),the morestringentvalueof

α

⁽=0.5)resultsinthemodelbeingappli- cableaslongas^T îs^less^than≈30%ofthefarfieldtemperature forfluidswithPr=0.7.Forthesameconditionsand

α

=1.0,our modelisapplicableforallvaluesof^T^.

In the left hand panel of Fig. 8, the impaction eﬃciency is shownasafunctionofangularpositionfordifferentStokesnum- bers for simulations with a conductivity ratio of =12. (The centre-lineisat270degrees.)ForthelargestStokesnumber(St= 0.9), all impaction occurs within an angle,

θ

max, that is smaller than 60 degrees from the centre line. This is consistent with the ﬁndings of Haugen and Kragset [3]without thermophoresis.

Whenthermophoresisisaccountedfor,however,theparticleswith smaller Stokes numbers impact the entire frontal surface of the cylinder. By increasing the strength of the thermophoretic force, whichis heredone by decreasingthe conductivityratioto unity, weseefromtherighthandpanelofFig.8thattheangularposition ofimpactionbecomesalmostuniformforthesmallerparticles.

Since thegas is an idealgas, the cylinderis surrounded by a boundarylayerofdensifiedgaswithasignificanttemperaturegra- dientfor cases withlarge temperaturedifferences. Alarger temperature differenceyields a stronger thermophoreticforce, which againresults in a higher impaction efficiencyfor small particles.

AscanbeseenfromFig.9,theresultscomputedfromourmathe- maticalmodel(dottedlines) ﬁtthesimulationresults(solidlines) wellforalltemperaturedifferencesstudiedhere(3K<^T<400 K).YetweseefromFig.7that^T=400Kisclosetothelimitof theapplicabilityofthemathematicalmodel,socautionisadvised whenusingthemodelinthisuppertemperaturerange.

LastlyweinvestigatetheeffectofReynoldsnumberontheim- pactioneﬃciency.Inthiswork, weincreasetheReynoldsnumber bydecreasingtheviscosity.Whiledoingthis,wealsoincreasethe resolutionin ordertoproperly resolvethe boundarylayer, which

Fig. 8. Front side impaction angle for = 12 (left panel) and = 1 (right panel). Simulations 0 and C1, respectively.

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Fig. 9. Impaction eﬃciency over Stokes number for different cylinder temperatures (simulations 0, dT3, dT10 and dT400 as listed in Table 4 ). A high temperature difference increases ηfor small particles. The dotted lines correspond to the impaction eﬃciency predicted by Eq. (25) for small Stokes numbers.

Fig. 10. Front side impaction efficiency over Stokes number for different flow Reynolds numbers. Simulations “0”, “R400” and “R1600”. Higher Reynolds numbers result in higher ηfor medium Stokes numbers, while for low Stokes numbers η is decreased. The dotted lines correspond to the impaction efficiency predicted by Eq. (25) for small Stokes numbers.

is thinnerforhigherReynolds numbers.Ifthethermal diffusivity ischangedlinearlywithviscosity,thePrandtlnumberiskeptcon- stant. Thisiswhatisdone inFig.10,fromwhichweseethat in- creasingtheflowReynoldsnumberresultsinhigherfrontsideim- paction efficiencies forintermediate Stokesnumbers inthe range 0.2<St<1. This is qualitatively consistent with the findings of Haugen andKragset [3] obtainedfor isothermal cases. From the results in Fig. 10 we also see that there is a clear but not dra- matic Reynolds number effect for small Stokes numbers. This is supportedbythemodelinEq.(25),wheretheimpactionefficiency for small Stokes numbers is inversely proportional to the square rootof theReynoldsnumber. Theprediction ofEq.(25)isrepre- sentedby the horizontaldottedlinesin Fig.10,accurately repro- ducingtheDNSresults.Wewouldalsoliketopointoutthat,since higherReynoldsnumbersareobtainedbydecreasingtheviscosity, theparticlesforlowStokesnumbersbecomequitesmallforcases withlargeReynoldsnumbers.Unlikeforisothermalcases,particle size doesnot,however,play anysignificantrole inthe impaction ofparticleswithsmallStokesnumberwhentheimpactionmech- anismisdominatedbythethermophoreticforce.

If the thermal conductivityof the ﬂuid is not changed when changingviscosity,thePrandtlnumberisdecreasedforincreasing Reynolds numbers, which is the case for the simulations shown inFig.11.Herewe seethatthe Reynoldsnumberdependenceon thedifference inimpactioneﬃciencyforthesmallerStokesnum-

Fig. 11. Front side impaction eﬃciency over Stokes number for different ﬂow Reynolds numbers. The Prandtl number is inversely proportional to the Reynolds number, such that it equals 0.7, 0.175 and 0.043 for Re = 10 0, 40 0 and 160 0, respectively. Simulations “0”, “RPv400” and “RPv1600”. Higher Reynolds numbers result in higher ηfor medium Stokes numbers, while for low Stokes numbers ηis decreased.

The dotted lines correspond to the impaction eﬃciency predicted by Eq. (25) for small Stokes numbers.

bersislargerthanforthecasewithconstantPrandtlnumber.This isalso inagreement withEq.(25) wherelower Prandtl numbers yieldlower impactioneﬃciencies,such thatboth thePrandtland Reynoldsnumbereffectsworkinthesamedirection.Bycomparing thesolid andthedottedlines,we seethat theDNSresults(solid lines)followsthemodelpredictions(dottedlines)nicely.

6. Conclusions

UsinghighorderDNS,theeffectofthermophoresis ontheim- paction efficiency of particles on a cylinder is studied for different values of the Reynolds number, conductivity ratio, temperature difference and particle Stokes number. It is found that the thermophoreticforce has an insignificanteffect onparticles with Stokesnumberslargerthan∼0.5,butisoftendominatingtheim- paction ratefor St0.3. For small particles (St0.2) we know thattheimpactionefficiencyscaleslinearlywiththeStokesnum- berforisothermal cases.Thisis,however,notthecasewhenim- paction iscontrolled by the thermophoreticforce. Forsuch cases theimpactionefficiencyisindependentofStokesnumberforSt 0.1.Furthermore,theimpactionefficiencyislargerforlowconduc- tivityratios,hightemperaturedifferencesandlow Reynoldsnum- bers.

An algebraicmodel (see Eq. (25)) that predicts the impaction efficiency due to thermophoresis has been developed based on fundamentalprinciples.Thevalidityofthemodelforawiderange of conditionshas been verified against highly accurate DNS. The model can therefore be used for accurate analytical predictions ofthe impactionefficiency.The developedmodelis validaslong asthe thermophoretic force is not very strong (see Eq. (26)and Fig.7).Outsideits rangeofvalidity, areliablemodeldoesnotyet exist.Thisshouldbethefocusoffutureresearch.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

NilsErlandL. Haugen: Conceptualization, Methodology, Soft- ware,Validation,Investigation,Resources,Datacuration,Writing–

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original draft, Writing – review & editing,Visualization, Supervi- sion,Fundingacquisition.JonasKrüger:Software,Validation,For- malanalysis, Investigation,Datacuration,Writing – originaldraft, Visualization. JørgenR. Aarnes: Conceptualization, Methodology, Software,Datacuration,Writing– originaldraft,Writing– review

&editing.EwaKarchniwy:Software,Datacuration,Writing– originaldraft,Writing– review&editing.AdamKlimanek:Resources, Writing – originaldraft, Writing – review & editing, Supervision, Fundingacquisition.

Acknowledgements

This research was supported by The GrateCFD project [grant 267957/E20], which is funded by: LOGE AB, Statkraft Varme AS, EGE Oslo, Vattenfall AB, Hitachi Zosen Inova AG and Returkraft AS together with the Research Council of Norway through the ENERGIX program. Computational resources were provided by UNINETT Sigma2AS [project numbersNN9405K]. We wouldalso like to acknowledge that this research was partly funded by the Research Council of Norway (NorgesForskingsråd) underthe FRINATEK Grant[grantnumber231444]andbythe grant“Bottle- necksforparticlegrowthinturbulentaerosols” fromtheKnutand AliceWallenbergFoundation,Dnr.KAW2014.0048.

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