Meshless method for solving coupled radiative and conductive heat transfer in refractive index medium

Transcription

1 Joural of Phyc: Coferece Sere PAPER OPE ACCESS Mehle ethod for olvg coupled radatve ad coductve heat trafer refractve dex edu o cte th artcle: Cheg-A ag et al 6 J. Phy.: Cof. Ser Related cotet - Mehle aaly of geoetrcally olear ea J M a D M e ad R H J - uercal aaly of heat coducto prole o rregular doa y ea of a collocato ehle ethod R Zaolo ad E ole - uercal aaly of heat coducto prole o 3D geeral-haped doa y ea of a RBF Collocato Mehle Method R Zaolo ad E ole Vew the artcle ole for update ad ehaceet. h cotet wa dowloaded fro P addre o 7//9 at 8:5

2 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 Mehle ethod for olvg coupled radatve ad coductve heat trafer refractve dex edu Cheg-A ag Haou Sadat ad Ja-Yu a School of Autoole Egeerg Har ttute of echology at eha et ehua Road eha 649 People Repulc of Cha ttut PPRME - CRS - Uverté de Poter ESMA Départeet Flude herque Couto - Capu Sud 4 aveue du recteur Peau - 86 POERS Cedex Frace E-al: taayu@htwh.edu.c; haou.adat@uv-poter.fr Atract. A dffue approxato ehle ethod (DAM) eployed a a ea of olvg the coupled radatve ad coductve heat trafer prole e-traparet refractve dex eda cotaed D ad D geoetre. he ehle approach for radatve trafer aed o the dcrete ordate equato. Cae of coed coductoradato are preeted cludg plae parallel la quare ecloure ad ecrcular ecloure wth a er crcle. he fluece of the refractve dex o the teperature dtruto ad heat fluxe vetgated. Reult otaed ug the propoed ehle ethod are copared wth thoe reported the lterature to deotrate the flexlty ad accuracy of the ethod.. troducto he coupled radatve ad coductve heat trafer prole e-traparet eda at hgh teperature are ay egeerg applcato uch a uclear egeerg olar eergy collector coolg of electroc copoet checal reactor ad o o []. Mot tude o coupled radatve ad coductve heat trafer prole have ee lted to ufor refractve dex eda ad eployed fte eleet [34] ad cotrol volue aed fte eleet ethod [5]. he the refractve dex of a e-traparet edu vare cotuouly the curved ray path detered y the Ferat prcple. A a reult the radatve eergy feld de the edu odfed. Becaue of the coplexty of olvg radatve trafer refractve dex edu the coupled radatve ad coductve heat trafer prole the refractve dex edu receved relatvely le atteto. he curved ray tracg techque [6-8] peudo-ource addg ethod coato wth the curved ray tracg techque [9] or ray-tracg/odal-aalyzg ethod [-] are aly eployed for aalyzg the radatve heat trafer ad tradtoal ethod uch a fte dfferece ethod or cotrol volue are ued for olvg eergy equato. Becaue of a large uer of ray to e lauched the ray tracg ethod coplex te coug ad dffcult to ettle. Mhra et al. [3] ued Lattce Boltza ethod (LBM) to olve the eergy equato ad dcrete trafer ethod (DM) to copute the radatve trafer. Zhu et al. [4] eployed P ethod to olve radatve trafer ad pectral ethod to pleet the patal dcretzato. hle DM or P ethod ot eay to e exteded to ultdeoal geoetre. order to overcoe thee dffculte oe reearcher propoed the dcrete ordate ethod (DOM) for the dcretzato of agular pace of radatve trafer equato (RE) refractve dex eda. he DOM frtly propoed y Leoer et al. [5] for D radatve heat trafer the la. Afterward aother kd of DOM wa derved y Lu et al. 3D Cartea [6] ad cyldrcal [7] coordate yte o who ay correpodece hould e addreed Cotet fro th work ay e ued uder the ter of the Creatve Coo Attruto 3. lcece. Ay further dtruto of th work ut ata attruto to the author() ad the ttle of the work oural ctato ad DO. Pulhed uder lcece y OP Pulhg Ltd

3 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 repectvely. Baed o the techque of DOM fte volue ethod (FVM) [6] fte eleet ethod (FEM) [8] ad pectral collocato ethod [9] have ee ued for the patal dcretzato. th artcle we eploy ehle ethod to olve radatve ad coductve heat trafer refractve dex eda. Dfferet fro tradtoal ethod lke FVM or FEM the ehle ethod ue a et of ode cattered the coputatoal patal doa wthout the eed for forato o the relatohp etwee the whch provde ore flexlty coplex geoetre []. o our et kowledge oly Lu [] ha ued a ehle local Petrov Galerk (MLPG) approach for olvg the coupled radatve (aed o DOM) ad coductve heat trafer a oe-deoal refractve dex la. But uercal tegrato eceary th approach whch creae the CPU te geerally. th paper we exted the ue of DAM propoed y ayrole [] whch ha ee ued uccefully for radatve heat trafer refractve dex eda [3] ad radato coupled wth coducto a hoogeou eda [4]. he artcle orgazed a followg. he detal of the theoretcal ad uercal calculato procedure are gve frt. o evaluate the accuracy ad flexlty of DAM D ad D tet cae are the codered. he reult are dcued ad copared wth other avalale lterature oluto.. he dcrete ordate equato For a aorg-ettg-catterg edu wth a varyg refractve dex the RE ter of the dcrete coordate ca e wrtte a [5] : d ( ) σ β( ) κ ( ʹ ) Φ( ʹ ) d( ʹ ) () d π 4 where the refractve dex; κ σ ad β are the coeffcet of aorpto catterg ad extcto repectvely; Φ ʹ the catterg phae fucto of the edu fro drecto ʹ to drecto. he gle-catterg aledo defed a ω σ β. he oudary codto for dffue wall ca e wrtte a ε ʹ ˆ ʹ d ʹ ˆ > () d d the lackody tety of the edu; ad ( ) ( ) ( ) ( ) ˆ ʹ < ε w π ( ) σ β( ) κ ( ʹ ) Φ( ʹ ) ( ʹ ) d ˆ (3) π 4 the lackody tety of the wall ad ˆ the ut ward where ε the wall evty w oral vector at the oudary locato. a Cartea coordate yte the Eq. () ca e wrtte t dvergece for a decred [6] : d ( ) where d β σ ( ) ξ k ( ) ( ) κ ( ʹ ) Φ( ʹ ) d( ʹ ) µ η ξ k co co k co (6) Ug a pecewe cotat agular (PCA) quadrature the total old agle dvded uforly polar ad azuthal drecto. he dcrete polar ad azuthal agle ca e expreed a follow : ( ) Δ (4) (5) (7)

4 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 where Δ π π Δ. ( ) Δ For each dcrete ordate the correpodg weght for ad (8) d co co (9) d () repectvely where ( ) ( ). Replacg the tegral ter y the dcrete ordate yeld to the dcrete for [6] : d ( ) ( ) ξ k ( ) d σ ( ) κ ( ) Φ( ) () β By dcretzg the dervatve wth repect to the polar ad azuthal agle we ota the dcrete ordate equato of the RE a 3D etraparet refractve dex eda [6] : d ( ) [ ax( ) ax( ) ax( ) d where ax ax ax ( ) β ] ( ) ax( ) ( ) ( ) ( ) ax( ) ( ) σ ( ) ( ) κ ( ) Φ( ) [ co ( αʹ co βʹ ) γ ] ʹ () (3) (4) αʹ co β ʹ β ʹ ( ) (6) the prevou equato αʹ β ʹ ad γ ʹ correpod to the copoet of vector ( ʹ β ʹ γ ʹ ) α grad. he oudary codto dcrete for are ε ( ) ε ( ) ˆ w ˆ > (7) π ˆ < he cdet radato gve y (5) 3

5 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 where he radatve heat flux q q G rx q ry rz ( ) d ( ) q r (8)!!! q q q k (9) rx ry ( ) d µ ( ) µ η d ξ d ( ) η ( ) ( ) ξ ( ) 3. Eergy equato he teady tate eergy equato for coupled radatve ad coductve heat trafer ca e wrtte a k Δ qr (3) For a refractve dex eda the dvergece of the radatve flux gve y 4 q r κ ( 4 σ G) (4) 3 where G the cdet radato. e defe the coducto-radato paraeter kβ σ. where he coductve heat flux q c rz 4 ref () () ()!!! q q q k (5) cx cy cz d q cx k dx (6) d q cy k dy (7) d q cz k dz (8) 4. Dffue approxato ehle ethod Coderg a ukow calar u defed a D feld at each odal pot the calculato doa the ecod order aylor expao of u at the pot tuated the vcty of etated y [6] : etated u ( ) ( ) α where P( ) a vector of polyoal a fucto defed a P( ) ( x x) ( y y) ( x x) ( x x)( y y) ( y y) ad α( ) a vector of coeffcet gve y : P (9) (3) 4

6 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 u u u u u x y! x xy! y whch are detered y zg the weghted dcrete L or defed a whch ( ) α α α 3 α 4 α 5 α 6 u (3) [ ] ( α ) ω ( ) u P( ) α( ) ω the ovg leat-quare weght fucto. t value decreae wth creag dtace etwee odal pot of doa of fluece. he zato of Eq. (3) gve: A ( ) ( ) B( ) where A (3) ad fro uty at odal pot to zero outde a gve α (33) ( ) ( ) P( ) P ( ) ω (34) ( ) ( ) P( ) u B ω (35) By vertg Eq. (33) the copoet of ( ) α ca e etated y ug the forato of eghorg odal pot : ( ) u u u u u * u [ A( )] ω ( ) P( ) u (36) x y! x xy! y th work the Gaua wdow: ω ( ) exp 3l σ f (37) ω ha ee ued the calculato. ( ) f > σ f defe the dtace etwee odal pot of ad. σ the dtace of fluece whch ued to elect a certa uer of eghorg odal pot f ( ) order to preerve a local character of the approxato ad alo to eure the atrx A ( ) ot gular. A tal value σ et frt. he the dtace f calculated. For D prole f the uer of odal pot that have a dtace of < σ greater tha 5 or aller tha 9 f ut the allowed rage. At each odal pot the coputatoal patal doa the dervatve appearg the equato (goverg equato or oudary codto) to e olved are replaced y ther dffue approxato Eq. (36). A a reult a algerac yte geerated fally ad ca e olved y ug the BCGSAB teratve ethod. σ decreaed or creaed utl ( ) 5. Dcretzato chee for goverg equato ad oudary codto M the followg the th le of the vere atrx [ A ] defed a a. 5.. Oe deoal prole f 5

7 For D prole the dcrete ordate equato of RE () ad oudary codto (3) (wth ˆ ) ca e wrtte a : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Φ σ κ β µ x ax ax ] ax ax [ (38) where µ co the drecto coe related to the x -ax. At the ae te Eq. (36) trafored to : ( ) [ ] ( ) ( ) ( ) u P A x u x u u! ω (39) wth ( ) ( ) ( ) x x x x P (4) By ug the dffue approxato Eq. (39) at each odal pot the followg algerac yte for the radatve (RAD) ad coductve (COD) equato are otaed repectvely: [ ] [ ] [ ] S R RAD (4) [ ] [ ] [ ] S C COD (4) where we have: ( ){ } ( ) ( ) ( ) P a RAD δ β µ ω ] ax ax [ ) ( (43) where δ Krocker yol. ( ) ( ){ } ( ) P a k COD! 3 ω (44) ad ( ) ( ) ( ) ( ) ( ) ( ) ( ) Φ σ κ R S ax ax (45) ( ) ( ) G S C 4 4 σ κ (46) 5.. wo deoal prole th cae the dcrete ordate equato of RE () ad oudary codto (3) (wth ˆ ) ca e wrtte: Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 6

8 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 µ σ κ where µ co ad η ( ) ( ) η [ ax( ) ax( ) x ax ax ax y ( ) ax( ) β ] ( ) ( ) ( ) ax( ) ( ) ( ) ( ) ax( ) ( ) ( ) Φ( ) (47) are the two copoet of the drecto vector. he coeffcet of the algerac yte for RAD ad COD equato are wrtte a: RAD( ) ω ( ){ µ a η a3 } P( ) [ ax( ) ad ax ( ) ax( ) ax( ) β ] δ S R COD ( ) ( ) [! k( a a )] 4! 6 (48) ω p (49) ( ) ax( ) ( ) ax( ) ( ) κ ax ( ) ( ) ax( ) ( ) σ ( ) Φ( ) 4 ( ) ( 4 σ G) S C (5) κ (5) 5.3. Boudary codto For odal pot lyg o the oudary wth ˆ > the followg coeffcet for Eq. (7) D ad D are otaed repectvely: RAD( ) δ (5) S R ( ) w ( ε ) ( ) ε ˆ D (53) ˆ < ε ( ) w ( ) S ˆ R ε D (54) π ˆ < he Drchlet oudary codto for eergy equato are troduced the ae way a Eq. (7) teratve proce he followg teratve proce choe to olve the o lear coupled yte. 7

9 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4. A fte uer of odal pot the doa ad o the oudare are choe frt. he the dervatve of all the pot are replaced y ther dffue approxato.. tal teperature feld et. 3. he dcrete ordate equato of RE olved. 4. he the eergy equato olved. 5. erate the terato proce f the covergece crtero atfed. Otherwe go ack to tep 3. ew old 5 he covergece crtero ued for the uercal calculato defed a. 6. Reult o facltate coparo of the reult otaed wth DAM ad thoe otaed the lterature the relatve error defed a : u ref ( ) ref ( ) E % (55) r 6.. Plae parallel la h cae coder a aorg ad ettg edu cotaed the plae parallel la wth a thcke L.. he teperature of left ad rght wall are K ad L 5 K repectvely. ew ax Fgure. eperature dtruto acro the la for dfferet lear refractve dex ε ε L he teperature dtruto acro the la are calculated y ug a dcretzato of odal pot ad 4 quadrature. he coeffcet of aorpto κ. - (optcal thcke τ ) ad the theral coductvty k. - K -. Frtly we coder the cae of lack urface ( ε ε L ) wth dfferet lear refractve dex aely ( x)..6 x / L ad ( x).8-.6 x / L. he fluece of refractve dex ( x) o the teperature dtruto how Fgure. t ca e ee that whe the gradet of refractve dex potve correpod to ( x)..6 x / L the teperature gloally hgher. Further the fluece of evty o the teperature dtruto x π x L.he the evte preeted Fgure for a uodal refractve dex ( ).8-.6 ( ) 8

10 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 of the two wall are ot equal the dtruto of teperature are greatly chaged coparg wth the tuato of two lack wall. he axu relatve error for the teperature dtruto.57%. Fgure. eperature dtruto acro the la for dfferet x π x L evty ( ).8-.6 ( ) Fgure 3. Radatve heat flux dtruto acro the la for dfferet lear refractve dex ε. ε. he radatve heat flux dtruto acro the la are calculated for a dcretzato of 5 odal pot ad 8 quadrature. he coeffcet of aorpto κ. - (optcal thcke τ ) ad the theral coductvty k. - K -. Frtly we coder lear refractve dex wth the ae expreo ( x)..6 x / L ad ( x).8-.6 x / L a efore ut wth dfferet evte ε. ad ε L. o the two wall. he fluece of refractve dex ( x) o the teperature dtruto how Fgure 3. ear the left wall wth ε. oce aga the radatve heat flux uch hgher for refractve dex whoe gradet potve; ut the pheoeo o the cotrary ear the rght wall. ext we coder the cae of lack urface wth dfferet uodal refractve L 9

11 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 dex aely ( x).8-.6 ( π x L) ad ( x)..6 ( π x L). he radatve heat flux how Fgure 4 ad preet a kd of partcular yetry fucto of refractve dex. he axu relatve error for the radatve heat flux.973%. All the reult have a good agreeet wth thoe y Lu [] who eployed MLPG approach. Fgure 4. Radatve heat flux dtruto acro the la for dfferet uodal refractve dex ε ε L c.8.6 h c.4. c Fgure 5. Square ecloure ad odal pot 6.. Square e coder a aorg ad ettg edu cotaed a quare a how Fgure 5. All the wall are lack. he left oe hot at teperature h ad the other three wall are cold at teperature c uch that h c.5. he aorpto coeffcet ad coducto-radato paraeter are et to κ. - ad. repectvely. e ue a dcretzato of odal pot ad 4 quadrature. he teperature dtruto refractve dex eda wth lear varato alog the two drecto x ad y are how Fgure 6 for a gradet value of ad repectvely. e ca fd that the refractve dex ( x y) coduct the axu teperature dtruto ad the

12 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 teperature decreae a the gradet value of refractve dex creae. e ca ee the reult otaed y DAM agree well wth thoe referece [8] y a fte eleet ethod wth a axu error of.854%. Fgure 6. Dtruto of the teperature alog ceterle y.5 for dfferet lear refractve dex (κ. - ε.) 6.3. Secrcular ecloure A ecrcular ecloure wth er crcle how Fgure 7 ad all the wall are lack. he ae patal dcretzato of 44 odal pot ued for the patal dcretzato. For the agular dcretzato dfferet fro forer cae whch S ethod ued here we ue a PCA (pecewe cotat agular quadrature) ethod ad a uer of dcrete drecto M M 4 ued. he edu aorg ettg ad catterg. he teperature of the outde ecrcle ad er crcle are K ad 5K repectvely. e ue the xed oudary codto o the otto a [7]: where h k ( )( ˆ ) h( ) εσ ( ) k w f (56) ˆ w f 3 K. y R c. R..4 x Fgure 7. Secrcular ecloure wth er crcle ad the dtruto of odal pot

13 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 Fgure 8. Effect of varou refractve dex o the deole coductve heat flux Ψ alog the otto wall c Fgure 9. Effect of varou refractve dex o the deole radatve heat flux Ψ alog the otto wall r For β ad ω.5 the deole radatve heat flux Ψ r qr σ ad coductve heat flux Ψ q σ for dfferet kd of refractve dex ( x y) [ ( x y ) R ]. 5.5 ad ( x y ) R c c 4 are how Fgure 8 ad Fgure 9. For ( x) our reult are copared wth thoe reported referece [78]. Although the relatve error E r gfcat ad equal to % ad.5% for radatve ad coductve heat flux repectvely the tred lar. Bede there are ot avalale reult other referece for the prole. For all kd of refractve dex the deole coductve heat flux Ψ c dtruto preet a yetrcal charactertc whch reach the peak the ceter of the otto wall ut the trough aroud the ceter. he tred o the cotrary for deole radatve heat flux Ψ. he peak ad the trough ecoe ovou for refractve dex ( x y ) R whch creae wth creag dtace fro the ceter of outer crcle. ale the CPU te eceary for covergece alo.5 r

14 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 gve for varou refractve dex. All the reult have ee otaed o a peroal coputer (3. GHz Proceor 8 GB RAM wthout parallelzato). ale. e eceary for dfferet refractve dex ( x y) te 57 ad 43 ec.5 ( x y ) R [ ( x y ) R ]. 5 6 ad 58 ec 9 ad 58 ec 7. Cocluo A dffue approxato ehle ethod (DAM) aed o the DOM of RE eployed for olvg radatve-coductve heat trafer refractve dex eda cotaed the D ad D geoetre. he reult are copared wth other echark reult ad how that DAM ha a good accuracy olvg radatve-coductve heat trafer aorg ettg ad catterg eda wth varato of refractve dex ultdeoal geoetre. he fluece of the refractve dex are vetgated o the teperature ad heat flux dtruto. he future work wll coder to exted the cae to coplex 3D geoetre ad alo to prove the calculato effcecy. 8. Referece [] Bouta L ad Beaïh R Appl. her. Eg [] ag F Q Shua Y a H P ad Yu C L 3 t. J. Heat Ma ra [3] Saka M Charette A ad Le Dez V 996 J. Quat. Spectroc. & Rada. ra [4] Rua L M e M Q H A ad a H P 6 J. Quat. Spectroc. & Rada. ra. 9 [5] alukdar P edorff F V r D ad Soo C J 8 Heat Ma ra [6] Be Adallah P ad Le Dez V J. Quat. Spectroc. & Rada. ra [7] Lu L H ad a H P 4 J. Quat. Spectroc. & Rada. ra [8] Lu L H 6 J. Quat. Spectroc. & Rada. ra [9] a L Huag Y a H P ad Zhag B t. J. Heat Ma ra [] Y H L a H P Luo J F ad Dog S K 5 J. Quat. Spectroc. & Rada. ra [] Y H L a H P Zhag H C ad Luo J F 6 uer. Heat ra. A [] Y H L e M ad a H P 8 t. J. Heat Ma ra [3] Mhra S C Krha A Gupta ad Chataya G R 8 t. J. Heat Ma ra [4] Zhu K Y Huag Y ad ag J 3 t. J. Heat Ma ra. 6-4 [5] Leoer D ad Le Dez V J. Quat. Spectroc. & Rada. ra [6] Lu L H 6 J. herophy. Heat ra [7] Lu L H Zhag L ad a H P 6 J. Quat. Spectroc. & Rada. ra [8] Y H L Zhe B He K a H P ad Zhou Y Kug Cheg Je u L Hueh Pao/J. Eg. herophy [9] Su Y S ad L B t. J. Heat Ma ra [] Zhag Y Y H L ad a H P 3 t. J. Heat Ma ra [] Lu L H a J Y ad L B 6 J. Quat. Spectroc. & Rada. ra [] ayrole B ouzot G ad Vllo P 99 C. R. Acad. Sc. Par [3] ag C A Sadat H ad Le Dez V Appl. Math. Model [4] Sadat H ag C A ad Le Dez V Appl.Math. Coput

15 Eurother Coferece 5: Coputatoal heral Radato Partcpatg Meda V OP Pulhg Joural of Phyc: Coferece Sere 676 (6) 4 do:.88/ /676//4 [5] Preedorfer R 965 teratoal Sere of Moograph o pure ad appled Matheatc Pergao Pre [6] Prax C Sadat H ad Daoura E 7 Appl. Math. Coput [7] Zhag Y Y H L ad a H P 3 J. Coput. Phy [8] ag C A Sadat H ad a J Y 4 uer. Heat ra. B Ackowledget h work wa upported y atural cetfc reearch ovato foudato Har ttute of echology (H.SRF. H.SRF.43) cetfc reearch foudato of Har ttute of echology at eha (H(H)3) ad Progra for ew Cetury Excellet alet Uverty (CE--5). 4