Homogenization of a conductive and radiative heat transfer problem

HAL Id: hal-00784061

https://hal.inria.fr/hal-00784061

Submitted on 4 Aug 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

transfer problem

Grégoire Allaire, K. El Ganaoui

To cite this version:

Grégoire Allaire, K. El Ganaoui. Homogenization of a conductive and radiative heat transfer problem.

Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2009, 7 (3), pp.1148-1170. �10.1137/080714737�. �hal-00784061�

CENTRE DE MATHÉMATIQUES APPLIQUÉES

UMR CNRS 7641

91128PALAISEAU CEDEX (FRANCE).Tél: 0169 33 4600. Fax: 01 69 3346 46

http://www.map.polytehnique.fr/

Homogenization of a ondutive

and radiative heat transfer

problem

Grégoire Allaire, Karima El Ganaoui

R.I. 639 September 2008

GRÉGOIRE ALLAIRE

†

AND KARIMA ELGANAOUI

‡

Abstrat. Thispaper isdevoted tothe homogenizationof a heat ondution problem ina

periodiallyperforateddomainwithanonlinearandnonloalboundaryonditionmodelingradiative

heattransfer intheperforations. Beause ofthe onsideredritialsalingit isessentialtousea

methodoftwo-saleasymptotiexpansionsinsidethe variationalformulationofthe problem. We

obtain a nonlinear homogenized problem of heat ondution with eetive oeients whih are

omputedviaaellproblemfeaturingaradiativeheattransfer boundaryondition. Werigorously

justifythishomogenizationproess forthelinearizedproblembyusingtwo-saleonvergene. We

perform numerialsimulationsin2-d: wereonstrutanapproximatetemperature eldbyadding

tothe homogenizedtemperature aorretorterm. Theomputednumerial errorsagreewiththe

theoretial preditederrorsand provethe eetiveness ofourmethod formultisale simulationof

ondutiveandradiativeheattransferproblemsinperiodiallyperforateddomains.

Keywords.Homogenization,two-saleonvergene,radiativetransfer,heatondution.

AMSsubjet lassiations.

1. Introdution. The goal of this paper is to theoretially and numerially

study the homogenization of a ondutiveand radiative heat transferproblem in a

perforatedperiodimedia. The motivation of this problem omes from the nulear

reator industry: an alternative onept to the usual pressurized water reators is

thatofgasooledreators. Typially,agraphitematrix(playingtheroleofneutron

moderator)isperiodiallyperforatedbylonghannelsontainingeithertheuranium

fueloragasoolantwhihishelium. Reallthatthessionnulearreationsprodue

alargeamountofheatwhihshouldberemovedfromthereatororebyaoolantin

ordertoativateasteamgenerator(throughaheatexhanger)andnallytoprodue

eletriity. Herewefousonlyontheheattransferprobleminsuhanheterogeneous

medium. Tosimplifytheexposition,weassumethatthegraphiteanduraniummatrix

isalreadyhomogenizedandanbeonsideredasasinglehomogeneousmaterial. In-

sidethismatrixheat istransmittedbysimplelinearondution. Ontheotherhand,

the helium heat ondutivity is ompletely negligible with respet to the radiative

transfer taking plae inside the hannels. We therefore fae a oupled problem of

heat ondution and radiation where the number of helium hannels is very large,

typially of the order of 10^{4. For} dimensioning purposes as well as safety studies manynumerialsimulationshavetobeperformedforwhihadiretapproah(mesh-

ing all the geometri details) is impossible, orat least muh too ostly. Therefore,

homogenizationisaneessaryingredientforthestudyofsuhdevies.

Inthisproblemthegoalofhomogenizationistwofold: rst,itmustyieldalear

denition of what is the homogenized problem, and seond, it has to give expliit

formulas for the eetive parameters as well as a reipe to approximate the exat

solution. Indeed,sinetheoriginalmodel isamixtureoftwodierenttypeofequa-

tions(ondutionandradiativetransfer),thepreiseformofthehomogenizedsystem

is not lear a priori. Conerning the seond point, the original problem is posed

in aperforatedmedium whilethe homogenizedproblem is posed in ahomogeneous

∗

ThisworkhasbeensupportedbytheFrenhAtomiEnergyCommission,DEN/DM2SatCEA

Salay.

†

CMAP,(gregoire.allairepolyte hniq ue. fr) .

‡

CMAP,(ganaouimap.polytehniq ue.f r).

medium, sotaking into aount orretorterms is of paramount importane if one

wantsageometriallysoundreonstrutionofanapproximatesolution.

Let us ome bak to the physial modeling of the original problem. The true

problemis three-dimensionalbut theheliumhannelsarelongparalleltubes,soho-

mogenization takes plae only in the ross setion. Therefore, it is not a severe

restritiontoonsideronlythetwo-dimensionalhomogenizationofarosssetionof

thegeometry (see Figure2.1) asweshall dobelow. Asusual in homogenizationwe

denote byε ^{the period. The matrixperforateddomain is} Ωε ^{where energytransfer}

is done by ondution. The tubes or holes are τε,i^{, with boundaries}Γε,i ^{whih are}

grey-diuse surfaes, and are lled by helium, assumed to be a transparentmedia

withoutheat ondution norabsorption of radiation. Under these assumptions, the

radiation equation anbeintegrated inside eah hole τε,i ^{to produe a ompliated}

(non linear and non loal) boundary ondition on the wall Γε,i^{. Setion 2.2 gives}

a preise desriptionof this boundaryondition. Let us simply gives theomplete

model when theemissivity is equalto one. Forgiven bulkand surfaeheat soure

termsf ^andg^,thetemperatureTε^{isasolutionof}















−div(Kε∇Tε) = f ⁱⁿ Ωε,

Kε∇Tε·n = g ^on ∂Ω,

−Kε∇Tε·n = σ

ε T_ε⁴(x)− Z

Γε,i

F(x, s)T_ε⁴(s)dγ(s)

!

on Γε,i,

(1.1)

where F(x, s) ^{is the so-alled view fator for the wall} Γε,i^{. The saling} ε^{−1 in the}

righthand sideof theboundaryonditionyieldsaperfetbalane, in thelimit asε

goestozero,betweenthebulkheatondution andthesurfaeradiativetransfer. A

dierentsalingwas studiedin[7℄.

Sine the seminal paper [12℄ it is known that the use of two-sale asymptoti

expansionsinperforateddomainsissometimesdeliate,espeiallywhentheboundary

onditionsarenonlinearandnonloal ashere. Indeed,thehomogenization of(1.1)

by the formal method of two-sale asymptoti expansions(as presented in [8℄, [9℄,

[11℄,[21℄)isnotompletelyobvious,allthemoreifoneworkswiththestrongfromof

theequations. Asexplainedin Setion3itismuhsimplerto performthetwo-sale

asymptoti expansionsin the variationalformulation of (1.1), symmetriallyin the

unknownand inthetest funtion (followinganideaofJ.-L. Lions[16℄). Asaresult

we obtain that the leading term T(x) ^{in the ansatz of} Tε(x) ^{is the solution of the}

followingnonlinearhomogenizedproblem

(

−div(K^∗(T)∇T) = ^mes(Y

∗)

mes(Y)f ⁱⁿ Ω,

K^∗(T)∇T·n = g ^on ∂Ω, ^(1.2)

whereK^∗(T)^istheeetiveondutivity,dependingonthemarosopitemperature

T^{,anddenedthroughaloalellproblem(3.3)whihisalinearizedondutiveand}

radiativetransferproblemintheunit ell(seeProposition 3.1).

In Setion 4 we give a rigorous justiation of suh an homogenization result

for the linearized versionof (1.1) (see Theorem 4.6). Our main toolsare two-sale

onvergene[2℄,[20℄andsuitableTaylorexpansionsofthetestfuntion oneahhole

boundaryΓε,i ^{inordertotakeadvantageof theviewfator}properties.

Eventually Setion 5is onerned with numerial simulations for this problem.

Followingalassialideainperiodihomogenization,weapproximatethesolutionTε

of (1.1)bythetworsttermsofitsansatz,i.e.,thehomogenizedsolutionT ^plusthe

so-alledorretorterm

Tε(x)≈T(x) +ε Xd i=1

ωi

T³(x),x ε

∂T

∂xi

(x), ^(1.3)

whereωi^{arethesolutionsoftheellproblems. Sine}T ^{isdenedinthefulldomain} Ω ^while Tε ^{is merely dened in the perforated domain} Ωε^{, the orretor term is}

ruialfor agoodapproximation. Wemakeomparisonsbetweentheexatsolution

Tε ^{(or,at least, aonvergednumerial}approximationof it,when available)and the reonstrution(1.3). WeobtainanumerialerrorestimateoftheorderofεⁱⁿL²(Ω)^,

as predited by homogenization theory [9℄. Of ourse, the gain in terms of CPU

timeand memorystorageis enormouswhen using (1.3)insteadofsolvingtheexat

problem(1.1)sinethehomogenizedproblem(1.2)requiresonlyaoarsemesh. Note

howeverthattheellproblemmustbesolvedfordierentvaluesof themarosopi

temperatureT^{. Finallyletus mentionthat aslightlysimplermodelis studiedin [6℄}

andthatmoredetailsanbefoundin[14℄.

2. Setting of the problem. Thegoalof thissetion isto denepreisely the

geometry of theperforated periodi medium, to introdue the model of ondutive

andradiativeheat transferproblemandtogivesomenotations.

Figure2.1. Refereneellandperiodidomain

2.1. Geometry. LetΩ^{beasmoothbounded opensetin} R^{d (}d= 2^or3^{in the}

appliations). WedeneaperiodiperforateddomainΩε^,whereε^{denotesitsperiod,}

byremovingfromΩ^{aolletionofholes}(τε,k)_{k=1,...,M(ε)}^{inaperiodimanner. Eah}

holeτ_ε^{k isequal,uptoa}translation, tothesameunit holeτ ^{resaledatsize} ε^{. The}

domain Ω ^{is also subdivided in} N(ε) ^{periodiity ells} (Yε,i)i=1,...,N(ε)^{, eah of them}

beingequal,uptoatranslation,to thesameunit ellY =Qd

j=1(0, ℓj)^{. Thenumber}

ofperiodiityellsisnotequaltothenumberofholessine,intheappliationtogas

ooledreators,thereareseveralholesperell(seeFigure2.1). WedenotebyY^{∗ the}

solid partof Y^{, i.e.,} Y^∗ =Y \τ^{, and by} Γ ^{the boundary of} τ ^{(by aslightabuseof}

languagewedenote byτ ^{anindividual holeaswellasall theholesontainedin the}

unitellY^{). Toavoidsomeunneessary}tehnialities(see[1℄fordetails),weassume that,ifaperiodiityellutstheboundaryof Ω^{, thenit doesnotontainanyhole.}

Theholesτε,k ^{orrespondtoheliumhannelsinourappliationwhereradiativeheat}

transfertakesplae,whileΩε^{orrespondstothesoliddomainwhereondutiontakes}

plae. Insummarywehave

Ωε= Ω\

M(ε)[

k=1

τε,k, ∂Ωε=∂Ω∪Γε ^with Γε=

M(ε)[

k=1

∂τε,k=

N(ε)

[

i=1

Γε,i, ^(2.1)

where Γε,i ^{denotesthe boundariesof theholes}τε,k ^{inside theell}Yε,i^{. Denotingby}

mesthemeasure(surfaeorvolume,dependingontheontext)ofaset,wereallthe

followingidentities

mes(Y)ε^d= mes(Ω)

N(ε) (1+O(ε)), mes(Γε,i) =ε^d−1mes(Γ), mes(Yε,i) =ε^dmes(Y).

Denotingbydγ(x)^{thesurfaemeasureon}Γε^{,wedenetheenterofmass}x0,i^ofΓε,i

by

x0,i= 1 mes(Γε,i)

Z

Γε,i

x dγ(x) ^orequivalently

Z

Γε,i

(x−x0,i)dγ(x) = 0.

Similarly, y0 ^{denotesthe enter of massofthe unit holeboundary}Γ^{. Wereall the}

followingobviousidentities.

Lemma 2.1. Asmoothfuntion f ^satises Z

Γε,i

fx ε

dx=ε^d−1 Z

Γ

f(y)dy, Z

Γε,i

fx ε

(x−x0,i)dx=ε^d Z

Γ

f(y)(y−y0)dy, Z

Γε,i

fx ε

(x−x0,i)⊗(x−x0,i)dx=ε^d+1 Z

Γ

f(y)(y−y0)⊗(y−y0)dy,

ε

N(ε)

X

i=1

mes(Γε,i)f(x0,i) = mes(Γ) mes(Y)

Z

Ω

f(s)ds+O(ε).

2.2. Boundaryonditions. Asalreadysaidtheholesareatuallyheliumhan-

nelswhere radiativeheat transfertakesplae. Sinehelium isassumed tobetrans-

parent(noheatondution norabsorptionofradiation),thisproessismodeledbya

boundaryonditionontheholesboundaries. Letus reallthemodelingofradiative

exhanges between grey-diuse surfaes [15, 17℄. A grey-diuse surfae emits and

absorbsradiationinthesamemannerinalldiretions. Partofthereeivedradiations

anbereeted: a surfaeisthus haraterized byits emissivity e^{whih takesval-}

uesbetween0(fullreetion)and1(noreetion). DenotingbyT ^thetemperature andbyR ^{theradiosity,i.e. theintensity ofemittedradiation,wehavethefollowing}

relationship

R(x) = eσT⁴(x) + (1−e)J(x), ^(2.2)

Figure2.2. DomainwitharadiativeavityΣ

whereσ^istheStefan-Boltzmannonstantand J ^isgivenby J(x) =

Z

Σ

F(x, s)R(s)dγ(s),

whereF(x, s)^{istheviewfator(ageometrialquantity)betweentwodierentpoints} x^and s ^{of a avity} Σ^{(see Figure 2.2). Thus, the radiosityis given asthe solution}

ofanintegralequationintermsofthetemperature. Forourappliation,theexpliit

formulaoftheviewfatorin2-dforaonvexavityis

F(x, s) = ns·(x−s)nx·(s−x) 2|s−x|³

where nz ^{denotestheunit normalat thepoint}z^{. However,our}mathematialstudy doesnotrelyonthisspei formulaandwesimplyneedthefollowingpropertiesof

thekernelF^{: forany}(x, s)∈Σ^2,itsatises

• F(x, s)≥0^,

• F(x, s) =F(x, s)^,

• R

ΣF(x, s)ds= 1^.

LetJ^{betheoperatorgoingfrom}L^p(Σ)^,1≤p≤+∞^{,intoitselfdened by} J(f)(x) =

Z

Σ

F(x, s)f(s)dγ(s). ^(2.3)

DenotingbyE^{theoperatoronsisting of}multiplyingbytheemissivityvaluee^,(2.2)

anberewritten

R= (Id−(Id−E)J)⁻¹EσT⁴.

Ontheavitywalltheenergybalanereads

q−R+J = 0, ^(2.4)

whereq ^{istheheat ux}transmittedbyondutionfrom thesolid Ω^{to theavity}Σ^,

fromwhihwededue

q= G(σT⁴),

whereG^{isalinearnon-loaloperatordened by}

G(ϕ) = [Id−J] [Id−(Id−E)J]⁻¹E(ϕ) ∀ϕ∈L^p(Σ). ^(2.5)

LetusreallsomepropertiesofJ^{denedby(2.3)(see[22℄).}

Lemma 2.2. The operatorJ ^{going from}L^p(Σ) ^toL^p(Σ)^,1≤p≤ ∞^,satises

• J(c) =c^,∀c∈R^;

• kJk ≤1^;

• J ^{isnonnegative:} ∀f ∈L^p(Σ), f≥0 ⇒ J(f)≥0^;

• J ^issymmetri(self-adjoint for p= 2^{)inthe sense that} Z

Σ

J(ϕ)ψ= Z

Σ

J(ψ)ϕ, ∀ϕ∈L^p(Σ), ψ∈L^p′(Σ), ^with 1 p+ 1

p^′ = 1.

We easily dedue from Lemma 2.2 that (Id−ςJ)^, 0 ≤ς < 1^{, is invertible (for} ς = 1^,(Id−ςJ)^{isnotinvertiblesine}ker(Id−J) =R^{). Inpartiularwededuethat} G ^{is well dened, symmetriand non negative (this is lear for} 0 <e ≤ 1 ^{and for} e = 0^wendG≡0^).

Remark 2.3. The operators dened by (2.3), (2.5) will be denoted by J_ε^, Gε

respetively,ifating onΓε^insteadof Γ^.

2.3. Governing equations. Let K^{bethe ondutivity tensorof the unit ell} Y^{∗. We assume}K^{to be symmetri,uniformlyoeriveand bounded (innorm} L^∞),

i.e.,thereexist twopositiveonstants0< α≤β ^suhthat

∀v∈R^d, ^fora.e. y∈Y^∗, α|v|²≤ Xd i,j=1

Ki,j(y)vivj≤β|v|². ^(2.6)

Asusual,K(y)^beingaY^{-periodifuntion,wedeneits}Yε^{-periodiextension}

Kε(x) = Kx ε

.

For given bulk and surfae soure terms f ^and g^{, we onsider the following mixed}

problemofondution andradiativeheattransferfortheunknowntemperatureTε





−div(Kε∇Tε) = f ⁱⁿ Ωε, Kε∇Tε·n = g ^on ∂Ω,

−Kε∇Tε·n = ¹_εGε(σTε⁴) ^on Γε,

(2.7)

where G ^{is the operator dened by (2.5). For} non-negativesoures, the boundary valueproblem(2.7)admitsauniquepositivesolutionaswasprovedin[22℄. Themain

diultyin(2.7)isthenon-linearandnon-loalboundaryonditiononΓε^{. Notealso}

theε^{−1 salingin theboundaryonditionwhihinsuresthat theradiativeondition}

will notdisappear when passingto the limit ε → 0 ^{and will be} represented in the homogenizedmodel.

2.4. Notations. Thesubsript # ^{in thedenition of funtional spaes on the}

unitellY ^{indiatesthatweonsider}Y^{-periodifuntions. Wedenoteby}L²(Ω;C#(Y))

thespaeofmeasurableandsquaresummablefuntionsofx∈Ω^{withvaluesintheBa-}

nahspaeofontinuousandY^{-periodifuntionsof}y^{. Wedenoteby}L²(Ω;H_#¹(Y^∗))

thespaeof measurableand squaresummablefuntions ofx∈Ω^{withvaluesin the}

Sobolev spae H_#¹(Y^∗) ^of Y^{-periodi funtions dened only on} Y^{∗. We all ell-}

problemaproblemthatwesolveonlyontheelementaryelloftheperiodidomain.

Cell-problemsusually takeinto aountthe mirostruturebehaviorand ontribute

totheeetiveparametersalulation. WedenotebyO(ε^p), p∈R^afuntionofε >0

suhthatthereexistsaonstantC^{notdependingon}ε^sothatwehave|O(ε^p)| ≤Cε^p

forallε >0^.