LOW ORDER POLYNOMIAL EXPANSION NODAL METHOD FOR A DeCART AXIAL SOLUTION

Size: px

Start display at page:

Download "LOW ORDER POLYNOMIAL EXPANSION NODAL METHOD FOR A DeCART AXIAL SOLUTION"

Gabriel Anthony
5 years ago
Views:

1 9 Inenaona Nucea Aanc Confeence - INAC 9 Ro de JaneoRJ az epembe7 o Ocobe 9 AOCIAÇÃO RAILEIRA DE ENERGIA NUCLEAR - AEN IN: LOW ORDER POLYNOMIAL EXPANION NODAL METHOD FOR A DeCART AXIAL OLUTION Jn-Young Cho Jae-eung ong and Hyun-Chu Lee Koea Aomc Enegy Reeach Inue 5 Deokjn-dong Yueong-gu Daejon 5-5 Koea jyoung@kae.e.k Koea Aomc Enegy Reeach Inue 5 Deokjn-dong Yueong-gu Daejon 5-5 Koea jong@kae.e.k Koea Aomc Enegy Reeach Inue 5 Deokjn-dong Yueong-gu Daejon 5-5 Koea hc@kae.e.k ATRACT Th pape appe a ow ode poynoma expanon noda (LPEN) mehod o he mpfed Pn (Pn) anpo equaon fo he DeCART (Deemnc Coe Anay baed on Ray Tacng) axa ouon. Fo a pefomance examnaon of he mpemened LPWN mehod he C5G7MOC -D benchmak and he C5G7 hexagona vaaon pobem ae oved. LPEN mehod expand he angua fux momen by ung up o he econd ode Legende poynoma. In he C5G7MOC -D benchmak DeCART how e han 4 pcm egenvaue and e % -D pn powe dbuon eo whch ae ma a he hghe ode noda mehod uch a NEM o ANM. In C5G7 hexagona vaaon pobem DeCART how he ma eo eve a n he C5G7MOX benchmak. Ao LPEN mehod eque e compung me and e memoy ze han he hghe ode noda mehod. Theefoe concuded ha he LPEN mehod moe appcabe o he DeCART ce baed axa ove han he hghe ode noda mehod n apec of accuacy and pefomance.. INTRODUCTION DeCART (Deemnc Coe Anay baed on Ray Tacng) [] a whoe-coe anpo code whch poduce ub-pn eve fux dbuon by ovng he anvee eakage couped ada and axa equaon [4]. The ada ouon of he DeCART code obaned by pefomng he pana ay acng cacuaon fo he MOC anpo equaon by ung axa anvee eakage whou paa homogenzaon. The axa ouon obaned by ovng he P N anpo equaon fo a homogenzed ce by ung ada anvee eakage. The cuen DeCART veon ove he axa mpfed Pn (P N ) equaon by noducng he hghe ode noda mehod uch a NEM [5] and ANM [6]. Ao DeCART noduce a ub-pane cheme whch dvde a hck MOC pane no evea ub-pane o educe he noda effec of he axa kene. The ada anvee eakage appoxmaon and he axa ouon ae cacuaed baed on hee ub-pane. The hghe ode noda mehod uch a NEM and ANM poduce a vey accuae ouon wh a eavey age meh ze by noducng he hghe ode poynoma o he anayc funcon. Howeve fo an effcen cacuaon hoe mehod pe-cacuae and oe he

2 coupng coeffcen befoe he man egenvaue cacuaon. Ao a hown n Fg. n cona o he mooh hape n he aemby baed noda cacuaon he axa anvee eakage n he ce baed cacuaon vae hapy a he odded pane bounday. Theefoe n he DeCART code he hghe ode noda mehod keep a ma meh ze by noducng he ub-pane cheme o ea he anvee eakage hape accuaey whch oe he me of he age meh ze of hee mehod. In h pape a owe ode poynoma expanon noda (LPEN) mehod noduced baed on he ub-pane cheme. The LPEN mehod appoxmae he neuon fux by he econd-ode poynoma and ue fa anvee eakage appoxmaon. Though LPEN eque moe numbe of ub-pane han he hghe ode noda mehod doe no eque pe-cacuaon of he coupng coeffcen due o mpcy. Theefoe h mehod eove boh he accuacy and he memoy pobem. In he nex econ he LPEN mehod decbed accuaey fo he P N equaon. In he foowng econ he C5G7MOX -D benchmak [78] and he C5G7 hexagona vaaon pobem [] ae oved and he accuacy and he pefomance of he LPEN mehod ae dcued. Rada Tanvee Leakage Unodded Regon Rodded Regon Refeco Regon Fne Meh (.484 cm) Noda Meh (.7 cm) Eevaon cm Fgue. Axa hape of Rada Tanvee Leakage. Low Ode Poynoma Expanon Noda Mehod fo P Tanpo Equaon In he LPEN mehod he one-dmenona angua fux momen ae expanded by ung he Legende poynoma a: ( u) ˆ φ ( u) φ ( u) + φ ( u a P () ) INAC 9 Ro de Janeo RJ az.

3 ( u) ˆ φ ( u) φ ( u c P ) whee φ and φ ae he zeo-h and he econd momen fo he angua fux. P (u) he Legende poynoma and a and c ae he expanon coeffcen. The expanon coeffcen of Eq. () can be obaned by ung he noda aveaged and he uface aveaged angua momen a: a φˆ c φˆ ( ˆ ˆ φ φ ) ( ˆ φ φˆ ) a c c ˆ ˆ ˆ φ + φ φ a ( ˆ φ ˆ + φ ) ˆ φ ( ) () whee ˆ φ ˆ φ ˆ φ and ˆ φ ˆ φ ˆ φ ae he node aveaged vaue and he vaue a he gh and ef uface fo he zeo-h and he econd angua fux momen epecvey. The cuen equaon a he gh uface can be expeed n em of he node aveaged momen and he uface momen a: J ˆ + ˆ 6 ˆ β 4φ φ φ J ˆ + ˆ 6 ˆ β 4φ φ φ () whee J and J ae he f and he hd angua momen a he gh uface and β and β ae defned a: D β D and D. h Σ 7 Σ z The noda neuon baance equaon fo he even ode angua momen can be wen a: ( Σ + γ ) ˆ φ Σ ˆ φ + 6γ ( ˆ φ ˆ + φ ) Σ ˆ φ + ( Σ + γ ) ˆ φ + 6γ ( ˆ φ + ˆ φ ) (4) whee Σ Σ + + h N ad ~ ( D Dˆ ) h N ad ~ ( D + Dˆ ) N ad ~ Σ Σ + D h N ad ˆ φ + h β γ hz ~ D ˆ φ INAC 9 Ro de Janeo RJ az.

4 ~ ~ and D and D ae he cuen coupng coeffcen n he odnay fne dffeence mehod ˆ he cuen coecve coeffcen n he coae meh fne dffeence mehod. D Fom he noda baance equaon of Eq. (4) he node aveaged even ode angua momen can be expeed n he max fom a: ˆ φ ˆ φ ˆ + ˆ. (5) φ φ 8 ˆ φ + ˆ φ y neng Eq. (5) o Eq. () he cuen equaon can be wen n he max fom a: J ˆ ˆ (6) Φ + Φ + whee J ˆ J Φˆ φ ˆ Φˆ φ J ˆ φ ˆ φ b b b8 b b b 6 b8 b b4 b5 7 b9 b ( ) b β 6 4 b β ( 6 ) b 6β4 b4 6β b5 6β b 6 β ( 6 8 4) b 7 β ( 6 8 ) b8 6β 7 b9 6β 5 b 6β 6. y ung Eq. (6) he cuen connuy condon a he neface beween node k and k+ can be wen n em of he uface fuxe and he ouce a: k k+ k k+ k+ k k+ ( + ) Φˆ + Φ Φˆ + ˆ. (7) k k In he P appoxmaon he cuen a he pobem bounday can be expeed n em of he uface fux and he ncomng paa cuen a: J αφ ˆ (8) jn whee and J and j 4 jn α j 8 7 J ae he f and he hd ncomng angua momen a he gh uface y ung he wo expeon of Eq. (6) and (8) he cuen connuy condon a he pobem bounday can be wen a: INAC 9 Ro de Janeo RJ az.

5 k k k k ( αφ ) ˆ + ˆ. (9) k k Φ jn The yem equaon fo a pn can be eabhed by coecng a he connuy condon of Eq. (7) and (9) a: whee α + + O O O K K K K + b K K α + K K Φˆ Φˆ Φˆ Φˆ L Φˆ [ ] T Φ ˆ () K K K K [ ] T jn L and K he oa numbe of node n he pobem doman. The yem equaon of Eq. () ncudng he hee bock dagona max of can be oved eay by appyng he bock LU facozaon cheme. If he uface fuxe ae obaned fom Eq. () he node aveaged fuxe of a he node can be obaned by ung Eq. (5). The ougong paa cuen a he pobem bounday whch ued a an ncomng paa cuen of he nex axa pane can be obaned a: jn J Ko αφ ˆ J. () Ko K In he LPEN mehod he coupng coeffcen ae no oed becaue hey can be poduced eay. Ao he anvee eakage hape no mpoan becaue he LPEN mehod ue he fa anvee eakage cheme fo a hn pane ze. Fuhemoe ony he node aveaged fuxe and he ougong paa cuen ae eaed a unknown. Theefoe he LPEN mehod eque vey ma compue memoy compaed o he NEM and ANM. Howeve he LPEN mehod eque abou hee me moe axa node o acheve he accuacy of he NEM and ANM.. Examnaon of he LPEN ouon To examne he ouon accuacy and he compuaona pefomance of he LPEN mehod he C5G7MOX -D benchmak fo ecangua coe and C5G7 hexagona vaaon benchmak fo hexagona coe ae oved. Theee -D benchmak con of pobem of one unodded and wo odded confguaon. The efeence ouon fo he ecangua and hexagona geomee ae gven by MCNP and he McCARD [9] code epecvey. Fg. how he DeCART ce mode fo C5G7MOX enchmak and fo he C5G7 hexagona vaaon pobem. The fue ce f dvded no 8 unfom co econ ng (5 fo fue and fo cooan egon) and hen each unfom co econ ng azmuhay INAC 9 Ro de Janeo RJ az.

6 dvded no 8 o 6 fa ouce egon fo ecangua and hexagona pobem. The efeco ce dvded no oa 8 and 5 fa ouce egon fo ecangua and hexagona pobem. The gap ce exng a he bounday of he hexagona aemby ae f dvded no 4 fa ouce egon and hen chopped by he aemby bounday ne. In ovng he benchmak pobem 4 azmuha ange fo 9± and 6± eco fo he ecangua and he hexagona geomey epecvey and opmzed poa ange and.5 cm ay pacng ae ued. In he coe cacuaon he 45± aemby edge ymmey opon fo he ecangua coe and he ± aemby cene ymmey opon fo he hexagona coe ae ued. (a) Fue Ce Mode (b) Refeco Ce Mode (c) Fue Ce Mode (d) Refeco Ce Mode (e) Aemby Gap Mode Fgue. DeCART Ce Mode fo C5G7MOX and Hexagona Vaaon Pobem. DeCART ove he -D pobem by empoyng he CMFD fomuaon baed on he homogenzed ce and MOC pane. The ada and he axa cuen coecve em of he -D CMFD fomuaon ae uppoed by he pana MOC and he LPEN kene whch pefom he pana MOC cacuaon and he cewe ub-pane cacuaon epecvey. In he MOC cacuaon he axa anvee eakage geneaed by he -D CMFD cacuaon ued. In he axa kene he ada CMFD and axa LPEN couped cacuaon pefomed baed on he homogenzed ce. The cuen geneaed n he MOC and he LPEN kene ae ued fo he cuen coecve em of he -D CMFD fomuaon. Tabe and how he ouon accuacy of he DeCART code accodng o he numbe of ub-pane fo he C5G7MOX -D benchmak and C5G7 hexagona vaaon pobem. The egenvaue and he pn powe dbuon eo appoach o a cean eo eve wh he INAC 9 Ro de Janeo RJ az.

7 nceae of he numbe of ub-pane and each he NEM accuacy when ung ubpane. In he C5G7MOX -D benchmak DeCART how abou and 5 pcm of egenvaue eo and abou.8 %.9 % and.5 % of -D oca pn powe eo fo he unodded odded A and confguaon. In he C5G7 hexagona vaaon pobem DeCART how abou and 5 of pcm egenvaue eo and abou.8 %.7 % and. % of -D oca pn powe eo fo he unodded odded A and confguaon. In he axay negaed - D powe dbuon DeCART how e han. % and abou.5 % eo fo he C5G7MOX -D benchmak and he C5G7 hexagona vaaon pobem. Pobem Unodded Rodded A Rodded Tabe. LPEN ouon Eo fo C5G7MOX enchmak Mehod NEM LPEN Np 5 5 ε k a pcm ε pp b % ε pd c % ε k a pcm ε p b % ε pd c % ε k a pcm ε p b % ε pd c % a Egenvaue Eo Refeence ae and 7777 fo Unodded Rodded A and wh e han 6 pcm of andad Devaon. b Maxmum D Pn Powe Eo c Maxmum D Pn Powe Eo Tabe. LPEN ouon Eo fo C5G7 Hexagona Vaaon Pobem Pobem Np 5 5 ε k a pcm Unodded ε pp b % ε pd c % ε k a pcm Rodded ε A p b % ε pd c % Rodded ε k a pcm ε p b % ε pd c % a Egenvaue Eo Refeence ae.7.89 and 6 fo Unodded Rodded A and wh e han pcm of andad Devaon. b Maxmum D Pn Powe Eo c Maxmum D Pn Powe Eo INAC 9 Ro de Janeo RJ az.

8 Fg. and how he axay negaed pn powe eo fo he Rodded confguaon. The neo pn powe dbuon how eavey e eo han he exeo one and he maxmum eo appea a he coe efeco bounday. In he ecangua pobem DeCART emae a hghe pn powe n he UO aembe and oppoey n he MOX aembe. In he hexagona pobem DeCART emae eavey hghe powe dbuon n he MOX aembe han n he UO aembe. Tabe ummaze he compung me beakup of he DeCART code fo he unodded confguaon accodng o he numbe of ub-pane. The compung me equed fo he MOC kene no eaed o he numbe of ub-pane and hey ae uned ou o be abou 8 and econd fo he ecangua and fo he hexagona coe. The compung me equed fo he -D CMFD and fo he axa LPEN kene nceae neay wh he numbe of ub-pane and hey ae uned ou o be abou 4 and econd fo he ecangua and fo he hexagona coe when ung ub-pane pe MOC pane. Theefoe concuded ha he LPEN mehod we deveoped and he LPEN kene we mpemened o he DeCART code pdecart pef 5 4 e - pef Fgue. Pn Powe Eo fo C5G7MOX Rodded Confguaon. INAC 9 Ro de Janeo RJ az.

9 p e DeCART p ef p ef Fgue. Pn Powe Eo fo C5G7 Hexagona Rodded Confguaon. Tabe. Compung Tme of LPEN Mehod fo C5G7MOX Unodded Pobem (INTEL QUAD(TM) QUAD. GHz econd) Pobem Np 5 5 Toa C5G7MOX CMFD+Axa enchmak MOC C5G7 Hex. Vaaon Toa CMFD+Axa MOC CONCLUION In h pape a ow ode poynoma expanon noda (LPEN) mehod whch expand he angua fux momen up o econd ode Legende poynoma fo he mpfed Pn (Pn) anpo equaon wa deveoped fo he DeCART axa ouon. In he pefomance examnaon DeCART howed e han 4 pcm egenvaue eo and e han % -D pn powe dbuon eo wh abou mnue of compung me when ung ub-pane pe MOC pane. Theefoe concuded ha he LPEN mehod we apped o he DeCART axa kene and poduce a good ouon whn an affodabe compung me. INAC 9 Ro de Janeo RJ az.

10 ACKNOWLEDGMENT Th wok wa uppoed by he Inenaona Nucea Enegy Reeach Inave (INERI) pogam jony funded by he Mny of Educaon cence and Technoogy of Koea and he Depamen of Enegy of he Uned ae. REFERENCE. H-G. Joo J-Y. Cho K-. Km C-C. Lee and -Q. Zee "Mehod and Pefomance of a Thee-Dmenona Whoe-Coe Tanpo Code DeCART" PHYOR 4 Chcago UA Ap. 5-9 (4).. J-Y. Cho K-. Km H-J. hm J-. ong C-C. Lee and H-G. Joo Whoe Coe Tanpo Cacuaon Empoyng Hexagona Modua Ray Tacng and CMFD Fomuaon" Jouna of Nucea cence and Technoogy 45 no (8).. N-Z Cho e a. Refnemen of he -D/-D Fuon Mehod fo -D Whoe-Coe Tanpo Cacuaon Tan. Am. Nuc. oc (). 4.. Koaka and T. Takeda Dffuon-Lke -D Heeogeneou Coe Cacuaon wh -D Chaacec Tanpo Coecon by Non-Lnea Ieaon Technque In. Conf. Nucea Mahemaca and Compuaon cence (M&C ) Ganbug Ap 6- CD-ROM AN (). 5. J-Y Cho K-. Km C-C. Lee -Q. Zee and H-G.. Joo "Axa PN and Rada MOC Couped Whoe Coe Tanpo Cacuaon" Jouna of Nucea cence and Technoogy 44 no (7). 6. J-Y. Cho K-. Km C-C. Lee -Q. Zee and H-G.. Joo "ub-pane cheme fo a Rada Tanpo and Axa Dffuon Code ICAPP 7 Nce Fance May -8 (7). 7. M. A. mh e a. "enchmak on Deemnc Tanpo cacuaon whou paa Homogenzaon (MOX Fue Aemby -D Exenon Cae)" OECD NEA Repo NEA/NC/DOC(5)6 (6). 8. M. A. mh e a. "enchmak on Deemnc -D MOX Fue Aemby Tanpo Cacuaon whou paa Homogenzaon" Poge n Nuc. Enegy 48 8 (6). 9. H-J. hm and C-H. Km "oppng Cea of Inacve Cyce Mone Cao Cacuaon" Nucea cence and Engneeng 57 no. -4 (7). INAC 9 Ro de Janeo RJ az.

Numerical Study of Large-area Anti-Resonant Reflecting Optical Waveguide (ARROW) Vertical-Cavity Semiconductor Optical Amplifiers (VCSOAs)

Numerical Study of Large-area Anti-Resonant Reflecting Optical Waveguide (ARROW) Vertical-Cavity Semiconductor Optical Amplifiers (VCSOAs) USOD 005 uecal Sudy of Lage-aea An-Reonan Reflecng Opcal Wavegude (ARROW Vecal-Cavy Seconduco Opcal Aplfe (VCSOA anhu Chen Su Fung Yu School of Eleccal and Eleconc Engneeng Conen Inoducon Vecal Cavy Seconduco