The effect of turbulence on mass and heat transfer rates of small inertial particles

0.48 MB·

by  Nils Erland L. Haugen

UnderconsiderationforpublicationinJ.FluidMech. 1
ff
The e ect of turbulence on mass and heat transfer
rates of small inertial particles
7
1
0
NilsErlandL.Haugen1,2,JonasKru¨ger1,DhrubadityaMitra3 andTereseLøvås1
2
n 1DepartmentofEnergyandProcessEngineering,NorwegianUniversityofScienceandTechnology,
KolbjørnHejesvei1B,NO-7491Trondheim,Norway
a
J 2SINTEFEnergyResearch,N-7465Trondheim,Norway
3Nordita,KTHRoyalInstituteofTechnologyandStockholmUniversity,Roslagstullsbacken23,
7
SE-10691Stockholm,Sweden
1

(Received)
]
n
y The effect of turbulence on the mass and heat transfer between small heavy inertial particles
d (HIP) and an embedding fluid is studied. Two effects are identified. The first effect is due to
- therelativevelocitybetweenthefluidandtheparticles,andamodelfortherelativevelocityis
u
presented.Thesecondeffectisduetotheclusteringofparticles,wherethemasstransferrateis
l
f
inhibited due to the rapiddepletion of the consumedspecies inside the dense particle clusters.
.
s This last effect is relevant for large Damko¨hler numbers and it may totally control the mass
c
i transfer rate for Damko¨hler numbers larger than unity. A model that describes how this effect
s
y shouldbeincorporatedintoexistingparticlesimulationtoolsispresented.
h
p Keywords:Reactingmultiphaseflow,Particle/fluidflow,Combustion,Turbulentreactingflows,
[ Turbulencesimulations

1
v
7
6
1. Introduction
5
4 Both in nature and in industrial applications,one regularlyfinds small inertial particles em-
0
beddedinturbulentflows.Bysmallinertialparticles,wemeanparticlesthataresmallerthanthe
.
1 smallestscalesoftheturbulenceandhavesignificantlyhighermaterialdensitythanthefluid.For
0 suchparticles,therewillbemomentumexchangebetweentheparticlesandtheturbulentfluid,
7
and,dependingontheconditions,theremayalsobeheatandmasstransfer.Thisisparticularly
1
soforchemicallyreactingparticles,buttherearealsoalargenumberofotherapplicationswhere
:
v heatandmasstransferbetweenparticlesandfluidareimportant.Here,themainfocuswillbeon
i
X reactingparticlesthatconsumeoneormoreofthespeciesintheembeddinggasthroughsurface
reactions.Relevantexamplesare;chemicalreactionsonthe surfaceof a catalytic particle,fuel
r
a oxidationonthesurfaceofaoxygencarryingparticleinaChemicalLoopingCombustion(CLC)
reactor,condensationofwatervaporonclouddropletsandcombustionorgasificationofchar.
The presence of turbulence in a fluid will enhance the transportpropertiesof the flow. This
means that the mean-field viscosity, diffusivity and conductivity may be drastically increased
from their laminar values. This effect has been studied for many years, and a large numberof
different models exist in the literature, such as the k-ǫ model (Jones&Launder (1972)) and
differentversionsoftheReynoldsStressModels(e.g.Pope (2003)).Turbulencemayalsomodify
gasphasecombustion,andeventhoughthisissomewhatmorecomplicated,asignificantnumber
ofmodelshavebeendevelopedduringthelastdecades.SomeexamplesaretheEddyDissipation
Model(Magnussen&Hjertager(1976)),theEddyDissipationConcept(Ertesvåg&Magnussen
(2000))andvariationsofProbabilityDensityFunction(e.g.Dopazo (1994))models.
Withtheaboveknowledgeinmind,itisinterestingtorealizethat,exceptfortherecentwork
2 NilsErlandL.Haugen1,2,JonasKru¨ger1,DhrubadityaMitra3andTereseLøvås1
ofKrugeretal. (2016),thereiscurrentlynomodeldescribingtheeffectofturbulenceontheheat
andmasstransferofsmallinertialparticles.Whenareactingparticleisembeddedinaturbulent
flow, the turbulencecan potentiallyinfluence the mass transfer, and hence the surface reaction
ratesintwoways.Thefirstwayisthroughparticleclustering,whereparticlesformdenseclus-
tersduetoturbulence,andwherethegasphasereactantswithintheclusterarequicklyconsumed
whiletherearenoparticlesthatcanconsumethereactantsintheparticlevoidsoutsidetheclus-
ters. The maineffectofthe clusteringisto decrease theoverallmasstransferrate.The second
way turbulence influence the mass transfer rate is by increasing the mean velocity difference
betweentheparticleandthegas.Thiseffectwillincreasethemasstransferrate.
Thesametwoeffectsarealsoactivefortheheattransfer.Thesimilaritybetweenheatandmass
transfercanbeseenbyconsideringtheexpressionsforthetransfercoefficientsofmass
ShD
κ= (1.1)
d
p
andheat
NuD
κ = th, (1.2)
th
d
p
where d is the particle radius, Sh and Nu are the Sherwood and Nusselt numbers and D and
p
D are the mass and thermaldiffusivities. For single spherical particles in flows with low and
th
mediumparticleReynoldsnumbers,theSherwoodandNusseltnumberscanbeapproximatedby
theempiricalexpressionsofRanz&Marshall (1952)
Sh =2+0.69Re1/2Sc1/3 (1.3)
RM p
1/3
Nu =2+0.69Re1/2Pr.
RM p
Awellknownexamplewherereactingparticlesareconsumedinaturbulentfluidisthecase
ofpulverizedcoalcombustion,whereturbulenceinfluencestheprocessinseveralwaysthatare
understood to varying degrees. The combustion of coal can be divided into four separate pro-
cesses;1)drying,2)devolatilization,3)combustionofvolatilesand4)burnoutoftheremaining
char.Processes1and2involvetheevaporationoffluidsandthermalcrackingofhydrocarbons,
whileprocess3involveshomogeneousreactions.Inprocess4,gasphasespeciesdiffusetothe
particlesurface andreactwith the solidcarbon.Thishappensvia adsorptionofe.g.an oxygen
radicaltoacarbonsiteontheparticlesurfaceandasubsequentdesorptionofcarbonmonoxide
intothegasphase.Thismakesprocess4dominatedbyheterogeneouschemicalreactions.Many
published studies utilize RANS based simulation tools that describe simulations of pulverized
coal conversionin the form of combustion or gasification with an Eulerian-Eulerianapproach
(Gaoetal. (2004)andZhangetal. (2005))oraLagrangian-Eulerianapproach(Silaen&Wang
(2010);Vascellarietal. (2014,2015);Klimaneketal. (2015);Chenetal. (2012,2000)).How-
ever,noneofthesepaperstaketheeffectofturbulenceontheheterogeneouscharreactionsinto
account. To the knowledgeof the authors, the only studies where accountis made for this ef-
fectarethepapersofLuoetal. (2012);Brosh&Chakraborty (2014);Broshetal. (2015)and
Haraetal. (2015)wheretheDirectNumericalSimulations(DNS)approachisused.InaDNS,
all turbulence scales are explicitly resolved on the computational grid, such that the effect of
turbulenceisimplicitlyaccountedfor.However,theDNSapproachisextremelycostlyandcan
thereforeonlybeusedforsmallsimulationdomains.Forsimulationsoflargescaleapplications,
theRANSorLESbasedsimulationtoolswillthereforebetheonlyapplicabletoolsforthefore-
seeablefuture.
Inthecurrentpaper,thesameframeworkaswasdevelopedbyKrugeretal. (2016)hasbeen
usedandextended.Theaimofthepaperistoidentifytheeffectofturbulenceonthemassand
Theeffectofturbulenceonmassandheattransferratesofsmallinertialparticles 3
heattransferofsolidparticles,andtodevelopmodelsthatdescribethiseffectforallDamko¨hler
numbers.
2. Mathematicalmodelandimplementation
In the current work, the so called point-particle direct numerical simulation (PP-DNS) ap-
proach is used. Here, the turbulent fluid itself is solved with the direct numerical simulation
(DNS) methodology,where all turbulent scales are resolved and no modelling is needed. The
particles are howevernotresolved,but rathertreated as pointparticles where the fluid-particle
momentum,massandheatinteractionsaremodelled.Thepointparticleapproachisasimplifica-
tionthatreliesheavilyonthequalityofthemodels.Thealternativeapproach,whichistoresolve
theparticlesandtheirboundarylayer,isextremelyCPUintensiveandcancurrentlynotbedone
formorethanafewhundredparticles,evenonthelargestcomputers(Deen&Kuipers (2014)).
Anumberofsimplificationsaremadeinthispaper.Thishasbeendoneinordertomakethe
simulationslessCPU intensive,and,evenmoreimportantly,toisolate thedominatingphysical
mechanisms. The particles are considered to be ever lasting, i.e. they are not consumed. The
reactionontheparticlesurfaceisconvertingreactantAtoproductB;
A→ B (2.1)
isothermally,i.e.;thereisnoproductionorconsumptionofheat,suchthatonlythemasstransfer
effectisconsidered.Asexplainedabove,theeffectontheheattransferratewillbesimilartothe
effectonthemasstransferrate.AsreactantAisconvertedproductB,thethermodynamicaland
transportpropertiesarenotchanged.
2.1. Fluidequations
Theequationsdeterminingthemotionofthecarrierfluidisgivebythecontinuityequation
∂ρ
+∇·(ρu)=0, (2.2)
∂t
andtheNavier–Stokesequation
Du
ρ =−∇P+∇·(2µS)+ρf +F. (2.3)
Dt
Here,ρ,u,µ = ρνandνarethedensity,velocityanddynamicandkinematicviscositiesofthe
carrierfluid,respectively.Thepressure Pandthedensityρarerelatedbytheisothermalsound
speedc ,i.e.,
s
P=c2ρ, (2.4)
s
whilethetrace-lessrateofstraintensorisgivenby
1 1
S= ∇u+(∇u)T − ∇·u. (2.5)
2 3
(cid:16) (cid:17)
Kinetic energy is injected into the simulation box through the forcing function f, which is
solenoidaland non-helicaland injectsenergyandmomentumperpendicularto a randomwave
vectorwhosedirectionchangeseverytime-step(Haugenetal. 2012;Krugeretal. 2016).Sim-
ilar kinds of forcing has also previously been used for particle laden flows by other groups
(Becetal. 2007). The energy injection rate is maintained at a level such that the maximum
Mach numberis always below 0.5. The domain is cubic with periodicboundariesin all direc-
tions.Themomentumexchangeterm,F,ischosentoconservemomentumbetweenthefluidand
thesolidparticles,i.e.,
1
F=− mkak (2.6)
V
cell Xk
4 NilsErlandL.Haugen1,2,JonasKru¨ger1,DhrubadityaMitra3andTereseLøvås1
whenV isthevolumeofthegridcellofinterestandmk and ak arethemassandacceleration
cell
(duetofluiddrag)ofthek’thparticlewithinthegridcell.
Theequationofmotionofthereactanthasthewell-knownadvection-reaction-diffusionform:
∂X
+∇·(Xu)= DM¯ ∇·(∇X)+R˜, (2.7)
∂t c
whereX,M¯ andDarethemolefraction,themeanmolarmassandthediffusivityofthereactant,
c
respectively.ThelastterminEq.(2.7),R˜,isthesinktermduetothegas-solidreactionsonthe
surfaceofthesolidparticles.
2.2. Particleequations
The N particlesthatareembeddedintheflow aretreatedaspointparticles,whichmeansthat
p
theyareassumedtobesignificantlysmallerthantheviscousscaleofthefluidandthediffusive
scaleofthereactant.Themotionofthek’thparticleisdescribedbytheequationsforposition
dXk
=Vk (2.8)
dt
andvelocity
dVk
= ak (2.9)
dt
whentheparticleaccelerationduetofluiddragisgivenbyak = 1 u(Xk)−Vk .Notethatgravity
τ
isneglectedinthiswork.Theparticleresponsetimeisgivenby(Shchiller&Niaumann (1933))
τ
τ= St (2.10)
1+ f
c
whenτ =Sd2/18νistheStokestime, f =0.15Re0.687isaReynoldsnumbercorrectiontermto
St p c p
theclassicalStokestime,S =ρ /ρisthedensityratio,ρ isthematerialdensityoftheparticles,
p p
|u(Xk)−Vk|d u d
p rel p
Re = = (2.11)
p ν ν
istheparticleReynoldsnumberandd istheparticlediameter.
p
2.3. Surfacereactions
Letus nowmodelthe reactiveterm. We assume thatthereactionsarelimited to the surfaceof
theparticlesandthatthereactionsarediffusioncontrolled,i.e.thatallreactantthatreachesthe
particlesurfaceisconsumedimmediately†.Thereactivetermcanthenbewrittenas
1
R˜ = AkκXK (2.12)
V p ∞
cell Xk
whereA =4πr2 istheexternalsurfaceareaoftheparticle,themasstransfercoefficientisgiven
p p
by
DSh
κ= (2.13)
d
p
andShistheSherwoodnumber.
Tocouplethereactiveparticlewiththecontinuumequationsweusethefollowingprescription;
forthek-thparticle,whichisatpositionXk,weset
Xk = X(Xk), (2.14)
∞
† Itispossibletorelaxtheassumptionofdiffusioncontrolledreactionsbyalsoaccountingforchemical
kineticsattheparticlesurface,seeKrugeretal. (2016).
Theeffectofturbulenceonmassandheattransferratesofsmallinertialparticles 5
i.e.;thefarfieldreactantmolefractionissetequaltothereactantmolefractionofthefluidcell
where the particle is. In the currentwork, the particle Sherwood number is determined by the
expressionofRanz&Marshall (1952)(seeEq.(1.3)intheintroduction),whichisincontrastto
theworkofKrugeretal. (2016)wheretheSherwoodnumberwassettoaconstantvalueof2,
whichcorrespondstotheSherwoodnumberinaquiescentflow.TheparticleReynoldsnumber
isgivenbyEq.(2.11)andtheSchmidtnumber,Sc = ν/D,istheratioofthefluidviscosityand
themassdiffusivity.
2.4. Thereactantconsumptionrate
Itisusefultodefineareactantconsumptionrateas
R˜
α=− =n A κ, (2.15)
p p
X∞!
when O represents the volume average of flow property O and n is the particle number den-
p
sity. If everything is assumed to be homogeneously distributed over the volume, the reactant
consumptionrateisgivenby
ShD
α =n A κ=n A (2.16)
hom p p p p
d
p
foragivenparticlesizeandnumberdensity.
In many RANS based simulation tools, where the local fluid velocity is not resolved, it is
commontoneglecttherelativevelocitydifferencebetweentheturbulenteddiesandtheparticles.
ThisimpliesthatSh=2.Sincetheeffectofparticleclusteringisalsoneglectedinsuchmodels,
themodelledreactantconsumptionratebecomes;
2D
α = lim α=n A . (2.17)
Sh,Da p p
Sh→2,Da→0 dp
Inthefollowing,α willbeusedfornormalization.
Sh,Da
It is useful to define the Damko¨hler number, which is the ratio of the typical turbulent and
chemicaltimescales,as
τ
Da= L (2.18)
τ
c
whereτ = L/u is the integraltime scale of theturbulence, L is the turbulentforcingscale,
L rms
u istheroot-mean-squareturbulentvelocityandthechemicaltimescaleis
rms
τ =1/α . (2.19)
c Sh,Da
Particlesinaturbulentflowfieldwilltendtoformclusterswithhigherparticlenumberdensity
thantheaverage(Squires&Eaton 1991;Eaton&Fessler 1994;Toschi&Bodenschatz 2009;
Woodetal. 2005).Ifthechemicaltimescaleisshortcomparedtothelife-timeoftheclusters,
the reactantconcentrationwithin the clusterswill be much lower than outside the clusters. On
the otherhand,if the particle numberdensityis low, the particleclusterswill nothaveenough
time to consume a significant fraction of the reactant during the life-time of the cluster, and
hence,thereactantconcentrationwillberoughlythesameinsideasitisoutsidetheclusters.By
assuming that the life-time of the clusters is of the orderof the turbulenttime scale, it is clear
thatthereactantconcentrationofparticleflowswithlowDamko¨hlernumberwillbehaveasifthe
particles were homogeneouslydistributed overthe volume, i.e.; for small Damko¨hlernumbers
thereisnoeffectofparticleclusteringonthereactantconsumption.
FromEqs.(2.16)-(2.19)itcanbededucedthatforthehomogeneouscase,andthenalsofor
allcaseswithlowDamko¨hlernumbers,thereactantconsumptionratewillscalelinearlywiththe
6 NilsErlandL.Haugen1,2,JonasKru¨ger1,DhrubadityaMitra3andTereseLøvås1
Damko¨hlernumberforagiventurbulentflowfield,suchthat
DaSh
α = . (2.20)
hom τ 2
L
When relaxing the restriction to small Damko¨hler numbers, the effect of particle clustering
eventuallycomesintoplay.Krugeretal. (2016)haveshownthatthereactantconsumptionrate
isgivenby
α α
α= c hom (2.21)
α +α
c hom
whenα isaclusterdependentdecayrate.(NotethatsinceKrugeretal.assumedtheSherwood
c
number to be 2, their α equals our α = Da/τ .) From this expression, the following
hom Sh,Da L
normalizedreactantconsumptionrateisfound
α α τ Sh
α˜ = Sh = c L . (2.22)
Sh α α τ +DaSh/2 2
Sh,Da c L
whenSh isgivenbyEq.(1.3) andthecorrespondingrelativevelocitybetweenthe particleand
thefluidisdeterminedbyamodel(whichwillbeobtainedinthenextsubsection).Fordiffusion
controlledreactions,themodifiedreactiondecayrate,asgivenbyEq.(2.22),isameasureofthe
relativemodificationtothemasstransferrateduetotheeffectofturbulence.Thismeansthata
modifiedSherwoodnumbercannowbedefinedthataccountsfortheeffectofturbulence;
Sh =2α˜. (2.23)
mod
InthelimitofsmallDamko¨hlernumbers,thisexpressionreducestoSh =Sh,asexpected.
mod
ByemployingthemodifiedSherwoodnumbergivenbyEq.(2.23),onecannowusethecom-
monexpressionforthereactantconsumptionrate,asgivenbyEq.(2.16),tofindtherealreactant
consumptionrate.In mostcases, however,oneneedstheparticleconversionraten˙ forindi-
reac
vidualparticles,which is closely connectedto the reactantdecay rate. For diffusioncontrolled
masstransfer,theparticleconversionrateisgivenbyn˙ =−κX C ,whereC isthemolarcon-
reac ∞ g g
centrationofthegasphaseandthemasstransfercoefficientisnowfoundbyusingthemodified
Sherwoodnumber(asgivenbyEq.(2.23))in;Eq.(1.1)
DSh
κ= mod. (2.24)
d
p
In many applications, the mass transfer rate is not purely diffusion controlled.This can be ac-
countedforbyincludingtheeffectofreactionkineticsattheparticlesurface.Thecorresponding
particleconversionratecanthenbeexpressedas(Krugeretal. 2016)
λκ
n˙ =− X C , (2.25)
reac λ+κ ∞ g
whereλisthesurfacespecificmolarconversionrate.Sincethereactionkineticsisonlydepen-
dent on the conditionsat the particle surface, the surface specific molar conversion rate is not
affected by the turbulence.This is, as we have alreadyseen, notthe case for the mass transfer
coefficient,whichisnowgivenbyEq.(2.24).Inthisway,allthecommonmachineryforcalcu-
latingparticlereactionratescanstillbeusedsincetheeffectsoftheturbulenceareincorporated
intothemodifiedSherwoodnumber.
3. Results
In all of the following,statistically stationaryhomogeneousand isotropic turbulenceis con-
sidered.TheReynoldsnumberisvariedbychangingthedomainsizewhilemaintainingconstant
Theeffectofturbulenceonmassandheattransferratesofsmallinertialparticles 7
Table1.Summaryofthesimulations.ThefluiddensityisunitywhiletheSchmidtnumberis0.2andthe
viscosity is 2×10−4 m2/s for all the simulations. For every simulation listed here, a range of identical
simulationswithdifferentDamko¨hlernumbershavebeenperformed.
Label L(m) N d ρ Re Sh St τ α α τ St/Sh
grid p p L c c L
1A π/2 643 3.4×10−3 50 80 2.5 1.0 1.6 0.9 0.63
2A 2π 1283 19×10−3 50 400 2.8 1.0 5 0.23 0.43
3A 8π 2563 11×10−3 500 2200 2.8 1.0 15 0.07 0.41
2AB 2π 1283 19×10−3 25 400 2.7 0.5 5 0.26 0.25
3AB 8π 2563 11×10−3 250 2200 2.6 0.5 15 0.09 0.26
2B 2π 1283 11×10−3 50 400 2.5 0.3 5 0.21 0.13
3B 8π 2563 11×10−3 150 2200 2.6 0.3 15 0.09 0.18
2C 2π 1283 19×10−3 5 400 2.4 0.1 5 0.55 0.12
3C 8π 2563 11×10−3 50 2200 2.4 0.1 15 0.20 0.13
2D 2π 1283 19×10−3 1.5 400 2.3 0.03 5 1.20 0.08
3D 8π 2563 11×10−3 16 2200 2.3 0.03 15 0.45 0.10
2E 2π 1283 19×10−3 0.5 400 2.2 0.001 5 4.10 0.10
viscosityandturbulentintensity.TheDamko¨hlernumberisvariedbychangingthenumberden-
sity of particles, while keeping everythingelse the same. All relevantsimulations are listed in
table1.
3.1. Themeanrelativeparticlevelocity
In order to predicta representative value of the particle Sherwoodnumber from Eq. (1.3), the
particleReynoldsnumberRe isrequired.FromEq.(2.11)itisclearthatthisalsorequiresthe
p
relativeparticlevelocityu ,whichwillbefoundinthissubsection.
rel
GivenaparticlewitharesponsetimethatequalstheStokestime;
Sd2
τ = p, (3.1)
p 18ν
such that τ < τ < τ , where τ is the Kolmogorov time scale and τ is the integral time
k p L k L
scale.Withrespecttotheparticle-turbulenceinteractions,theturbulentpowerspectrummaybe
dividedintothreedistinctregimes,basedontherelationbetweentheparticleresponsetimeand
theturbulenteddyturnovertimeτ .Thefirstregimeisdefinedasthesectionoftheturbulent
eddy
power spectrum where the turbulent eddies have turnover times that are much larger than the
response time of the particles, i.e. where τ ≫ τ . All the turbulent eddies in this regime
eddy p
will see the particles as passive tracers, which follow the fluid perfectly. I.e., there will be no
relativevelocitybetweentheparticlesandtheeddies.Thethirdregimeisdefinedasthepartof
thepowerspectrumwheretheturbulenteddieshavemuchshortertimescalesthantheparticles,
i.e.whereτ ≪τ .Theeddiesinregimethreewillseetheparticlesasheavybulletsthatmove
eddy p
instraightlines,withoutbeingaffectedbythemotionoftheeddies.Hence,thevelocityofthese
eddies will contribute to the relative particle-fluid velocity. The second regime is now defined
as the relatively thin band in-between regimes one and three, where τ ≈ τ . These are the
eddy p
eddiesthatare responsibleforparticleclustering,sincetheyareabletoacceleratetheparticles
8 NilsErlandL.Haugen1,2,JonasKru¨ger1,DhrubadityaMitra3andTereseLøvås1
Figure1. Theparameterβ,relatingtherelativeparticlevelocitytothesubscalevelocityasdefinedin
Eq.(3.9),isshownasafunctionofStokesnumber.
Figure2.Leftpanel:kineticenergyspectrumfordifferentReynoldsnumbers.Rightpanel:relativeparticle
velocityasafunctionofStokesnumber.
to a levelwhere they are thrownout of the eddy due to their inertia. In the following,we will
refertoatypicaleddyinregimetwoasaresonanteddy,andwedefinethescaleofthiseddyas
ℓ.Theresonanteddiesareidentifiedbytheirtimescale,τ ,whichisoftheorderoftheparticle
ℓ
responsetime,τ .Forconvenience,wesetthetwotimescalesequal,suchthat
p
τ =τ . (3.2)
ℓ p
Basedonthedefinitionsabove,itisclearthatthelargestturbulenteddiesthatyieldarelative
velocitybetweenthe fluidandtheparticles,aretheresonanteddies.By assumingKolmogorov
scaling, the velocity of the resonant eddies is known to be u = u (ℓ/L)1/3, which can be
ℓ rms
combinedwiththeaboveexpressionforthetimescalestoyield
k =k St−3/2 (3.3)
ℓ L
whentheparticleStokesnumberisdefinedas
τ
p
St= (3.4)
τ
L
andk =2π/landk =2π/Larethewave-numbersoftheresonanteddiesandtheintegralscale,
ℓ L
respectively.InobtainingEq.(3.3),ithasalsobeenusedthattheturnovertimeoftheresonant
eddiesisτ =l/u ,whilethatoftheintegralscaleeddiesisτ = L/u .
ℓ ℓ L rms
Since all scales smaller than ℓ will induce a relative velocity between the particles and the
fluid,itisreasonabletoassumethattherelativevelocitybetweenthefluidandtheparticleswill
beacertainfractionβoftheintegratedturbulentvelocityu˜ ofallscalessmallerthanℓ,suchthat
ℓ
u =βu˜ (3.5)
rel ℓ
Theeffectofturbulenceonmassandheattransferratesofsmallinertialparticles 9
whenu˜ isdefinedas
ℓ
1 kη
u˜2 = E(k)dk (3.6)
2 ℓ Zkℓ
and k = 2π/η is the wave-number of the Kolmogorov scale (η = (ν3/ǫ)1/4), where ǫ is the
η
dissipationrateofturbulentkineticenergy.IntegrationofEq.(3.6)yields
k−2/3−k−2/3
u˜ℓ =urmsvtkℓ−2/3−kη−2/3 (3.7)
L η
forE(k)=cǫ2/3k−5/3whenithasbeenusedthatthetotalturbulentkineticenergyisgivenby
1 kη
u2 = E(k)dk, (3.8)
2 rms Z
k1
wherek isthewavenumberofthelargestscaleinthesimulation.CombiningEqs.(3.3)and(3.7)
1
withEq.(3.5)finallyyields
Stk−2/3−k−2/3
urel =βurmsvt k−L2/3−k−η2/3 . (3.9)
L η
Theunknownconstantinthisequation,β,canbedeterminednumericallyfromEq.(3.5),i.e.β=
u /u˜ .Here,u isfounddirectlyfromDNSsimulations,whileu˜ iscalculatedfromEq.(3.7).
rel ℓ rel ℓ
Itisseenfromfigure1thatβiscloseto0.41formostStokesandReynoldsnumbers.Themain
exceptionisforlow ReynoldsandStokesnumbers,whereβissignificantlylarger.Thiscanbe
understoodbyinspectingtheleftpaneloffigure2,whereitisseenthatforRe=180andSt<0.1,
we are alreadyfarinto thedissipativesubrange,whereourmodelis notexpectedtobe correct
sinceitreliesonaKolmogorovscaling.
It is surprising to see that Eq. (3.9) reproduces the relative particle velocity for such low
Stokes numbers, even for the smaller Reynolds numbers. This may be explained by reconsid-
ering Eq. (3.2), where we assumed that the resonanteddiescorrespondto the eddies thathave
exactly the same turnover time as the response time of the particles. This is just an order of
magnitudeestimate,andamorecorrectexpressionwouldprobablybe
τ =γτ , (3.10)
ℓ p
where γ is of the orderof unity.More work should, however,be devotedto understandingthe
couplingbetweentheparticlesandtheturbulenteddies.Inparticular,amoreexactdefinitionof
theresonanteddiesisneeded.We neverthelessbelievethatβisauniversalpropertyoftheHIP
approximationandtheNavier-StokesequationsthatwillhaveaconstantvalueforallReandSt
aslongastheresonanteddiesarewithintheinertialrange.
In the right panel of figure 2, the average relative particle velocity, as found from the DNS
simulations(symbols), is comparedwith the predicted valuesfromEq. (3.9) (solid lines). It is
seenthatthefitisrathergoodformostReynoldsandStokesnumbers.Thissupportstheuseof
Eq.(3.9)forpredictingtherelativeparticlevelocity.
3.2. Theclustersize
Thetypicalsizeoftheclustersℓisassumedtobethesizeoftheresonanteddies.FromEq.(3.3)
thisyieldsaclustersizeof
l= LSt3/2. (3.11)
Itcanbeseenfromfigure3thattheparticlenumberdensitydistributiondoesindeedshowmore
smallscalevariationforthesmallerStokesnumbers.Thishasbeenquantifiedinfigure4where
10 NilsErlandL.Haugen1,2,JonasKru¨ger1,DhrubadityaMitra3andTereseLøvås1
Figure3.ParticlenumberdensityforSt=1(upperleft),St=0.3(upperright),St=0.1(lowerleft)and
St=0.03(lowerright)(runs3A,3B,3Cand3Dintable1).
the power spectrum of the particle number density is shown. Here we see that the spectrum
peaks at large scales for St = 1 while the peak is located at much smaller scales for smaller
Stokes numbers. The peak in the spectrum does not, however, follow Eq. (3.11) as accurately
asexpected.Thereasonforthisismostlikelythatpowerspectraarenottherightdiagnosticsto
studythesizeofparticleclusters,butitmayalsobepartlybecauseof:1)poorstatisticsduetotoo
fewparticles(thesmallerclustersarenotfilled withparticles),2)theconstantinthedefinition
oftheresonanteddiesnotbeingunity(seee.g.Eq.(3.10)),or3)finiteReynoldsnumbereffects.
The power spectrum P can be integrated to yield a measure of the strength in the particle
numberdensityfluctuations,givenbytheroot-mean-square(rms)particlenumberdensity;
n = Pdk. (3.12)
rms
Z
ItisfoundthatthermsparticlenumberdensityisdecreasingwithStokesnumber.Morespecif-
ically,n is1.6,1.5,1.2and0.8forStokesnumbersof1,0.3,0.1and0.03,respectively.This
rms
meansthatthehighdensityregimeshavehigherparticlenumberdensitiesforlargerStokesnum-
bers.
3.3. Reactantconsumptionrate
Thenormalizedreactantconsumptionrateisshowninfigure5.Thesymbolscorrespondtothe
resultsfromtheDNSsimulations,whilethesolidlinesaregivenbyEq.(2.22).Here,theStokes
numberisfoundbyusingthemodelfortherelativevelocity,asgivenbyEq.(3.9),intheexpres-
sionfortheSherwoodnumber(Eq.(1.3)).Thevalueoftheclusterdecayrate,α ,istheonlyfree
c
parameteranditischosenbyabestfitapproach.Thevaluesofα arefoundintable1.
c
The valueof α˜ for small Damko¨hlernumbersequalsthe Sherwoodnumberdividedby two,