ON THE CONVERGENCE OF THE DISCRETE ORDINATES AND FINITE VOLUME METHODS FOR THE SOLUTION OF THE RADIATIVE TRANSFER EQUATION. P. J.

Transcription

1 ON THE CONVERGENCE OF THE DSCRETE ORDNATES AND FNTE VOLUME METHODS FOR THE SOLUTON OF THE RADATVE TRANSFER EQUATON P. J. Coeho Mechaica Egieeig Depatet tituto Supeio Técico Techica Uiveity of Libo Av. Rovico Pai Liboa Potuga

2 NTRODUCTON Thea adiatio pay a key oe i ay eevat egieeig pobe (atopheic phyic oa eegy coveio yte idutia heatig cooig ad dyig pocee fie cobutio yte) Radiative heat tafe i paticipatig edia i goveed by a itegodiffeetia equatio ko a the adiative tafe equatio (RTE). Aog the ueica ethod ued to ove thi equatio the dicete odiate ethod (DOM) ad the fiite voue ethod (FVM) ae pobaby the ot idey ued oe. cotat ith fuid fo pobe goveed by the Navie-Stoke equatio hee oy the patia dicetizatio i eeded the RTE equie both a patia ad a agua dicetizatio. The pupoe of the peet ok i to ivetigate the covegece of the DOM ad FVM o to tet pobe hee a aaytica outio of the RTE i avaiabe.

3 RADATVE TRANSFER EQUATON Radiative Tafe Equatio (RTE) d η d σ η κη η + κη bη σ η η + ( ) ( ) i Φ i dωi 14 4 π η π 4444 η (1) Deceae due to aboptio (2) ceae due to eiio (3) Deceae due to catteig (4) ceae due to catteig i

4 RADATVE TRANSFER EQUATON Bouday coditio fo adiative tafe equatio (Diffuey eittig ad efectig opaque uface) ( ) ε ( ) ( ) ( ) ρ + b. π. < 0 ( ) dω cidet adiatio: G π η dω dη 0 4 Radiative heat fux vecto q dω π η dη 0 4

5 DSCRETE ORDNATES METHOD FUNDAMENTALS The dicete odiate ethod (Chadaekha Radiative Tafe 1950 Cao e Lathop i Coputig Method i Reacto Phyic Geepa Kebe e Oket ed. Godo & Beach 1968 Fivead J. Heat Tafe Vo. 106 pp ) DOM tafo the equatio of adiative tafe ito a et of iutaeou patia diffeetia equatio The tota oid age age of 4π aoud a poit i pace i dicetized ito a et of oid age ad the adiatio iteity i aued to be cotat ove each dicete oid age Each patia diffeetia equatio decibe the vaiatio of the adiatio iteity i pace aog the diectio aociated ith a dicete oid age tega ove oid age ae appoxiated by ueica quadatue

6 DSCRETE ORDNATES METHOD FUNDAMENTALS Geea equatio of adiative tafe fo a eittig abobig ad catteig ediu: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Ω Φ + + d d d b 4 4 π π σ κ β Dicete odiate equatio: Thee equatio ae ao vaid o a pecta bai fo a ogey ediu ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) b L Φ + + π σ κ β k j i µ η ξ + + Dicete odiate equatio: hee i the quadatue eight aociated ith diectio :

7 DSCRETE ORDNATES METHOD FUNDAMENTALS Radiative heat fux i the ediu: ( ) ( ) 1 q ( ) ( ) G 1 Net adiative heat fux at a uface: cidet adiatio: ( ) ( ) ( ) ( ) < b q 0 π ε Net adiative heat fux at a uface: Divegece of adiative heat fux (ouce te of eegy equatio): b 1 4π κ q

8 DSCRETE ORDNATES METHOD Spatia dicetizatio ( ) ( ) ( ) ( ) ( ) k A k da dv dv k k A V V P ( ) ( ) ( ) ( ) ( ) ( ) b P b P P P V dv V κ β κ β ( ) ( ) ( ) ( ) Φ Ω Φ P P V V d π σ π σ π ( ) ( ) ( ) Φ + P P P P P b P k V V k A k k 1 4 π σ β κ

9 TEST CASE 1 To-dieioa quae ecoue ith back a Eittig-abobig ediu uity eiive poe A a ae back ad cod uifo gid S 8 quadatue Soutio accuacy evauated uig the L 1 o of the eative eo of the icidet adiatio The eo i deteied ith efeece to the aaytica outio of the dicete odiate equatio athe tha the outio of the RTE Thi ao the actua evauatio of the patia dicetizatio eo egade of the agua dicetizatio eo

10 TEST CASE 1 E G 1 N N G i G G i 1 i ef i ef Aveage eative eo of the icidet adiatio

11 TEST CASE 2 To-dieioa quae ecoue ith back a Eittig-abobig ediu uity eiive poe A a ae back; top a i hot othe ae cod uifo gid S 8 quadatue Soutio accuacy evauated uig the L 1 o of the eative eo of the icidet adiatio The eo i deteied ith efeece to the aaytica outio of the dicete odiate equatio athe tha the outio of the RTE Thi ao the actua evauatio of the patia dicetizatio eo egade of the agua dicetizatio eo

12 TEST CASE 2 Aveage eative eo of the icidet adiatio

13 TEST CASE 2 cidet heat fux o botto bouday (Quai-exact outio: Cobie ad Scheke JQSRT Vo. 31 pp ) S 8 S 16 N θ N φ 100

14 TEST CASE 2 Fae catteig (ao efeed to a fae diffuio o ueica eaig) i a eo that aie fo the patia dicetizatio he the diectio of popagatio of adiatio ake a age ith the gid ie Fae catteig ay be educed by uig a fie gid ad/o high ode patia dicetizatio chee Ray effect i a eo that aie fo the agua dicetizatio hich appoxiate a cotiuouy vayig fuctio (the adiatio iteity) by a tepie oe i.e. the adiatio iteity fied at a give poit i pace i epeeted by the adiatio iteitie at a fiite ube of diectio The ay effect i idepedet of the patia dicetizatio i.e. it eai eve if o patia dicetizatio eo i peet The ay effect ay be educed by uig a fie agua dicetizatio

15 TEST CASE 2 Agua dicetizatio Spatia dicetizatio Exact RTE Dicete odiate diffeetia equatio Dicete odiate agebaic equatio Ray effect Fae diffuio Fae catteig (fae diffuio ueica catteig ueica eaig) eo ted to ooth the adiatio iteity fied Ray effect eo ted to ehace dicotiuitie o gadiet of the adiatio iteity fied Fae catteig ad ay effect ted to copeate each othe

16 TEST CASE 2 Gid efieet ad/o oe accuate patia dicetizatio chee educe fae catteig but have o ifuece o ay effect copeatio effect diappea Agua efieet educe ay effect but ha o ifuece o fae catteig copeatio effect diappea both cae outio eo of DOM i ae copaed to exact outio of DOE but ay be age copaed to exact outio of RTE Both gid efieet (o oe accuate patia dicetizatio chee) ad agua efieet eed to caied out iutaeouy. Ateativey odify the tadad DOM.

17 CONCLUSONS The outio of the RTE fo gey edia uig the DOM (o FVM) equie both patia ad agua dicetizatio. Spatia ad agua dicetizatio eo ted to copeate each othe f oy patia efieet i caied out the ueica outio appoache the aaytica outio of the dicete equatio but doe ot covege to the aaytica outio of the RTE. Thi ea that the outio accuacy i ot ipoved ad ay eve get ot ith patia gid efieet. Both patia ad agua efieet ae eeded to eue that the ueica outio covege to the aaytica outio of the RTE. f the tepeatue of the bouday i dicotiuou the adiatio iteity fied i ao dicotiuou ad the ode of accuacy i oe tha the foa ode of accuacy of the dicetizatio chee.