Décomposition Orthogonale aux valeurs Propres (POD) appliquée à un modèle bidimensionnel de stockage d'énergie par adsorption pour une application au bâtiment

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(1)Science Arts & Métiers (SAM) is an open access repository that collects the work of Arts et Métiers Institute of Technology researchers and makes it freely available over the web where possible.. This is an author-deposited version published in: https://sam.ensam.eu Handle ID: .http://hdl.handle.net/10985/6619. To cite this version : Marie DUQUESNE, Alain SEMPEY, Stéphane GINESTET, Jean TOUTAIN, Elena PALOMO Décomposition Orthogonale aux valeurs Propres (POD) appliquée à un modèle bidimensionnel de stockage d'énergie par adsorption pour une application au bâtiment - 2012. Any correspondence concerning this service should be sent to the repository Administrator : archiveouverte@ensam.eu.

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(7) Congrès français de thermique, SFT2012, Talence, 29 mai - 1er juin 2012. !. Décomposition Orthogonale aux valeurs Propres (POD) appliquée à un modèle bidimensionnel de stockage d'énergie par adsorption pour une application au bâtiment. Marie DUQUESNE1*, Alain SEMPEY1, Stéphane GINESTET2, Jean TOUTAIN1, Elena PALOMO DEL BARRIO1 ". #$%&!!'()*+,-./!012/!#23!4154/!67889::!;,<+$=+/!6),$=+>!! ?@3A/!012/!#23!4154/!6!889::!;,<+$=+/!6),$=+>!! B)CD!+C!2+C%+)D!E,)%D;+=F/!012/!#23!4154/!67889::!;,<+$=+/!6),$=+>! 1 #$%&+)D%CG!*+!;(-<(-D+/!0@ABH#$%&+)D%CG!E,-<!A,I,C%+)/!J2K?!LB!8:1M/!! "84!,&+$-+!*+!3,$N-+%<!H!8":MM!;(-<(-D+!?+*+.!:9! O P,-C+-)!=())+DQ($*,$C!R!S,)%+>*-T-+D$+U+$D,S>+-V! Résumé : J+! DC(=W,N+! CF+)S(=F%S%T-+! Q,)! ,*D()QC%($7*GD()QC%($! *,$D! -$! ,*D()I+-)! =X<%$*)%T-+! %$CGN),$C!<+!=(-Q<+!YG(<%C+!"8Z7+,-!,![,%C!<\(I]+C!*+!Q<-D%+-)D!S(*G<%D,C%($D!P"K/!1KV!T-+!<\($!)+C)(-&+! *,$D!<,!<%CCG),C-)+>!J,!)GD(<-C%($!*-!S(*^<+!I%*%S+$D%($$+<!*G&+<(QQG!*,$D!=+CC+!GC-*+!%SQ<%T-+!_:::! GT-,C%($D!*%[[G)+$C%+<<+D!$($!<%$G,%)+D!+C!=(-Q<G+D!NG$G),$C!*+D!C+SQD!*+!=,<=-<D!=($DGT-+$CD>!J,!E`K! $(-D!,!Q+)S%D!*+!S($C)+)!T-+!=+!S(*^<+!+DC![()C+S+$C!)G*-=C%I<+>!! Nomenclature. !!. Q)(Q()C%($!*\,*D()I,C!*,$D!<+!<%C! =(+[[%=%+$C!C+SQ()+<!*+!<,!I,D+!E`K! *%[[-D%&%CG!*-![<-%*+/!$%&'()* +! Q)(Q()C%($!*\,*D()I,$C!*,$D!<+!<%C! ,* [,=C+-)!Q)G7+.Q($+$C%+<! -! =F,<+-)!DQG=%[%T-+/*.&/0()&1()* 2 * =,Q,=%CG! =,<()%[%T-+! *-! =(-Q<+! ,*D()I,C7 ! ,*D()I,$C/!.&/0()&1()** 3/! G$+)N%+!DC,C%DC%T-+!,-!D+$D!*+!<,!E`K* 34! G$+)N%+!*\,=C%&,C%($/!1* /! Q+)SG,I%<%CG!*+!<\,*D()I,$C/*$%* 5*** <($N-+-)!*-!<%C/!$* 6*** S,DD+!S(<,%)+!*+!<\,*D()I,C/!/0&$78()* 9* Q)+DD%($/!:"* ;* T-,$C%CG!*\+,-!*,$D!<+!<%C/!/0&$(<* =* =($DC,$C+!*+D!N,Y!Q,)[,%CD/!.&$78()&1()* >! ),X($/!$* ?! C+SQG),C-)+/!1* @! C+SQD/!'* A! &%C+DD+!*-!N,Y!*,$D!<\,*D()I,$C/!$!&'()* B! &%C+DD+!S(X+$$+!*-![<-%*+/!$!&'()* C! S(*+!*+!<,!I,D+!E`K* "! "#!. D! <($N-+-)/!$* EF$,78G'*0>G-'* * &,<+-)!Q)(Q)+!*+!<,!I,D+!E`K* Q()(D%CG!! * !! =($*-=C%&%CG!CF+)S%T-+!H&$()&1()! ! =F,<+-)!*\,*D()QC%($/!.&/0()* ! &%D=(D%CG!:"&'* ! S,DD+!&(<-S%T-+/!/0&$(<* ! =($DC,$C+!D-%&,$C!<,!<(%!*\B))FG$%-D* I4JK-G'*G@*GL97'"4@'* "! QF,D+!,*D()IG+! ,! <%C!*\,*D()I,$C! G! GT-%<%I)+! #! [<-%*+! 0! QF,D+!N,Y+-D+! K4! +$C)G+! $! C-I+!SGC,<<%T-+! 9! Q,)C%=-<+D!*\,*D()I,$C! :MN! )+=($DC)-%C+!,&+=!<,!E`K! '! ,*D()I,$C!D(<%*+! O! %$%C%,<!. 1. Introduction L$!6),$=+/!<+!D+=C+-)!*-!IaC%S+$C!+DC!)+DQ($D,I<+!*\+$&%)($!-$!T-,)C!*+D!GS%DD%($D!*+!?`1! +C!*+!9_b!*+D!=($D(SS,C%($D!G$+)NGC%T-+D!c"d>!E(-)![,%)+![,=+!,-!)G=F,-[[+S+$C!=<%S,C%T-+!. - 583 -.

(8) Congrès français de thermique, SFT2012, Talence, 29 mai - 1er juin 2012. ! +C! e! <\GQ-%D+S+$C! *+D! G$+)N%+D! [(DD%<+D/! %<! [,-C! =($D(SS+)! S(%$D! *\G$+)N%+! +C! -C%<%D+)! *+D! D(-)=+D! [,%I<+S+$C! GS+CC)%=+D! *+! ?`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aC%S+$C! +DC! *G&+<(QQG>!L$[%$/!<,!KG=(SQ(D%C%($!`)CF(N($,<+!,-.!&,<+-)D!E)(Q)+D!PE`KV/!+DC!-C%<%DG+!Q(-)! +$&%D,N+)!*+!)G*-%)+!<,!C,%<<+!*-!S(*^<+!*G&+<(QQG!+C!*\+$![,=%<%C+)!<,!D%S-<,C%($>!. 2. Sélection du type de stockage Af%<! +.%DC+! -$! N),$*! $(SI)+! *+! S(*+D! *+! DC(=W,N+! P=[>! [%N-)+! "V/! <+! DC(=W,N+! CF+)S(=F%S%T-+!Q,)!D()QC%($!Q)GD+$C+!<+D!=,),=CG)%DC%T-+D!)+T-%D+D!=($=+)$,$C!<+D!N,SS+D!*+! C+SQG),C-)+D!+C!*+!)G&+)D%I%<%CG!Q(-)!-$+!,QQ<%=,C%($!,-.!IaC%S+$CD>!'%+$!T-f%<!D(%C!<GN^)+S+$C! S(%$D! Q+)[()S,$C! +$! C+)S+D! *+! *+$D%CG! G$+)NGC%T-+! T-+! <+! DC(=W,N+! =F%S%T-+/! %<! +DC! =(SQ,C%I<+! ,&+=! <+D! C+SQG),C-)+D! Q)(*-%C+D! e! Q,)C%)! *+! <fG$+)N%+! D(<,%)+! PPO* Q* %%OR2! F()D! =($=+$C),C%($V>! L$[%$/! <,! N),$*+! =(SQ,=%CG! *+D! ,*D()I,$CD! <+-)! =($[^)+! *+D! T-,<%CGD! +DD+$C%+<<+D!+$!C+)S+D!*+!=(gC!+C!*+!&(<-S+>!!. ! SK0A>G*)*T*6"@U>K"AL*"J'7>,"4@'*G@*@F9G'*JG*'@7-/"0G*V%W* ! J+! =F(%.! *-! =(-Q<+! ,*D()I,$C7,*D()I,C! e! %$CGN)+)! ,-! D+%$! *\-$! DXDC^S+! *+! DC(=W,N+! +DC! Q)%S()*%,<!=,)!D($!+[[%=,=%CG!+$!*GQ+$*>!;(-C!*\,I()*/!<+!=F(%.!*\-$!,*D()I,C!+DC!)GN%!Q,)!D+D! Q)(Q)%GCGD!QFXD%=(7=F%S%T-+D!c8d!+C!c9d>!J\+,-!+DC!Q,)[,%C+S+$C!Dg)+!Q(-)!<\h(SS+/!*%DQ($%I<+! +$!T-,$C%CG/!,&+=!*+!I($$+D!Q)(Q)%GCGD!CF+)S%T-+D!+C!,QQ,),iC!=(SS+!%*G,<+!C,$C!,-.!$%&+,-.! FXN%^$+/! DG=-)%CG! +C! +$&%)($$+S+$C! T-\G=($(S%T-+! +C! Q),C%T-+>! K+! $(SI)+-.! ,*D()I,$CD! ,DD(=%GD! e! <\+,-! D($C! e! *%DQ(D%C%($/! <+-)D! =,),=CG)%DC%T-+D! Q)%$=%Q,<+D! D($C! Q)GD+$CG+D! *,$D! <+! C,I<+,-! ">! J+D! D%<%=+D! SGD(Q()+-D+D! ()N,$%DG+D! +C! <+D! S+C,<7()N,$%=! [),S+j()WD! ($C! *+D! ,[[%$%CGD! C)^D! )+DC)+%$C+D! Q(-)! <\+,->! J,! *+$D%CG! G$+)NGC%T-+! CFG()%T-+! *+D! YG(<%C+D/! S%$G),-.! S%=)(Q()+-.! ,QQ,)C+$,$C! e! <,! [,S%<<+! *+D! ,<-S%$(D%<%=,C+D/! +DC! *\+$&%)($! "k:! /HXY$<! D(%C! <+! *(-I<+!*+!<\+,-!+$!DC(=W,N+!Q,)!=F,<+-)!D+$D%I<+!+C!<+!C,I<+,-!"!S($C)+!T-+!<+!=(-Q<+!YG(<%C+7 +,-!,!*+!C)^D!I($$+D!=,Q,=%CGD!*\,*D()QC%($!+C!*+!)GNG$G),C%($/!-$+![,%I<+!C(.%=%CG/!-$!Q)%.!I,D!. - 584 -.

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(14) Congrès français de thermique, SFT2012, Talence, 29 mai - 1er juin 2012. ! +C! -$+! N),$*+! DC,I%<%CG! e! <($N! C+)S+>! B%$D%/! %<! ! D+SI<+! =($DC%C-+)! -$+! *+D! D(<-C%($D! <+D! Q<-D! ,*,QCG+D!e!-$!DC(=W,N+!*+!<\G$+)N%+!e!I,DD+!C+SQG),C-)+!Q(-)!-$+!,QQ<%=,C%($!,-.!IaC%S+$CD>! !. A()I,$CD!. ?,Q,=%CG! *\,*D()QC%($!. ;,-.! *\,*D()QC%($!. 3GNG$G),C%($!. E)%.!. ;(.%=%CG!. AC,I%<%CG!e! <($N!C+)S+!. !. Z0Y0[*. !. !. !. !. !. l+<!*+!D%<%=+!. :>9:7:>94!. mm!. mmm!. m!. m!. mmm!. nG(<%C+D!. :>1:7:>94!. mmm!. mmm!. m!. m!. mmm!. B<-S%$(7! QF(DQ,C+D!. :>"47:>49!. mm!. mm!. mm!. m!. m!. A%<%=+D! SGD(Q()+-D+D!. :>947:>k9!. mm!. m!. mm!. m!. m!. B)N%<+D!. :>147:>9k!. mmm!. m!. m!. m!. m!. ?",8G"A*)*T*:>K4-K9"8G'*-">"-@U>K'@K;AG'*JG'*$"@U>K"AL*"J'7>,"4@'*"''7-KU'*Q*8\G"A*V]W*. 3. Modélisation bidimensionnelle de l'adsorbeur J+D! DXDC^S+D! e! D()QC%($! D($C! =(SQ(DGD! *\-$! ,*D()I+-)/! *\-$! G&,Q(),C+-)! +C! *\-$! =($*+$D+-)>!J+D!*+-.!*+)$%+)D!GC,$C!*+D!G=F,$N+-)D!*%QF,D%T-+D!=<,DD%T-+D!!<,)N+S+$C!GC-*%GD! *,$D! <,! <%CCG),C-)+/! $(-D! $(-D! D(SS+D! [(=,<%DGD! D-)! <,! S(*G<%D,C%($! I%*%S+$D%($$+<<+! *-! DC(=W,N+!*+!<\G$+)N%+!*,$D!<\,*D()I+-)/!G<GS+$C!=<G!=,)!=($C+$,$C!<+!S%<%+-!Q()+-.>!! 3.1. Description du problème physique J\,*D()I+-)!GC-*%G!+DC!=<,DD%T-+/!*+!$(SI)+-.!,-C+-)D!($C!C),%CG!<+!SoS+!CXQ+!*+!DXDC^S+D! c_d>!0<!+DC!=($DC%C-G!*\-$![<-%*+!=,<(Q()C+-)!T-%!=%)=-<+!*,$D!-$!C-I+!SGC,<<%T-+!+$C(-)G!Q,)!-$! <%C! *+! YG(<%C+! "8Z! =X<%$*)%T-+! P[%N-)+! 1V>! B%$D%/! <()D! *+! <,! QF,D+! *+! DC(=W,N+! *+!<\G$+)N%+/! <+! [<-%*+!=F,-[[+!<\,*D()I,$C!Q,)!<\%$C+)SG*%,%)+!*-!C-I+!SGC,<<%T-+!+C!+$N+$*)+!<,!*GD()QC%($!*+! <\,*D()I,C/!<,!&,Q+-)!*\+,-/!T-%!D\G&,=-+!Q,)!<+D!*+-.!+.C)GS%CGD!*-!<%C>!!. SK0A>G*%*T*E-XU$"*JA*9>7,8^$G*9XF'K;AG*U@AJKU*. - 585 -.

(15) Congrès français de thermique, SFT2012, Talence, 29 mai - 1er juin 2012. ! 3.2. Hypothèses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quations de conservation J+D! C),$D[+)CD! =(-Q<GD! *+! =F,<+-)! +C! *+! S,DD+! D-)&+$,$C! ,-! D+%$! *+! <\,*D()I+-)! ($C! GCG! GC-*%GD!Q(-)!*G&+<(QQ+)!-$!S(*^<+!D\%$DQ%),$C!*+D!C),&,-.!*+!A-$!+C!,<>!cM/kd>!J+D!G&(<-C%($D! DQ,C%(7C+SQ()+<<+D! *+D! &,)%,I<+D! *\GC,C! D($C! D%S-<G+D! Q+)S+CC,$C! ,%$D%! <\,$,<XD+! *+! D($! =(SQ()C+S+$C! *X$,S%T-+>! J+! S(*^<+! I%*%S+$D%($$+<! C),$D%C(%)+! )G,<%DG! ,! Q(-)! &,)%,I<+D! <,! C+SQG),C-)+! *,$D! <+! [<-%*+/! ;[/! <,! C+SQG),C-)+! ;! +C! <,! Q)+DD%($! Q! *,$D! <\,*D()I,$C>! J+D! GT-,C%($D!T-+!<\($!)GD(-C!D($C!R! <\GT-,C%($!*+!=($D+)&,C%($!*+!<\G$+)N%+!*-![<-%*+!D\G=(-<,$C!e!-$+!&%C+DD+!S(X+$$+!B!*,$D! -$!C-I+!*+!),X($!3"/! !!!! > ; > F1 I 2? # 1B ?" < G J " # ? # : ,&+=! 2@ ?@ = =" H JK 2D !. 2? #. !. ? # > / D / : ?:. ! ? # > / :/ @ ?K4!!+C!? # > /"/ @ !<%I)+ !. !!!!!!!!!! P8V! !. !!!!. <\GT-,C%($!*+!=($D+)&,C%($!*+!<,!S,DD+!*,$D!<\,*D()I,$C/! 2# 2; ! 7 #A + 2@ 2@. ! , " 9 9 !+C!+ " 9 " , ! ! ! ! !. - 586 -. !. !. ,&+=!!!. 9 > / :/ @ 9 > /"/ @ 9 > / D / : 9: !. 29 =1 / D / @ 2>. 29 =8 / D / @ 2>. :!. !!!!!!!! P9V!.

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(21) Congrès français de thermique, SFT2012, Talence, 29 mai - 1er juin 2012. ! <\GT-,C%($!*+!=($D+)&,C%($!*+!<\G$+)N%+!*,$D!<\,*D()I,$C/!. 2 2:?. 7. #A- ? 7> 7? + _ 22;@. 2@ 2: ! # A-0 + # ' -' ;-" 0. . "#$% . . ? > / D / : ?:. !. 2? > / :/ @ 2D. 2? > /"/ @ 2D. P4V!. :!. . K+! Q<-D/! ! =($[()SGS+$C! ,-.! FXQ(CF^D+D/! %<! $\X! ,! Q,D! *\,==-S-<,C%($! *\G$+)N%+! *,$D! <+! SGC,</! <,! =($*%C%($! *+! ),==()*+S+$C! *-! [<-.! +$C)+! <\%$C+)[,=+! [<-%*+pSGC,<! +C! =+<<+! SGC,<p,*D()I,$C!D\+.Q)%S+!*($=!R!!. 2? # =" / D / @ . !. 2>. 2? =1 / D / @ $ " ? # =" ? =1 ! $ " ? # =" ? =1 !+C!. # =" <$ =" =1 =1 <$ =" =1 2>. P_V!. 3.4. Isotherme d'adsorption. J,!=%$GC%T-+!*\,*D()QC%($!*\-$!N,Y!Q,)!-$!D(<%*+!Q()+-.!](-+!-$!)q<+!Q)GQ($*G),$C!*,$D!<+! [($=C%($$+S+$C! *\-$! DXDC^S+! *+! DC(=W,N+! e! ,*D()QC%($>! J+! S(*^<+! =%$GC%T-+! GC-*%G! D\,QQ-%+!! ! D-)!<+!S(*^<+!NG$G),<%DG!*+!J,$NS-%)!e!8!=(-=F+D!c9d!R! ! 2; ( ; ;G ! !! 2@. e!<\GT-%<%I)+/!. ; ;G ! 9 ! . !!!!!! !. !!!!!!! ,&+=!. 2; 2;G 29 2;G 2? !. 2@ 29 2@ 2? 2@. ;G ;' / 4. ;' /"," 9. " ," 9. . ;' /1,1 9. . ;' /8,8 9. " ,1 9 " ,8 9 " " " ":/4 "/4 1/14 8/84 ? ? ?. ,4 ,:/4 +.Q 34 p ? ! 4 "/ 1/8 !. PMV! . !?(SQC+!C+$-!*+D!N),$*+-)D!=,),=CG)%DC%T-+D!*+D!YG(<%C+D!+C!*+D!=($*%C%($D!*\-C%<%D,C%($/!<+! C+SQD!=,),=CG)%DC%T-+!!*+!<,!=%$GC%T-+!*+!C),$D[+)C!*+!S,DD+!*+!$(C)+!Q)(I<^S+!+DC!*+!<\()*)+!*+! <,! S%=)(D+=($*+>! B%$D%/! $(-D! Q(-&($D! =($D%*G)+)! T-+! <+! DXDC^S+! GC-*%G! +DC! e! <\GT-%<%I)+! CF+)S(*X$,S%T-+!e!C(-C!%$DC,$C>! 3.5. Résultats numériques. J+!D=FGS,!*+!)GD(<-C%($!D\,QQ-%+!D-)!-$+!SGCF(*+!*\()*)+!9!I,DG+!D-)!-$!D=FGS,!=(SQ,=C! =+$C)G/!Q)G=%D!+C!DC,I<+!c5d>!J+!DXDC^S+!+DC!*%D=)GC%DG!+$!"::!S,%<<+D!<($N%C-*%$,<+D/!1:!S,%<<+D! ),*%,<+D! *,$D! <+! [<-%*+! +C! 1:! *,$D! <\,*D()I,$C>! B%$D%/! 1:::! GT-,C%($D! D($C! e! )GD(-*)+! Q(-)! =F,T-+! &,)%,I<+/! D(%C! -$! DXDC^S+! *+! _:::! GT-,C%($D! ,-! C(C,<>! #$+! D(<<%=%C,C%($! PG=F+<($! *+! C+SQG),C-)+! ,QQ<%T-G! +$! Yr:! R! ?K4`ab<* 1V! +DC! ,QQ<%T-G+! ,-! DXDC^S+/! *($C! <\GC,C! %$%C%,<! +DC! e! <\GT-%<%I)+!P?O`%b<*1!+C!:O`cO*:"V>!J\G&(<-C%($!*-!DXDC^S+!+DC!,<()D!GC-*%G+!D-)!-$+![+$oC)+! C+SQ()+<<+!*+!8!F+-)+D>!J\%$CGN),C%($!*-!DXDC^S+!+DC!)G,<%DG+!Q,)!-$+!SGCF(*+!*+!CXQ+!l+,)!e! ()*)+!+C!Q,D!&,)%,I<+D!c":d>!J,!C(<G),$=+!*-!=,<=-<!+DC!*+!<\()*)+!*+!":7_>!J\()*%$,C+-)!-C%<%DG!+DC! -$!0$C+<!P3V!?()+!P;2V!0M/!?E#!5M4U8>88!lhY/!1>:1!lhY/!"1l(!*+!3B2>!! J,!)GD(<-C%($!$-SG)%T-+!*+!=+!DXDC^S+!%SQ<%T-+!<\%$CGN),C%($!C+SQ()+<<+!*+!_:::!GT-,C%($D! *%[[G)+$C%+<<+D!NG$G),$C!*+D!C+SQD!*+!=,<=-<D!=($DGT-+$CD!P"F8:V>!E(-)!Q,<<%+)!=+CC+!*%[[%=-<CG/! <+D! Q(%*D! )+<,C%[D! *+D! *%[[G)+$CD! SG=,$%DS+D! *+! C),$D[+)C! *+! S,DD+! +C! *+! =F,<+-)! ($C! GCG! =(SQ,)GD>! ?+Q+$*,$C! <\%SQ()C,$=+! )+<,C%&+! *+! =F,T-+! SG=,$%DS+! *+! C),$D[+)C! $+! Q+)S+C!. - 587 -.

(22) Congrès français de thermique, SFT2012, Talence, 29 mai - 1er juin 2012. ! ),%D($$,I<+S+$C! Q,D! *+! Q(-&(%)! D%SQ<%[%+)! QFXD%T-+S+$C! <+D! GT-,C%($D/! *\(s! <\,QQ<%=,C%($! *\-$+!SGCF(*+!*+!)G*-=C%($>!. 4. Décomposition orthogonale aux valeurs Propres J,! E`K! +DC! ,QQ<%T-G+! e! $(C)+! S(*^<+! ,[%$! *f,QQ)(.%S+)! <+! DXDC^S+! =)GG! *+! *%S+$D%($! G<+&G+! $G=+DD%C,$C! -$+! %$CGN),C%($! C+SQ()+<<+! Q,)! -$! ,-C)+! DXDC^S+! *+! *%S+$D%($! $+CC+S+$C! Q<-D! [,%I<+! c""d>! L<<+! $(-D! Q+)S+C! *($=! *+! *GC+)S%$+)! D%! <+! S(*^<+! +DC! [()C+S+$C! )G*-=C%I<+! ,[%$!*\G&%C+)!<+D!C+SQD!*+!=,<=-<D!Q)(F%I%C%[D>!! J,! E`K! =($D%DC+! e! =($D%*G)+)! <+D! &,)%,I<+D! =(SS+! -$+! =(SI%$,%D($! <%$G,%)+! *+! S(*+D! Q)(Q)+D!C!+C!*+!=(+[[%=%+$CD!C+SQ()+<D!"!C(-C!+$!=($D+)&,$C!<+D!=,),=CG)%DC%T-+D!+C!<+!=(-Q<,N+! *-!S(*^<+!R! > / D / @ !!?:MN > / D / @ ? > / D : C$ > / D "$ @ ! /. !. PkV!. $ ". ?+CC+!*G=(SQ(D%C%($!+DC!(IC+$-+!e!Q,)C%)!*f-$!+$D+SI<+!*+!*($$G+D!Q)(&+$,$C!D+<($!<+!=,D! *+!D%S-<,C%($D!$-SG)%T-+D!(-!*f+.QG)%+$=+D>!J+D!S(*+D!Q)(Q)+D!*G[%$%DD+$C!,<()D!-$+!I,D+!*+! Q)(]+=C%($! ()CF(N($,<+! )+Q)GD+$C,C%&+! *+D! )G,<%D,C%($D! <+D! Q<-D! Q)(I,I<+D! +C! (QC%S,<+! ,-! D+$D! G$+)NGC%T-+! c"1d>! #$! [,%I<+! $(SI)+! *+! S(*+D! +DC! ,<()D! D-[[%D,$C! Q(-)! *G=)%)+! =())+=C+S+$C! <\G&(<-C%($! *X$,S%T-+! *-! DXDC^S+>! B[%$! *+! *GC+)S%$+)! =+! $(SI)+! W! *+! S(*+D! e! =($D+)&+)! $(-D!,&($D!=($D%*G)G!Q<-D%+-)D!=)%C^)+D!R!!. J\G$+)N%+!DC,C%DC%T-+!,-!D+$D!*+!<,!E`K/!G$+)N%+!=-S-<G+!LW!*+D!S(*+D!=($D+)&GD/!*(%C!oC)+! D-QG)%+-)+!e!55b!+C!<,!*+)$%^)+!&,<+-)!Q)(Q)+!PV!=($D+)&G+!*(%C!oC)+!%$[G)%+-)+!,-!=+$C%^S+! *+! <,! Q)+S%^)+>! ?($D+)&+)! <+D! 8! Q)+S%+)D! S(*+D! $(-D! ,DD-)+! <+! )+DQ+=C! *+D! 1! =)%C^)+D! Q)G=G*+$CD!P=[!C,I<+,-!1V>!!. : /. 3/ . !. K " ,. K. K ". K. :. !. P5V!. 2(*+D!. LW!P;[V!. !P;[V!. LW!P;V!. !P;[V!. LW!PQV!. !PQV!. "!. 5M/1:51!. "/kk8>":7_!. 5k/k"MM!. 1/"5M>":74!. kk/1:44!. 5/M"_>":79!. 1!. 55/4M91!. :/:9_>":7_!. 55/4k:4!. :/:"M>":74!. 55/:_9k!. "/"5_>":79!. 8!. 55/k918!. :/::4>":7_!. 55/k4_5!. :/::_>":74!. 55/M98M!. :/:M4>":79!. 9!. 55/5_8:!. :/::1>":7_!. 55/549:!. :/::1>":74!. 55/51MM!. :/:1:>":79!. 4!. 55/55k4!. t!":7k!. 55/5M85!. t!":7k!. 55/5M""!. :/::4>":79!. ?",8G"A*%*T*34G>0KG*-A$A8UG*G@*B"8GA>'*9>79>G'*'G874*8G'*$7JG'!. J,!)+=($DC)-=C%($!*+D!&,)%,I<+D!e!Q,)C%)!*+!<,!=(SI%$,%D($!<%$G,%)+!*+!S(*+D!Q)(Q)+D!+C!*+! =(+[[%=%+$CD!C+SQ()+<D!+DC!=(SQ,)G+!,-.!&,)%,I<+D!)G+<<+D!Q(-)!<,!C+SQG),C-)+!+C!<,!Q)+DD%($! *,$D! <\,*D()I,$C! N)a=+! ,-! =,<=-<! *+! <\G=,)C7CXQ+! +$C)+! <+D! &,)%,I<+D! )G+<<+D! +C! )+=($DC)-%C+D! *,$D!<\,*D()I,$C>!! !. - 588 -. G9 . EE 9 9 1! >J>JD !+C!G EE 1! >J>JD 1. :MN. ?. EE ? ? 1! >J>JD ! EE 1! >J>JD 1. :MN. P":V!.

(23) 32+6?7*6%2=%-7()8,)61-59)!"

(24)

(25) "%0)2')

(26) 1%- )6.9-2

(27)

(28) Congrès français de thermique, SFT2012, Talence, 29 mai - 1er juin 2012. ! J,![%N-)+!8!S($C)+!T-+!8!S(*+D!D+SI<+$C!D-[[%D,$CD!Q(-)!)+=($DC)-%)+!<,!C+SQG),C-)+!*,$D! <\,*D()I,$C>! L$! +[[+C/! <,! C+SQG),C-)+! GC,$C! =(SQ)%D+! +$C)+! %b<* 1! +C! ab<* 1/! <\G=,)C7CXQ+! (IC+$-!+$C)+!<+D!C+SQG),C-)+D!)G+<<+!+C!)+=($DC)-%C+!+DC![,%I<+>!E,)!=($C)+/!<,!Q)+DD%($!%$%C%,<+! +DC! cO* :"/! ($! Q+-C! *($=! *G*-%)+! *+! <,! [%N-)+! 9! T-+! 4! S(*+D! D+SI<+$C! $G=+DD,%)+D! Q(-)! )+=($DC)-%)+!Q)G=%DGS+$C!<,!Q)+DD%($!*,$D!<\,*D()I,$C>!. SK0A>G*<*T*U-">@(@F9G**G?*. SK0A>G*a*T*U-">@(@F9G**G9*. !. K,$D! <+! =,*)+! *+! $(C)+! GC-*+/! <\G$+)N%+! DC(=WG+! +DC! <,! N),$*+-)! T-%! %SQ()C+>! B-.! C)(%D! =)%C^)+D! Q)G=G*+$CD! =<,DD%T-+S+$C! -C%<%DGD! ,&+=! <,! E`K/! $(-D! ,](-C($D! *($=! =+! =)%C^)+! D-QQ<GS+$C,%)+>! J\G$+)N%+! L! DC(=WG+! *,$D! <+! S,CG)%,-! +C! <\G=,)C7CXQ+! +L! +$C)+! <+D! G$+)N%+D! )G+<<+!+C!)+=($DC)-%C+D!D\G=)%&+$C!R!!. 2d 3 E + _ J@ !,&+=!d EE ; 1! >J>JD +C!G3 2@ : @. !!!. SK0A>G*]*T*27$9">"K'74*G4@>G*U4G>0KG'* '@7-/UG'*>UG88G*G@*>G-74'@>AK@G*. EE 3 3 1! >J>JD ! EE 1! >J>JD 1. :MN. P""V!. SK0A>G*c*T*U-">@(@F9G*G3*. J,![%N-)+!4!S($C)+!T-+!=($D+)&+)!8!(-!4!S(*+D!%$*-%C!*+D!+))+-)D!,-.!C+SQD!=(-)CD!*,$D!<,! )+=($DC)-=C%($! *+! <\G$+)N%+! DC(=WG+>! E,)! =($DGT-+$C/! Q(-)! T-+! <,! )+=($DC)-=C%($! D(%C! D-[[%D,SS+$C! Q)G=%D+! +C! $(C,SS+$C! ,-.! C+SQD! =(-)CD/! k! S(*+D! *(%&+$C! oC)+! =($D+)&GD! Q(-)! =F,T-+!&,)%,I<+>!. - 589 -.

(29) Congrès français de thermique, SFT2012, Talence, 29 mai - 1er juin 2012. !. 5. Conclusion K,$D!=+C!,)C%=<+/!-$!CXQ+!*+!DC(=W,N+!+C!-$!DXDC^S+!C+=F$%T-+!($C!GCG!DG<+=C%($$GD>!J\GC-*+! *+D! C),$D[+)CD! =(-Q<GD! *+! =F,<+-)! +C! *+! S,DD+! ,-! D+%$! *+! <\,*D()I+-)/! G<GS+$C! =<G! *+! =+! DXDC^S+/!,!*($$G!<%+-!e!-$!S(*^<+!I%*%S+$D%($$+<>!J\,$,<XD+!*+D!Q(%*D!)+<,C%[D!*+!=F,=-$!*+D! SG=,$%DS+D!+$C),$C!+$!]+-!,!S($C)G!T-+!=+!S(*^<+!$+!Q(-&,%C!oC)+!D%SQ<%[%G!QFXD%T-+S+$C>! J,!E`K!$(-D!,!*($=!Q+)S%D!*+!*GC+)S%$+)!T-+!k!S(*+D!Q(-)!=F,T-+!&,)%,I<+!D-[[%D+$C!Q(-)! *G=)%)+! <+! =(SQ()C+S+$C! *X$,S%T-+! *+! <\,*D()I+-)! GC-*%G>! @(-D! +$&%D,N+($D! e! Q)GD+$C! *+! )G*-%)+! <+! S(*^<+! ,[%$! *\+$! [,=%<%C+)! <,! D%S-<,C%($! +C! *\-C%<%D+)! <,! E`K! =(SS+! SGCF(*+! *\,$,<XD+! QFXD%T-+! *+D! *($$G+D! Q(-)! )G,<%D+)! -$+! GC-*+! ,QQ)([($*%+! *+D! *%[[G)+$CD! SG=,$%DS+D!*+!C),$D[+)CD!=(-Q<GD!*+!=F,<+-)!+C!*+!S,DD+>! Références c"d ;,I<+! )($*+! *-! l)+$+<<+! *+! <\L$&%)($$+S+$C! uJ-CC+)! =($C)+! <+D! =F,$N+S+$CD! =<%S,C%T-+Du/! E)(N),SS+!u2(*+)$%D+)!<+!IaC%S+$C!+C!<,!&%<<+u/!P1::MV>! c1d v,=(I!w,$!'+)W+</!A(<,)!CF+)S,<!DC(),N+!C+=F$%T-+D/!3+D+,)=F!=(SS%DD%($+*!IX!;F+!@+CF+)<,$*D! BN+$=X![()!L$+)NX!,$*!CF+!L$&%)($S+$C!@`wL2/!Q)(]+=C!x!"98>_1:7584>k!P`=C(I)+!1:::V>!! c8d A>!2,)C+,-/!B*D()QC%($/!+*91_8!0@3A/!18P85V/!P1::4V>! c9d y>!?F+W%)(-/!LC-*+!+C!,$,<XD+!*\-$+!S,=F%$+![)%N()%[%T-+!D(<,%)+!e!,*D()QC%($/!;F^D+!*+!K(=C(),C! +$!A=%+$=+D!+$!EFXD%T-+/!#$%&+)D%CG!*+!2+$C(-)%7?($DC,$C%$+/!P1::kV>! c4d z>0>! B)%DC(&/! ?F,<<+$N%$N! ([[+)D! ([! S,C+)%,<! D=%+$=+! [()! ,*D()QC%($! DC(),N+! ([! CF+)S,<! +$+)NX/! L-)(CF+)S!A+S%$,)!${58/!P"_7"k!@(&+SI)+!1:""V>! c_d J>n>! nF,$N/! A(<,)! L$+)NX/! B! CF)++7*%S+$D%($,<! $($7+T-%<%I)%-S! S(*+<! [()! ,$! %$C+)S%CC+$C! ,*D()QC%($!=((<%$N!DXDC+S/!E78">*34G>0F/!P1:::V/!_5P"V/!1M784> cMd J>!2>!A-$/!@>!'+$!BS,)!+C!6>!2+-$%+)/!@-S+)%=,<!DC-*X!($!=(-Q<+*!F+,C!,$*!S,DD!C),$D[+)!%$!,$! ,*D()I+)!j%CF!+.C+)$,<![<-%*!F+,C%$N/!_G"@*=G-7BG>F*EF'@G$'*e*2_:/!P"554V/!"4P"V/!"5715>! ckd @>! '+$! BS,)/! J>2>! A-$/! 6>! 2+-$%+)/! @-S+)%=,<! ,$,<XD%D! ([! ,*D()QC%&+! j,&+! )+N+$+),C%&+! F+,C! C+SQ+),C-)+!Q-SQ/!!998KGJ*?XG>$"8*340K4GG>K40/!P"55_V/!"_P"V/!9:479"k>! c5d A>! h,S*%! +C! ,<>/! 2+CF(*! ([! <%$+Dp+.,SQ<+! %SQ<+S+$C,C%($p*DD:99/! E-X78">9GJK"/! 1P":V! R! 1k45/ ! *(%!R!":>9195pD=F(<,)Q+*%,>49:"/!P1::MV/!FCCQRppjjj>D=F(<,)Q+*%,>()Np,)C%=<+p2+CF(*|([|<%$+D> c":d J>6>! AF,SQ%$+/! 2>y>! 3+%=F+<C! +C! v>B>! }%+)Y+$W,/! A(<&%$N! 0$*+.7"! KBLD! %$! 2B;JB'! ,$*! A%S-<%$W/!EI!6*=GBKGf/!P"555V/!9"/!48k7441>! c""d B>! A+SQ+X/! ?>! 0$,)*/! ?>! lF%,-D/! ?>! B<<+)X/! 6,DC! D%S-<,C%($! ([! C+SQ+),C-)+! *%DC)%I-C%($! %$! ,%)! =($*%C%($+*! )((SD! IX! -D%$N! Q)(Q+)! ()CF(N($,<! *+=(SQ(D%C%($/! +AK8JK40* "4J* 34BK>74$G4@/! P1::5V/!99P1V/!1k:715k> c"1d J>! A%)(&%=F/! ;-)I-<+$=+! ,$*! CF+! *X$,S%=D! ([! =(F+)+$C! DC)-=C-)+D/! E,)C! "! R! ?(F+)+$C! DC)-=C-)+D>! E,)C! 1! R! AXSS+C)%+D! ,$*! C),$D[()S,C%($D>! E,)C! 8! R! KX$,S%=D! ,$*! D=,<%$N>* dA"@G>8F* 7#* !998KGJ* 6G-X"4K-'/!P"5kMV/!94/!4_"745:> Remerciements ?+D!C),&,-.!($C!GCG!+$!Q,)C%+![%$,$=GD!Q,)!<fBN+$=+!@,C%($,<+!*+!<,!3+=F+)=F+!PB@3V!,-!C),&+)D!*-! Q)(N),SS+! h,I%C,C!%$C+<<%N+$C!+C!D(<,%)+!QF(C(&(<C,~T-+!PQ)(]+C!A020@;hL?!${B@37:k7hB'0A`J7 :""V>!. - 590 -.

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