Modeling heat transfer in gas-particle mixtures: Calculation of the macro-scale heat exchange in Eulerian–Lagrangian approaches using spatial averaging

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Belerrajoul, Mohamed and Davit, Yohan and Quintard, Michel

and Simonin, Olivier and Duval, Fabien Modeling heat

transfer in gas-particle mixtures: Calculation of the

macro-scale heat exchange in Eulerian–Lagrangian approaches

using spatial averaging. (2019) International Journal of

Multiphase Flow, 117. 64-80. ISSN 0301-9322

Official URL:

https://doi.org/10.1016/j.ijmultiphaseflow.2019.04.029

Modeling

heat

transfer

in

gas-particle

mixtures:

Calculation

of

the

macro-scale

heat

exchange

in

Eulerian–Lagrangian

approaches

using

spatial

averaging

Mohamed

Belerrajoul

a,b

_,

_Yohan

_Davit

a

_,

_Michel

_Quintard

a

_,

_Olivier

_Simonin

a

_,

_Fabien

_Duval

b,∗

a CNRS, Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, Toulouse, France b Institut de Radioprotection et de Sûreté Nucléaire (IRSN), Saint Paul Lez Durance 13115, France

Keywords: Euler-Lagrange methods Upscaling Volume averaging Heat transfer Closure problems

a

b

s

t

r

a

c

t

Heat transfersindilutegas-particle mixtures areoftenmodeledusinghybrid Euler–Lagrange descrip-tions,treatingthecarrierfluidviaanEulerianrepresentationandfollowingeachparticleinaLagrangian framework.Oneofthefocalissuesinthesemodelsisthe calculationofthemacro-scaleheattransfer betweenthecontinuousphaseandparticles.Inthestandardapproach,theheattransferforeachparticle isconsideredtovarylinearlywiththeaveragetemperaturedifferencebetweentheparticleandthefluid. Here,weusethemethodofvolumeaveragingwithclosuretofiltertheheattransferequations atthe micro-scaleand deriveaclosedformoftheheattransferrate,whichissignificantlydifferentfromthe standardcase.Theprimarydifferenceisthattheheattransferforagivenparticledoesnotonlydepend onthetemperatureoftheparticlebutalsodependsonthetemperaturesofallotherparticleswithin theaveragingvolume.Thisyieldsamatrixofheatexchangecoefficientsthatcapturesindirectparticle– particleexchangesatmacro-scale.Usingsimplemodelcases,wevalidateourapproach,compareittothe standardheattransfermodelandshowthatitdegeneratestowardthestandardmodelonlyinspecific cases.

1. Introduction

We are interested in dust explosion hazard in the context of nuclear safety and, in particular, in modeling flame propagation fordiluteanddispersedgas-particleflows withparticlesizes typ-icallyrangingfrom10−6 to10−4m andavolume fractionof par-ticlesupto10−3.Such configurationscorrespondto accident sce-nariosthatmayhappen duringoperationsofdecommissioning of uraniumnatural graphite gasreactors (D’Amico etal., 2016) that involvegraphiteparticlesorlossofvacuumaccidentinthevessel offusionreactors(DenkevitsandDorofeev,2005;Janeschitz,2001) thatinvolveeithertungstenorberylliumparticles.Inthiscontext, animportantmodelingissueistoidentifythemechanismsthat in-fluencetheseverityoftheexplosionandmayresultintherelease ofradioactiveortoxicmaterials.

Our global strategy follows the ones proposed in e.g.,

Cannevière (2003) and Neophytou and Mastorakos (2009) for liquid fuel droplets or in Cloney et al. (2018) and Park and Park(2016) forcoal particles. The idea isto usedetailed

numer-∗ _{Corresponding author.}

E-mail address: [email protected] (F. Duval).

ical simulations to analyzethe flame structure andcalculate the laminarburningvelocitythatisthenusedinasecondstepto de-terminetheturbulentflamevelocity(seeforinstanceProust,2017; Silvestrini et al., 2008). Detailed numericalsimulations are often computed via a hybrid Euler–Lagrange description,which uses a filteredEulerianrepresentationofthefluidphaseandaLagrangian trackingoftheparticles(Boivinetal., 2000;Capecelatroand Des-jardins,2013;Croweetal.,2011;Simoninetal.,1993).Inthis con-text,the macro-scaleenergybalance equationsarewritten classi-callyas

∂

t



α

β

(

ρ

cp

)

_β



T_β



β



+

∇

·



α

β

(

ρ

cp

)

_β



T_β



β



u_β



β



=

∇

·



α

β

λ

∗β

∇

Tβ



β



−1_V N_V  p=1 Q_βp (1)

(

mcp

)

p dTp dt =Qβp (2)

The first equation corresponds to the heat transfer equation for the filtered temperature



T_β



β of the continuous

β

-phase, where

λ

∗_β,



u_β



β, (

ρ

cp)_β and

α

_β are respectively an effective

Nomenclature

Romanletters

Ap surfaceareaofaparticle,m2

A_βσ interfacial area of the

β

₋

σ

contained withintheaveragingvolume,m2

av interfacialareaperunitvolume,m−1

Bip=hpdp/

λ

σ particleBiotnumber

b_β vectorthatmaps

∇

T_β



β ontoT_β,m cp specificheatcapacity,Jkg−1K−1 dp particlediameter,m

hpj heatexchangecoefficients,Wm−2K−1

K_β effective thermal dispersion tensor, Wm−1K−1

l_β characteristic length (micro-scale)forthe

β

-phase,m

L_β macroscopiccharacteristiclength,m mp particlemass,kg

g weightingfunction

N_V numberofparticles containedinthe averag-ingvolumeV

n_βσ unitnormalvectorfromthe

β

-phasetowards the

σ

-phase

Q_βp macro-scale heat transfer rate between the

continuousphaseandparticlep,W r positionvector,m

rg radiusoftheaveragingvolume,m sp scalarthatmaps



T_β



β− TjontoTβ T_η

η

₌

β

_,

σ

_,temperatureinthe

η

-phase,K Tp particletemperature,K



T_β



β intrinsic average temperature for the

β

-phase,K



T_β deviationtemperatureforthe

β

-phase,K T0

β undisturbedtemperatureforthe

β

-phase,K t∗ characteristictimeassociatedwiththe

micro-scalediffusion,s

u_β velocityinthe

β

-phase,ms−1



u_β



β intrinsic average velocity for the

β

-phase, ms−1



u_β deviationvelocityforthe

β

-phase,ms−1 V averagingvolume,m3

xp particleposition,m Greekletters

λ

β thermalconductivityforthe

β

-phase,Wm−1K−1

ω

σ volumetricheatsource,Wm−3

ρ

η

η

=

β

,

σ

,densityforthe

η

-phase,kgm−3

α

η

η

=

β

,

σ

,volumefractionofthe

η

-phase

δ

βσ Diracdistributionassociatedwiththe

β

−

σ

inter-face

γ

β indicatorfunctionforthe

β

-phase

β

continuousphase

σ

dispersedphase

ξ

Laplacevariable,s−1 Superscripts

∞ quasi-steady

∗ _{timeconvolutionproduct}

capacityandthevolumefractionofthe

β

-phase.The lasttermin

Eq.(1)correspondstothemacro-scaleheattransferratebetween the continuous phase and the N_V particles contained within the averaging volume V. Forthe pth-particle,the corresponding heat transferreads

Q_βp=



Ap

n·

λ

β

∇

T_βdS (3)

whereAp isthe surfaceofparticle pandn denotes theoutward

unitnormalvectorontheparticlesurfaceAp.

The second equation in the Euler–Lagrange description

Eq. (2) describes the averaged particle temperature Tp, where

(mcp)p is the product of the mass and heat capacity of the pth-particle. The model for the macro-scale heat transfer Q_βp is usuallybasedonthedescriptionofheattransferinthecaseofan isolated particle (Michaelides andFeng, 1994). To solve the heat transfer problem, this description uses a decomposition of the surrounding phase temperature into an undisturbed temperature andadisturbanceduetothepresenceoftheparticle.Generalizing fromanisolatedparticletoasetofparticles,theundisturbed tem-peratureisoftenviewedasamacro-scaletemperature



T_β



β and the resulting heat exchange reads (Ling et al., 2016; Michaelides andFeng,1994) Q_βp=  Ap n·

λ

β

∇

T_β



βdS+Aphp





T_β



β− Tp



+Ap  t 0 Kp

(

τ

)

d





T_β



β− Tp



d

τ

d

τ

(4)

InEq.(4),thefirsttermreferstotheundisturbedheatfluxseenby theparticle.Itrepresentsthefluxthatwouldhaveenteredthe vol-umeoccupiedbythecontinuousphaseintheabsenceofthe parti-cle.Thesecondtermcorrespondstothequasi-steadyheattransfer fromtheparticletothesurroundingphaseduetotemperature dif-ference.Theheat exchangecoefficient hp betweenthecontinuous

phase and the particle is usually expressed in terms of the par-ticleNusseltnumber asNup=hpdp/

λ

_β forwhich numerous

em-piricalfunctionsofthe particle Reynoldsnumberhave been pro-posed(seeforinstanceRanzandMarshall,1952;Whitaker,1972) with the asymptotic limit Nup=2 when the particle Reynolds

numbereffect can be neglected. The last termin Eq.(4) is non-local and captures history effects due to the transient develop-mentofthetemperatureintheparticlevicinity. The kernel func-tionKp(

τ

)thatappearsinthehistoryintegral hasbeencalculated

analyticallyinMichaelidesandFeng(1994)foranisolatedparticle without a velocity difference betweenthe continuous phase and the particle, and the results have been extended to finite parti-cleReynoldsnumberinBalachandarandHa(2001) andFengand Michaelides (1996). Usual simplifications of thisproblem include neglecting the non-local contribution to decrease the computa-tionalcost andreplacing the first termby the time derivative of the undisturbed temperature (Ling et al., 2016; Michaelides and Feng,1994).

In this work, our goal is to propose alternative expressions ofthe heat transfer rateusing an up-scaling methodology based onspatial averagingwith closure(CarbonellandWhitaker, 1984; Davitet al., 2013;Quintard andWhitaker, 2000). This methodol-ogyallowsustoderivemacro-scaleequationsthat,unlikestandard Euler–Lagrange approaches, take into account heat transfer be-tweenparticlesintheheatexchangeQ_βpthroughamatrixofheat exchange coefficients. To be more clear about heat transfer be-tweenparticles,thesehavenottobeconfusedwithdirectparticle– particle exchanges that take place duringcollisions and that are negligibleinthediluteregimeconsideredinthiswork.Here,heat transferbetweenparticlestake placethrough thecontinuous car-rier phase and thus will be referred as indirect particle–particle exchanges.Theresultingheatexchangecoefficientsinvolvebotha quasi-steadyandanon-localcontributionthatcaptureshistory ef-fects.Ourapproachfurtherprovidesclosureproblemsthatlinkthe effectivetransport coefficientstothemicro-scalegeometry.To re-covermorestandard formulations,the problemcan besimplified

averageof

ψ

_β isdefinedas



ψ

β



β=g_g∗_∗

ψ

_γ

β β =



ψ

β



α

β (11)

Second, we now average the micro-scale heat Eq. (5) on the averaging volumeV, by multiplyingthe micro-scaletransport Eq. (5)by

γ

_βg

(

x_{− r}

)

_,andthenintegratingit overthephysical space to finally apply the general transport theorem (Whitaker, 1985), whichleadsto

∂

t



(

ρ

cp

)

_β



T_β





+

∇

·



(

ρ

cp

)

_βT_βu_β



=

∇

·



λ

_β

∇

T_β



+



n_βσ·

λ

β

∇

Tβ

δ

βσ



(12) where

δ

_βσ isthe Dirac deltafunction associatedto the interface betweenthetwophases.

Next,weintroducetheperturbationdecompositionof

ψ

_β into anaverageandadeviation

ψ

_β (seeforinstanceGray,1975)

ψ

β=



ψ

β



β+

ψ

β (13)

Finally, following Carbonell and Whitaker (1984),

Gray (1975) and Quintard and Whitaker (1993a), we write theaveragedequation,Eq.(12),as

∂

t



α

_β

(

ρ

cp

)

_β



T_β



β



+

∇

·



α

_β

(

ρ

cp

)

_β



T_β



β



u_β



β



=

∇

·



α

β

λ

β

∇

Tβ



β+



λ

βTβnβσ

δ

βσ





−

∇

α

β·

λ

β

∇



Tβ



β+



nβσ·

λ

β

∇

Tβ

δ

βσ



−

∇

·



(

ρ

cp

)

_β



T_βu_β



β



(14)

where we haveignored the variations ofthe physicalproperties withintheaveragingvolume.

The Lagrangian descriptionof thedispersed phase isobtained by integratingthemicro-scaleheattransferEq.(8)overeach par-ticle(Croweetal., 2011).The boundaryconditionEq.(6)leadsto thefollowingequationfortheaveragedparticletemperatureTp

(

mcp

)

p dTp dt =  Ap n·

λ

β

∇

T_βdS+

ω

pVp (15)

where Vp is the volume of the particle. At this stage of the

de-velopments,wehaveobtainedthemacro-scaleEuler–Lagrange de-scriptionthatcorrespondstoEqs.(14)and(15)oftheheattransfer problem. However, they are not in a closed form since the tem-perature deviations T_β are still present. To getrid ofthese devi-ations, it is necessary to propose an estimation of the deviation from its boundaryvalue problem. In orderto obtain the govern-ingequationforthedeviationT_β,weintroducethedecomposition

Eq.(13) in themicro-scale Eq.(5), andthen we subtract the av-eragedEq.(14)dividedby

α

_β.Themicro-scaletransportequation forthedeviationmaybewrittenas

(

ρ

cp

)

_β

∂

tT_β+

(

ρ

cp

)

_βu_β·

∇

T_β+

(

ρ

cp

)

_βu_β·

∇

T_β



β =

∇

·



λ

β

∇

Tβ



−

α

β−1

∇

·



λ

βTβnβσ

δ

βσ



−

α

−1 β



nβσ·

λ

β

∇

Tβ

δ

βσ



+

α

β−1

∇

·



(

ρ

cp

)

_β



T_βu_β





(16)

The corresponding boundary conditions are also obtained by in-troducing the decompositionEq. (13)in the boundaryconditions

Eqs. (6) and (7). As a result, one obtains the following set of boundaryconditionsoneachparticlesurfaceAp,1≤ p≤ NV 

T_β =T_σ−



T_β



β (17)

λ

β

∇

Tβ· nβσ =



λ

σ

∇

T_σ−

λ

β

∇



Tβ



β



· nβσ (18) Theseequationscanbesimplifiedthroughanevaluationofthe or-dersofmagnitudeofthedifferentterms.Introducing the decom-positionEq.(13)intheboundaryconditions(7)andintheno-slip

condition

(

u_β=0

)

at the

β

−

σ

interfaceleadsto an estimation oftheorderofmagnitudeofthevelocitydeviationu_β andofT_βas (seeCarbonellandWhitaker,1984;QuintardandWhitaker,1993a; QuintardandWhitaker,2000)

u_β =O





u_β



β



(19)  T_β =O



l_β L_β



Tβ



β



₍₂₀₎

Furthermore, according to the developments in Quintard and Whitaker(1993a,b),thespecificareacanbeestimatedby



n_βσ

γ

_β

δ

_βσ



=Aβσ V =O



_α

β l_β



(21)

wherel_β isthemicro-scalecharacteristiclength.Eqs.(19)–(21) al-lowustoobtainthefollowingestimates

λ

β

∇

2T_β =O



_λ

β l_βL_β



Tβ



β



₍₂₂₎

α

−1 β

∇

·



λ

βTβnβσ

δ

βσ



=O



_λ

β L2 β



T_β



β



(23)

α

−1 β

∇

·



α

β

(

ρ

cp

)

_β



T_βu_β





=O



ρ

βcp



β l_β L2 β



T_β



β



u_β



β



(24)



ρ

βcp



βuβ·

∇

Tβ =O



ρ

βcp



β 1 L_β



Tβ



β



uβ



β



(25)

Usingthelength-scaleconstraintl_β L_β furtherleadsto

α

−1

β

∇

·



λ

βTβnβσ

δ

βσ



λ

β

∇

2Tβ (26)

α

−1

β

∇

·



α

β

(

ρ

cp

)

_β



T_βu_β







ρ

_βcp



_βu_β·

∇

T_β (27)

Therefore, the micro-scale transport equation for the deviation

Eq.(16)canberewrittenas

(

ρ

cp

)

_β

∂

tT_β+

(

ρ

cp

)

_βu_β·

∇

T_β+

(

ρ

cp

)

_βu_β·

∇

T_β



β

=

∇

·



λ

β

∇

Tβ



−

α

β−1



nβσ ·

λ

β

∇

Tβ

δ

βσ



(28) Aspreviouslydiscussed,thethermalconductivityinthesolid,

λ

_σ, ismuchlarger thanthe thermalconductivityinthe

β

-phaseand thustheboundaryconditionEq.(6)yields

|

∇

T_σ

|

=

λ

β

λ

σ

∇

Tβ

∇

T_β

(29)

Thisimpliesthatthetemperaturefieldcanbeconsidereduniform withineachparticleandthattheboundaryconditionEq.(18)can be eliminated from the micro-scale boundary value problem for thedeviationwhileboundaryconditionEq.(17)reads



T_β=Tp−



T_β



β atAp,1≤ p≤ NV (30)

One way to proceed further would be to solve simultaneously the macroscopic problem and the problem for the deviation T_β, whichinmanywaysismorecomplexthansolvingtheinitialone

Eqs.(5)–(8).Thereisanotherwaytoproceedthatconsistsin build-ing an approximate form of the deviations T_β by mapping it to macroscopicquantities through closurevariables. As we will see, this makes it possible to fully uncouple micro- and macro-scale problems.

3.2.Unsteadyclosure

IntheboundaryvalueproblemforthedeviationEqs.(17)–(28), we can identify the macroscopic quantities

(

Tp−



T_β



β

|

xp

)

with

1≤ p≤ NV and

∇

Tβ



β. These terms can be treated as macro-scopicsourcetermsresponsibleforgeneratingthespatialdeviation T_β. Inthis way,deviations canbe mapped to thesesource terms (CarbonellandWhitaker,1984;Quintardetal.,1997;Quintardand Whitaker,1993a; 2000; ZanottiandCarbonell, 1984)through clo-surevariables.

To build theformof alocal solutionforthe spatialdeviation, wefirstfollowDavitandQuintard(2012)andMoyne(1997)to ex-pressthe deviationin the formof a time convolution according to _Tβ=−NV p=1 sp∗

∂

t





T_β



β

|

xp− Tp



+b_β· ∗

∂

t

∇

T_β



β (31)

wherespandb_β aretheclosurevariables,whichdependonspace

xandtime t.Thetime convolution,denotedhereby ∗,isdefined forthefunctionsaandbas

(

a∗ b

)

(

t

)

= t

0

a

(

t−

τ

)

b

(

τ

)

d

τ

(32)

The closurevariables are solutions of unsteady closure problems giveninAppendix A.2, inwhichwe haveassumed thermal equi-libriumatt<0. Theseclosureproblemsneed tobe solved in ge-ometriesrepresentative ofthe structure of interest and, in many cases, this is done in unit cells with periodic boundary condi-tions.Effectsofusingperiodicunitcellsonthemacroscopicresults andthevalidityofthemethodinorderedanddisorderedsystems are further discussed inQuintard et al.(1997) andQuintard and Whitaker (1993a). For disordered media, the periodicity condi-tions can be used when the size of the unit cell is sufficiently large compared to the correlation lengths of the heterogeneities (Quintardetal.,1997).

FortheEulerianfield, weinjectEq.(31)intheaveraged trans-portEq.(14)andobtain

∂

t



α

β

(

ρ

cp

)

_β



T_β



β



+

∇

·



α

β

(

ρ

cp

)

_β



T_β



β



u_β



β



=

∇

·



K_ββ· ∗

∂

t



∇

T_β



β



− N  p=1 g

(

x− xp

)

Q_βp + N  p=1

∇

·



d_βp∗

∂

t





T_β



β

|

xp− Tp



(33)

where K_ββ is the effective thermal dispersion defined by

Eq.(121) andaccountsforthermalconduction,tortuosity and hy-drodynamicthermaldispersion.Thecoefficientd_βpisanadditional

velocity-like coefficient and is defined by Eq. (122). Finally, Q_βp

representsthe macro-scaleheat transfer betweenthe continuous phaseandtheparticlepandisdefinedby

Q_βp=  Ap n·

λ

β

∇

Tβ



βdS+Ap NV  k=1 hpk∗

∂

t





T_β



β

|

xk− Tk



− v_βp· ∗

∂

t



∇

T_β



β



(34)

The effective propertiesh_pk definedby Eq.(93) and v_βp defined byEq.(98)arerespectivelytheeffectiveheatexchangecoefficients andthevelocity-likecoefficient.

For the Lagrangian description ofthe dispersed phase, substi-tutingEq.(31)intoEq.(15)andusingthedefinitionEq.(34)ofthe macro-scaleheattransferyields foraveragedparticletemperature Tp

(

mcp

)

p

dTp

dt =Qβp+

ω

pVp (35)

In thesemacro-scale Euler–Lagrangeequations, both the ther-maldispersiontensor,K_ββ,andtheheatexchangecoefficients,hpk,

appearinginEqs.(33)–(35)looklike classical terms.This contrasts

with the velocity-like v_βp andadditional velocity-like d_βp terms thatdonotappearintheclassicalmacro-scaleEuler–Lagrange de-scription (Crowe et al., 2011; Ling et al., 2016; Michaelides and Feng,1994;Simonin,1996;ZhangandProsperetti,1997).Thesecan becalculatedfromtheclosureproblemsgiveninAppendixA.2and their contributions have been studied in the literature (Quintard etal., 1997;Zhang andHuang, 2001) for heat transfer inporous mediawithathermalnon-equilibriummodel.Inthiswork,we as-sumethatthecontributionsofthenon-classicaltermscanbe ne-glectedcomparedtotheclassicalterms.Indoingso,weobtainthe followingsimplifiedmodel

∂

t



α

β

(

ρ

cp

)

_β



T_β



β



+

∇

·



α

β

(

ρ

cp

)

_β



T_β



β



u_β



β



=

∇

·



K_ββ· ∗

∂

t



∇

T_β



β



− NV  p=1 g

(

x− xp

)

Q_βp (36)

(

mcp

)

p dTp dt =Qβp+

ω

pVp (37) where Q_βp=  Ap n·

λ

β

∇

T_β



βdS+Ap NV  k=1 hpk∗

∂

t





T_β



β

|

xk− Tk



(38) 3.3. Steady-stateclosure

Eqs.(36)–(38)are obtainedby expressingthespatialdeviation inthe formofa time convolution. Themain consequenceis that the resultingeffectivetransport coefficientsin themodel are not onlydependingonthephysicalpropertiesandthemicro-scale ge-ometry butthey also depend upon time. Inaddition, the macro-scale Euler–Lagrange equations with unsteady closure problems must degenerate into the quasi-steadyversion when the macro-scopictimeissignificantlygreaterthanthecharacteristictime as-sociated to the relaxation of the micro-scale conduction process (DavitandQuintard,2012;Moyne,1997).Thisconstrainthasbeen discussed inprevious works(Quintardet al., 1997; Quintardand Whitaker,1993a)andcanbewrittenas

λ

βt

(

ρ

cp

)

_βl2_β

 1 (39)

Whenthisisverified,theunsteadyterminthetransport equa-tionforT_β canbe neglected,sothat theboundaryvalueproblem forthedeviationcanbetreatedasquasi-steady.Followingprevious worksCarbonell andWhitaker(1984), Quintard etal.(1997) and

QuintardandWhitaker(1993a,2000),wecanthenexpressthe de-viationintermsofthemacroscopicsourcetermsas

 T_β=− NV  p=1 s∞p





T_β



β

|

xp− Tp



+b∞_β ·

∇

T_β



β (40)

Here,themappingvariabless∞_p andb∞_β aresolutionsofthe quasi-steadyproblemsgivenin AppendixA.2.This asymptoticbehavior corresponds to thetransitionform sp andb_β to the limitu

(

t

)

s∞p

andu

(

t

)

b∞_β intheconvolutionproductdefinedbyEq.(31),where u(t)istheHeavisidefunction.Thesedecompositionscanbewritten asfollows(DavitandQuintard,2012)

sp− u

(

t

)

s∞p =s∗p (41)

b_β− u

(

t

)

b∞_β =b∗_β (42)

where s∗_p andb∗_β represent the contribution ofhistory effects in theunsteadyclosureproblemandverifyrespectively

lim t→+∞s ∗ p=0,and _tlim →+∞b ∗ β=0 (43)

The mapping variables associated withthe historyeffects s∗_p and b∗_β are solutions of the closure problems related to the his-tory effects (AppendixA.3). Theseclosure problemsare obtained by introducing the decompositions Eqs. (41) and (42) into the unsteady closure problems (Appendix A.1), then subtracting the quasi-stationary closure problems (Appendix A.2) multiplied by u(t).

Similarly,introducingthedecompositionsEqs.(41)and(42) re-spectivelyinEqs.(93)and(121)yieldsthefollowingexpressionfor theeffectivetransportcoefficientshpkandKββ

hpk=u

(

t

)

h∞pk+h∗pk (44)

K_ββ=u

(

t

)

K∞_ββ+K∗_ββ (45)

where h∗_pk and K∗_ββ are respectively the effective heat exchange andtheeffectivethermaldispersioncoefficientsrelatedtohistory effects.The definitionsofthesecoefficientsaregiveninappendix

Eqs.(115)–(125).

Lastly, by introducing Eqs. (44) and (45) into the averaged transport equation forthe

β

-phase Eq.(36)andthe equation for theaveragedparticletemperatureEq.(38),themacro-scaleEuler– Lagrangeequationswithunsteadyclosureproblemscanbewritten as

∂

t



α

β

(

ρ

cp

)

_β



T_β



β



+

∇

·



α

β

(

ρ

cp

)

_β



T_β



β



u_β



β



=

∇

·



K∞_ββ·

∇

T_β



β



− NV  p=1 g

(

x− xp

)

Q_βp +

∇

·



K∗_ββ· ∗

∂

t



∇

T_β



β



(46)

(

mcp

)

p dTp dt =Qβp+

ω

pVp (47) where Q_βp=  Ap n·

λ

β

∇

T_β



βdS+Ap NV  k=1 h∞_pk





T_β



β

|

xk− Tk



+Ap NV  k=1 h∗_pk∗

∂

t





T_β



β

|

xk− Tk



(48)

In the averaged transport equation for the

β

-phase Eq. (46), thefirsttermcorrespondstothequasi-steadyconductiveterm;the secondtermtothemacro-scaleheattransferbetweenthe continu-ousphaseandparticlesintheaveragingvolume;andthelastterm takestheformofa historyintegral that accountsformemory ef-fectsof theunsteady thermalconduction.The appearanceof this term is a directconsequence of the decomposition inthe quasi-steadyandmemorycontributionsofthe effectivethermal disper-sion.

Inthemacro-scaleheatexchangeEq.(48),thefirsttermlooks like,atleastformally,theundisturbedheatfluxcontributionwhile thesecond termcorrespondstothequasi-steadythermaltransfer fromtheparticletothecontinuousphase.Thelasttermrepresents the unsteady thermal diffusion due to the temporal variation of the thermalboundarylayer aroundthe particlesin theaveraging volume. Thiscontribution accountsfortheeffect ofthe past his-torytemperaturechangesofthe

β

-phaseandparticleswithinthe averaging volume to the current temperaturechange ofthe pth -particle.

4. Theoreticalcomparisonwithstandardmodels

When comparing the classical Euler–Lagrangemodel withthe proposed model (46)–(48), themain differencesare the convolu-tion terminEq. (46)andtheexpression ofthe macro-scaleheat

Fig. 2. Comparison of Chang’s unit cell (dashed line) with a spatially periodic sys- tem (solid line).

exchangetermQ_βp.Inthissection,wereviewthesedifferencesin detailforthecasesofisolatedparticlesandcloudsofparticles.

4.1. Isolatedparticle

As mentioned in the introduction, the modeling of heat ex-changebetweenthefilteredcontinuousphaseandtheparticlesis usually based on the description of heat transfer for an isolated particleinaninfinitemedium(Croweetal.,2011;Lingetal.,2016; MichaelidesandFeng,1994;ZhangandProsperetti,1997).This de-scriptionuses a decomposition ofthe surrounding

β

-phase tem-perature into an undisturbed thermal field T0

β, which is not in-fluenced by thepresence of theparticle, anda disturbancefield, whichisentirelyduetotheinfluenceoftheheattransferfromthe particle.

In order to compare our model to the standard results in the case of an isolated particle, we consider the case of a sin-gle isolated spherical particle with no macroscopic gradient. We consider an immobile sphere in a spherical unit cell, as repre-sentedinFig. 2,which isa sphericalversion of Chang’s unit cell (Chang, 1983). This unit cell hasbeen extensively studied in the literature (Quintard and Whitaker, 1993b; 1995) as it allows to obtain an analytical solution of the closure problems. Following

Quintard and Whitaker (1995), the correspondence between the Chang’s unitcell andthespatially periodicsystemisobtainedby requiring that the volume fraction be equal. This leads to l3

β=

(

4/3

)

π

R3_{whereRistheradiusoftheChang’s unitcellillustrated}

inFig.2.Theradiusoftheparticle,rp,andthecharacteristiclength

oftheunitcell,R,arelinkedby

R= rp



1−

α

β



1/3

(49)

The closure problem associated to Chang’s unit cell for the mappingvariables∞p canbewrittenas

0=

λ

β

∇

2s∞p −

α

−1_β Ap V h∞p , forrp≤ r≤ R (50) s∞_p =1, atr=rp (51) n·

∇

s∞p =0, atr=R (52) Average:



s∞_p



β=0 (53)

frame attachedto the continuousphase allows usto further re-move the convective term in Eq. (46) and the undisturbed heat flowinEq.(48).

Inthiscase,themacroscopicdescriptionisa0Dproblemwith

α

β

(

ρ

cp

)

_β d



T_β



β dt =− 1 V NV  p=1 Q_βp (72)

(

mcp

)

p dTp dt =Qβp+

ω

pVp (73) where Q_βp= NV  k=1 Aph∞_pk





T_β



β

|

xk− Tk



+ NV  k=1 Aph∗_pk∗

∂

t





T_β



β

|

xk− Tk



(74)

Here,wehaveusedtheclassicalboxweightingfunctiongdefined by

g

(

x

)

=



₁

V, x∈V

0, x∈/V (75)

Inthisframeworkwithnoaveragegradient,onlytheclosure prob-lemsfors∞p ands∗pneedtobesolved.

As stated previously, the accuracy of the macro-scale models isassessed bycomparingresultstocomputationsofthecomplete micro-scaleproblem.Intheremainderofthissection,these micro-scalesimulations aretermedDirect NumericalSimulations (DNS). Weconsider two different studies. Inthe first study,the validity ofboth theunsteady andquasi-steady modelsis addressedfora micro-scaleone-dimensionalgeometryforwhichtheclosure prob-lemsaswell asthemacroscopicequationscan besolved analyti-cally.Inthe second study,we consider onlythe quasi-steady ap-proximationandwe assess thevalidity of a diagonal approxima-tionofthe heat exchangematrix fortwo-andthree-dimensional geometries.DNSofthemicro-scaleproblemhavebeenperformed usingthe CALIF3_{S software(CALIF)that employs unstructured}

fi-nite volume discretization. The interested reader is referred to

Piar etal.(2013) fora detailedpresentation ofthe finitevolume discretizationbased on the SUSHI scheme forthe approximation ofthediffusivefluxes.

5.1.Validityofthequasi-steadyclosure

In this section, we compare the unsteady and quasi-steady models.The micro-scalestructure isa one-dimensional geometry illustratedinFig.5andconsistsinanarrayofthreeparticleswith thesamediameterdp andseparatedbythedistancel_β.

Theaveragingvolumeisassumedtobespatiallyperiodic(Davit andQuintard,2015;QuintardandWhitaker,1993a).Further,taking intoaccount thesymmetry withrespect tothe centerof particle 1,thematrixofheatexchangecoefficientsissymmetrich∞_pk=h∞_kp. Moreovertheconsidered geometryallows usto solveanalytically thecoefficientsandto determinethem allfromonly h∞₁₁ andh∞₁₂ (seeAppendixC.1).

Themacro-scaleEulerian–Lagrangianordinarydifferential equa-tions are non-dimensionalized with the following dimensionless

variables ˆ

∇

=l_β

∇

; ˆt=

λ

β l2 β

(

ρ

cp

)

_β t ; ˆ

ω

p= l2 β Tr

λ

_β

ϕ

ω

p;

ϕ

=

(

ρ

cp

)

p

(

ρ

cp

)

_β ; ˆ h∞_pk=l 2 βAp

λ

βVh ∞ pk ; ˆTη= T_η− Tr Tr ,with

η

=

β

,k (76)

whereTrreferstosomereferencetemperature.Thecorresponding

macro-scaleequationscanbeexpressed inmatrixformaccording to

d dtˆ[Tˆ]=H

∞_{· [}Tˆ]+[

ω

ˆ] (77)

where [Tˆ] and[

ω

ˆ] are columnvectors andH∞ _{is a four-by-four}

matrix.Allthreecanberepresentedexplicitlyas

[Tˆ]=

⎛

⎜

⎝



_Tˆ_β

_

β ˆ T1 ˆ T2 ˆ T2

⎞

⎟

⎠

; [

ω

ˆ]=

⎛

⎜

⎝

0 ˆ

ω

1 ˆ

ω

2 ˆ

ω

2

⎞

⎟

⎠

(78) H∞₌ 3

α

σ

ϕ

⎛

⎜

⎜

⎝

−ασϕ αβ hˆ∞ ασϕ 3α_βhˆ∞ α3σαϕ_βhˆ∞ α3σαϕ_βˆh∞ ˆ h∞ −ˆh∞₁₁ −ˆh∞₁₂ −ˆh∞₁₂ ˆ h∞ −ˆh∞₁₂ −ˆh∞₁₁ −ˆh∞₁₂ ˆ h∞ −ˆh∞₁₂ −ˆh∞₁₂ −ˆh∞₁₁

⎞

⎟

⎟

⎠

(79) wherehˆ∞₌hˆ∞₁₁₊2ˆh∞₁₂.

Byusingthematrixexponentialexp(.)(seee.g.,Magnus,1954), the solutionof the lineardifferential equation system Eq.(77)is givenby [Tˆ]

(

tˆ

)

=



 ˆt 0 exp

(

uH∞

₎

_du



· [

ω

ˆ]+eˆtH∞ · [Tˆ0] (80)

wherethevector[Tˆ0]representsthevector[Tˆ]attheinitialstate.

ThematrixH∞_{isdiagonalizablewiththreedistincteigenvaluesa} 0, a1 anda2 definedby

{

a0,a1,a2

}

=

⎧

⎨

⎩

0,− 3

hˆ∞₁₁− ˆh∞₁₂



α

σ

ϕ

,−3 ˆ h∞



1

α

σ

ϕ

+ 1

α

β

⎫

_⎬

⎭

(81)

We apply successively the Cayley–Hamilton theorem and Sylvester’s formula to exp

(

uH∞

₎

_{(for example, see Horn and} Johnson,1990;Cvetkovicetal.,1997).Accordingly,Eq.(80)reads

[Tˆ]

(

tˆ

)

=F0· [Tˆ0]− 2  i=1 1 ai Fi· [

ω

ˆ]+F0· [

ω

ˆ]tˆ + 2  i=1 eaitˆF i·

₁ ai [

ω

ˆ]+[Tˆ0]



(82)

whereFiare theFrobenius covariantsofthe matrixH∞,defined

by

Fig. 6. Comparison of the quasi-steady closure with the reference solution obtained by averaging Direct Numerical Simulations (DNS) results of the micro-scale problem. Table 1 Model parameters. Parameter λσ λβ (ρcp)σ (ρcp)β ασ Tr ωˆ 1 ωˆ 2 ωˆ 2 Value 10 3 ₁₀ 3 ₁₀−3 _{300 0 10 5} Fi= 2  i=0, j=i 1 ai− aj



H∞_{− a} jI



(83)

where_{I is}theidentitymatrix.

The micro-scaleproblem,Eqs.(5)–(8),issolvednumericallyto obtaintheT_β andT_σ fields,whichcanbeaveragedtoproducethe referencevaluesforthemacro-scalemodels.Comparisonsare car-ried out forscenarios ofdust explosion typical ofnuclear safety (Table1). Results in Fig. 6show that thevalues obtainedby the Euler–Lagrange model with the quasi-steady closure are in very goodagreement withthereferencevalues,particularlyfora time greaterthan0.5. Forshortertimes(i.e.tˆ<0.5),a goodagreement betweenresultsisobservedfortheaveraged

β

-phasetemperature, aswellasthetemperatureofparticles2and2 .However,the tem-peratureoftheinertparticle(particle1)decreasesinitiallyandthis isinclearcontradictionwiththemicro-scaleresultsFig.6andin violationofphysicalprinciples.

Todetermine the originofthis differenceand whetherit isa consequenceofthequasi-steadyclosure,wenowconsiderthe so-lution of the problem with history effects and compare the ref-erence solution, the quasi-steady model and the fully transient formulations. The closure problems for the memory variable are solved analytically intheLaplace domain(AppendixC.2) andthe Eulerian–Lagrangianmodelwiththeunsteadyclosureisrewritten intheLaplacedomainas

ξ

[Tˆ]− [Tˆ0]=



H∞+

ξ

H∗



· [Tˆ]+[

ω

ˆ] (84)

wheretheoverbardenotesthetransformedvariableintheLaplace domain. _H∗ is defined by analogy to the matrix _H∞ using the memoryheatexchange coefficients.The solutionofthesystemof linear equations (84) is straightforward and the temperatures in physicalspaceareobtainedbyusingnumericalinverseLaplace al-gorithms.ResultsinFig.7showthatthemodelwiththeunsteady closureisinperfectagreementwiththereferencesolution.The

av-eragedtemperaturesarealsoveryclosetothetwopreviousresults iftˆ<0.5andthethreetypesofresultsaresimilarfortˆ>0.5.

Thiscomparisondemonstratesthat ourmodelsaccurately pre-dicttheaveraged temperatures,even intheshort-time limit.The quasi-steadyassumption is validwhen the constraintEq. (39) is verified,whichmeansthatthehistoryeffectscanbeneglected.In thenext studycase,weconsideronlythequasi-steadymodeland assumethatthisconstraintisverified.

5.2.Validityofthediagonalizationfortheheatexchangecoefficients matrix

We now assess the accuracy of the quasi-steady macro-scale modelforatwo-dimensionalmicro-scalegeometryandtheimpact ofadiagonalapproximationoftheheatexchangecoefficients ma-trixEq.(79).Such adiagonalizationconsistsinneglecting the in-directexchangebetweenparticlesinEqs.(72)–(74).Toclarifythis point,considerthefollowingexpressionoftheheatexchange

Q_βp=Aph∞p





T_β



β

|

xp− Tp



+ NV  k=1 Aph∞pk

(

Tp− Tk

)

(85)

whereh∞p correspond to lumped coefficientsalreadydefinedand

thatarerecalledbelow

h∞_p = NV

 k=1

h∞_pk (86)

Note that this expression is only valid when there is no aver-agedtemperaturegradient,thusallowingustoidentifythe macro-scale continuous phase temperature



T_β



β

|

x_k with



T_β



β

|

xp. The

most straightforward way to obtain a diagonal approximation is thereforeto neglect the sum inEq. (85), which is oftenreferred to aslumping the coefficientsin the matrix. When this is done, the diagonal coefficients can be calculated from a single closure problemcorresponding to the sumof all mapping variables (see

AppendixD).Thisgreatlysimplifiesthedeterminationoftheheat exchangecoefficientsandallowsustoobtainanalyticalexpressions forsimpleunitcellsliketheoneillustratedinFig.2.

Forinstance,thelumped coefficientscanbesolved analytically forthecylindricalversionoftheChang’s unitcellshowninFig.2. Asfor thesphericalcase, we compare theanalytical solution ob-tained for Chang’s unit cell with the numerical solution of the

Fig. 10. Comparison of averaged DNS results with the macro-scale predictions for both the quasi-steady and the quasi-steady lumped approximation – two-dimensional case NV = 3 × 3 .

Table 2

Dimensionless heat exchange coefficients ˆ hpk = h ∞pk d p /λβfor different values of N V and ασ= 10 −3 for a periodic arrangement of in-line cylinders (  denotes particles

on the same x -axis, ⊥ between the first and the second rows and ⊥⊥ between the first and the third rows).

NV hˆ 11 ˆ h12 hˆ 12⊥ hˆ 13 hˆ 13⊥ ˆ h13⊥⊥

1 0.737

3 × 3 0.537 0.01 0.039

5 × 5 0.492 −0 . 028 0.0017 0.0186 0.0218 0.025

exceptforthediagonalcoefficient h∞pp whichisofthesameorder

ofmagnitude.Alsonotethat thevalueofh∞_p doesnotdependon N_V since thelumped coefficientscan be obtainedfromsumming mappingvariablessdefinedinAppendixDandsolvedforthe spa-tiallyperiodiccellshowninFig.8.

Next, we compare the average temperature



T_β



β and the temperatures of particles 1, 2 and 2 obtained from micro-scale simulations to those obtained from the macro-scale Eulerian– Lagrangian description usingthe heatexchange coefficientsgiven in Table 2 and the same parameter values reported in Table 1. Recall from Eq. (85) that the lumped approximation is formally equivalenttothefullmacro-scaleheatexchangecoefficientsmatrix when there is no temperature difference between particles. The proposed case involving active and inert particles leadsto large temperaturedifferencesbetweenparticlesand,therefore,allowsus to test the validity of thelumped approximation.Parameters are listedinTable1andresultsareplottedinFig.10.Wefindthatthe quasi-steadymodelisingoodagreementwiththereference micro-scale simulations fora dimensionless time greater than 3 (recall thatthetheoreticalconstraintistˆ 1).

Thelumpedapproximationyieldsagoodagreementforthe av-eraged

β

-phase temperature



T_β



β butan underestimationofthe temperaturedifferencebetweenparticles.Inordertogetmore

in-sightforlargertemperaturedifferencesbetweenparticles, we ex-tend the previous caseto N_V =5 with only particle1 remaining activeand

ω

ˆ1=100.TheresultsreportedinFig.11show thatthe

lumped approximationstill yields agood agreementfor



T_β



β at ˆ

t>4but failsto capturedifferences observedin DNSresults for particles1and3.

We further remark that other diagonal approximations may producebetterresults.Inthisspecificconfiguration,Fig.12shows thatwe obtainslightlybetter resultsifwe substitutethelumped coefficients by the diagonal coefficients h∞_pp reported in Table 2. Thissuggeststhat,dependingonthesituationofinterest,different diagonalizations may yield better results. This is something that needsto beaddressedinfutureworks,especiallywhen consider-ingaveragegradientsoftemperature.

Wemove finally tothe three-dimensionalcaseandassess the validity of the lumped approximation for the three-dimensional version ofthesystem illustrated inFig.9 that consistsina peri-odicarrangementofin-linesphericalparticles. Aspreviously,itis assumedthatthereis noaveragedtemperaturegradientandthat thesystemishomogeneous exceptinthex-direction.As aresult, hereagaina singlerowofparticlesalongthex-directionis repre-sentativeoftheentiresystem.Wedonotrepeathereallthecases studiedpreviouslyinthetwo-dimensionalcaseandwerestrict to the caseN_V=3× 3× 3. The results reported inFig. 13 in which the

β

-phase average temperature



T_β



β iscompared to the tem-peraturesofparticles reflectthesametrends asalreadyobserved intheprevioustwo-dimensionalcase.Hereagain,thequasi-steady approximationleadstoagoodpredictionofthemacro-scale tem-peratureswhilethelumpedapproximationfailstoaccurately pre-dict the temperatures of particles. It is instructive to note that in this special case of ordered distribution of spherical particles withlarge temperaturedifference betweenparticles andwith no averaged temperaturegradientin thecontinuousphase, both the quasi-steady and the quasi-steady lumped approximation yields

Fig. 11. Comparison of averaged DNS results with the macro-scale predictions for both quasi-steady and quasi-steady lumped approximations – two-dimensional case NV = 5 × 5 .

Fig. 12. Comparison of averaged DNS results with the macro-scale predictions for both quasi-steady and quasi-steady diagonal approximations – two-dimensional case NV = 5 × 5 .

thesameaveragedtemperatureforthecontinuousphase.This be-haviorwasexpectedfromEqs.(68)to(69)asmacro-scaleEulerian description isequivalent to the lumped approximation. This also meansthatinthiscase,thelumpedapproximationaccurately pre-dicttheaveragedtemperatureforthedispersedphasedefinedby

Eq.(67).

6. Discussionandconclusion

Using the method of volume averaging with closure, we havederived a novel macroscopicEuler–Lagrange modelfor heat transfer in gas-particle mixtures. The primary difference be-tween our model andthe classical one is an expression of heat

Fig. 13. Comparison of averaged DNS results with the macro-scale predictions for both the quasi-steady and the quasi-steady lumped approximation – three-dimensional case N V = 3 × 3 × 3 .

transfers that captures indirect particle–particle interactions. We recall herethatindirect exchangesreferto heattransfer between particlesthroughthecontinuouscarrierphaseandhavenottobe confused with direct particle–particle exchanges that take place duringcollisionsindensesuspensions.Bycomparingourtheoryto micro-scalecomputationsinsimplemodelcasesofparticleclouds, we found that the generalformulationwith time convolutionsis able to predict the exact macroscopic temperatures forany time in a case with no average gradient. This shows that the most generalformulationof theheat transfercapturesboth history ef-fectsand heatexchangesbetweenparticles. We havealso shown that the quasi-steadymodel accurately capturesthe temperature fieldsfor sufficientlylong times.Thisillustrates theusefulnessof the quasi-steady approximation which is, in practice, frequently adoptedwhentheconstraintEq.(39)isverified.

We havefurthershownthatthe standardformulationscan be recovered from a diagonalization approximation of the exchange matrix. In particular, a lumped matrix can be used and the di-agonal coefficients can be obtainedby solving the closure prob-lemforaunitcellwithanisolatedparticle,similarlytothe classi-cal approach.Thisapproximationisformallyequivalenttothefull macro-scalemodelwhenthetemperaturedifferencesbetween par-ticlescanbeneglected.Thisapproximationwastestedinthecase oftwo- andthree-dimensional systems.The simplifiedversion of the model was able to accurately describe the average tempera-ture



T_β



β.However,thelumpedapproximationfailedincapturing temperaturedifferencesbetweenparticles,whichisduetothefact thattheindirectexchangebetweentheparticlesisneglected.

Theconsideredmodelcasesofparticleclouds,evenifvery sim-ple, have led us to a better fundamental understanding of the models. The validity of the various macro-scale approximations havebeenassessedincaseswithlargetemperaturedifference be-tweenparticleswithintheaveragingvolumebutnoaverage gradi-entsoftemperature.Therefore,limitationsofthestandard descrip-tion, whichneglectboth indirectexchange betweenparticles and memoryeffects,mustbefurtherevaluated inpractical caseswith macroscopicgradients.

Suchpracticalcaseswouldrequiremorerealisticconfigurations such as random arrangements of spherical particles investigated in Sun et al. (2015) and Thiam et al. (2019) for the closure of

Eulerian–Eulerianheattransferusingdirectnumericalsimulations. Thiswouldrepresenta highlychallengingtasksincedifferent ar-rangementsthatcorrespondtothesamevolumefractionmaylead tovery differentheat transfercoefficientsvalues.One couldthen adoptthe methodology followed in the Eulerian–Eulerian frame-work by using ensemble average of a number of realizations to getrelationshipsfortheeffectivepropertiessuchash_pk=h_pk

(

α

_β

)

. However,itisnotclearatthisstageonhowthismethodologymay apply in the Eulerian–Lagrangian framework (Kriebitzsch et al., 2013)andthiscallstofuturework.

AppendixA. Closureproblems

Inordertoobtaintheclosureproblemsforthemappings vari-ables,weintroduceEq.(40)inthelocalproblemforthedeviation 

T_β andproceedinthestandardwayinthevolumeaveraging the-ory.Indoingso,weobtainN_V identicalclosureproblemsIkforthe

variablesskandtheclosureproblemIIforvariablebβ.These

prob-lemsareasfollows.

A1.Unsteadyclosureproblems

– ProblemI_kfors_kwith1_{≤ k≤ NV}



ρ

βcp



β



∂

tsk+uβ·

∇

sk



=

λ

β

∇

2sk −

α

−1 β NV  p=1 g

(

x− xp

)

Aphpk,inthe

β

-phase (88) sk=1,atAk (89) sk=0,atAj with j=k (90) Periodicity:sk

(

x+r

)

=sk

(

x

)

(91) Initialcondition:sk=0,att=0 (92) where: hpk=− 1 Ap  Ap n·

λ

β

∇

skdS (93)

– ProblemIIforb_β



ρ

βcp



β



∂

tbβ+uβ·

∇

bβ+u _β



=

λ

β

∇

2b_β −

α

−1 β NV  k=1 g

(

x− xp

)

v_βp,inthe

β

-phase (94) b_β=0,atAp,with 1≤ p≤ NV (95) Periodicity:b_β

(

x+r

)

=b_β

(

x

)

(96) Initialcondition:b_β=0,att=0 (97) where: v∞_βp=  Ap n·

λ

β

∇

bβdS (98)

A2.Quasi-steadyclosureproblems

– ProblemI∞_k fors∞_k with1_{≤ k≤ NV}



ρ

βcp



βuβ·

∇

s∞k =

λ

β

∇

2s∞k −

α

−1 β NV  p=1 g

(

x− xp

)

Aph∞_pk,inthe

β

-phase (99) s∞_k =1,atAk (100) s∞_k =0,atAjwith j=k (101) Periodicity:s∞_k

(

x+r

)

=s∞_k

(

x

)

(102) Average:



s∞_k



β=0 (103) where: h∞_pk=−1 Ap  Ap n·

λ

β

∇

s∞k dS (104) – ProblemII∞ forb∞_β



ρ

βcp



β



u_β·

∇

b∞_β +u_β



=

λ

β

∇

2b∞β −

α

−1 β NV  p=1 g

(

x− xp

)

v∞_βp,inthe

β

-phase (105) b∞_β =0,atAp,with 1≤ p≤ NV (106) Periodicity:b∞_β

(

x+r

)

=b∞_β

(

x

)

(107) Average:



b∞_β



β=0 (108) where: v∞_βp=−  Ap n·

λ

β

∇

b∞_β dS (109) A3.Memoryeffectsclosureproblems

– ProblemI∗_kfors∗_kwith1≤ k≤ NV



ρ

βcp



β



∂

ts∗k+uβ·

∇

s∗k+

δ

(

t

)

s∞k



=

λ

β

∇

2s∗k −

α

−1 β NV  p=1 g

(

x− xp

)

Aph∗pk,inthe

β

-phase (110) s∗_k=1− u

(

t

)

,atAk (111) s∗_k=0,atAj with j=k (112) Periodicity:s∗_k

(

x+r

)

=s∗_k

(

x

)

(113) Initialcondition:s∗_k=0,att=0 (114) where: h∗_pk=−1 Ap  Ap n·

λ

β

∇

s∗_kdS (115) – ProblemII∗forb∗_β



ρ

βcp



β



∂

tb∗_β+uβ·

∇

b∗_β+

δ

(

t

)

s∞j +

(

1− u

(

t

)

)

u β



=

λ

β

∇

2b∗β−

α

β−1 NV  p=1 g

(

x− xp

)

v∗_βp,in the

β

-phase (116) b∗_β=0,atAp,1≤ p≤ NV (117) Periodicity:b∗_β

(

x+r

)

=b∗_β

(

x

)

(118) Initialcondition:b∗_β=0,att=0 (119) where: v∗_βp=−  Ap n·

λ

β

∇

b∗βdS (120)

AppendixB.Definitionsoftheeffectivetransportcoefficients

K_ββ=

α

β

λ

βI+



λ

βnβσ· bβ

δ

βσ



−

(

ρ

cp

)

_β



u _βb_β



(121) d_βp=

(

ρ

cp

)

_β



spu _β



−



λ

βspnβσ

δ

βσ



(122) K∞_ββ=

α

β

λ

βI+



λ

βnβσ· b∞β

δ

βσ



−

(

ρ

cp

)

_β



u _βb∞_β



(123) d∞_βp=

(

ρ

cp

)

_β



s∞pu _β



−



λ

βs∞pnβσ

δ

βσ



(124) K∗_ββ=



λ

βn_βσ· b∗ β

δ

βσ



−

(

ρ

cp

)

_β



u _βb∗_β



(125) d∗_βp=



ρ

βcp



β



s∗pu β



−



λ

βs∗pnβσ

δ

βσ



(126)

AppendixC. Heatexchangecoefficientsforone-dimensional unitcells

C1. Quasi-steadyclosureproblems

TheclosureproblemsI∞_k forthemappingvariabless∞_k with1_≤ k≤ NV readforone-dimensionalunitcellsas

0=d2s∞k dx 2 −

α

β−1 NV  p=1 ˆ h∞_pk,inthe

β

-phase (127) s∞_k =1,atx =xk± rk (128) s∞_k =0,atx =xp± rpwith p=k (129) Periodicity:s∞_k

(

x +L

)

=s∞_k

(

x

)

(130) Average:



s∞_k



β=0 (131) where: ˆ h∞_pk= ds∞k dx

|

(xp−rp)− ds∞_k dx

|

(xp+rp) (132)