The finite element models of thin-walled branched structures in heat transfer problems

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1 03 ISSN ECHANIKA. 0 olum 8(): h fini lmn modls of hin-walld branchd srucurs in ha ransfr problms S. urskinė Šiauliai Univrsiy 9 išinskio g Šiauliai Lihuania Sigia@fm.su.l hp://dx.doi.org/0.5755/j0.mch Inroducion A prsn nginrs ar ofn facd wih h ask of dsigning hin-walld branchd srucurs working undr condiions of innsiv ha inrchang wih h nvironmn. In h juncion on of h plas of such srucurs gra mpraur oscillaions ak plac which has an influnc on h analysis of h srss srain sa of h srucur []. In solving his problm firs of all w hav o valua accuraly h valus of h mpraur in h pla juncion on. ransin ha conducion in an anisoropic marial is dscribd by h diffrnial quaion k x k y k Q c x y hr kx ky k ar anisoropic hrmal conduciviy cofficins; is h mass dnsiy of h marial; c is h spcific ha capaciy; Qx y is h ha gnrad wihin h body; x y is h mpraur; is h im. o solv h Eq. () iniial and boundary condiions ar inroducd 0 x y () () k x n x k y n y k n q 0 (3) x y x y q x y is h ha flux α is h convcion cofficin ovr h surfac is h fluid mpraur. Bcaus h analysd srucur is of a complx gomrical shap i is convnin o solv h problm of h mpraur fild by h fini lmn mhod. All mahmaical dpndncs of h mhod ar asily formalisd and gnrad by compur chnologis. An Eq. () wih iniial condiions () and wih boundary condiions (3) in accordanc wih [] is wrin as h sysm of diffrnial quaions n n n ar h dircion cosins C K F () F ar rspcivly hrmal conduciviy marix ha capaciy marix and vcor of hrmal load. h fini lmn modl of h xamind srucur is composd of fini lmns of svral yps: plan K C and (riangular or quadrangular) and connciv quadrangular prismaic fini lmns. ha is why h problm of modlling pla juncion ons wih fini lmns appars. I is analysd in rsarch works [3-5]. Plan fini lmns ar joind wih riangular or quadrangular prismaic connciv fini lmns. Pracical calculaions hav shown ha in hos cass whn connciv fini lmns join ohr fini lmns in h nods ha ar in h middl-surfac of h plas i is ncssary o us h middl nods of h prism bas insad of cornr nods of h prism. o calcula mpraur valus in such nods spcial ransformaion marics ar ncssary. In his work ransformaion formulas of coordinas ha ransla h middl sid nods of h bass of a quadrangular prism o h cornr nods of h prism and formulas ha ransla h mpraur valus of h cornr nods ino h mpraur valus of h middl sid nods of h bass of h fini lmn ar found. h problm is solvd for h connciv riangular prismaic fini lmn [3]. h aim of h prsn aricl is o assss h ransin mpraur fild of h pla juncion on of a hinwalld branchd srucur of an anisoropic marial whn h boundary condiions on h laral surfacs of h pla can b xprssd in hr diffrn forms: convcion ha sourc and ha flow. h juncion on of h plas is going o b modlld by a connciv quadrangular prismaic fini lmn h main mahmaical xprssions of which for h isoropic marial ar dscribd in [].. arics of h connciv quadrangular prismaic fini lmn o mak a discr modl of h juncion on of h plas a connciv quadrangular fini lmn of an qual cross-scion is going o b usd; wih a quadrangl as h bas (Fig. ). h lmn has 8 dgrs of frdom. Z S 8 S 5 5 S Y S 6 S 3 b 7 6 Fig. h gomry of h connciv fini lmn h 3 a S X

2 0 h mpraur insid h fini lmn changs linarly 8 Nii (5) i N [N] is h marix of h shap funcions of h lmn { } is mpraur valus in h nods of h lmn. h shap funcions saisfying propris in h nods of h lmn N N x y (6) x y 0 whn i i i i i i j j j ar nrd as follows S is h cross-scional ara of h connciv fini lmn h is h high of h lmn. h valus of a and b ar calculad by using h formula o calcula h disanc bwn wo poins. h conribuion of h connciv fini lmn o h marics K C and vcor F of h Eq. () is xprssd by h following formula K B DBd N NdS (8) S C c N N d (9) F QN d qn ds N ds (0) S S N b xa yh N b xa yh N 3 b xa yh N b xa yh N 5 b xa y N 6 b xa y N 7 b xa y N 8 b xa y (7) hr is h volum of h fini lmn S is h laral ara of h lmn [B] is a marix of h drivaivs of h shap funcions of h lmn in h dircions x y and [D] is a marix of h hrmal conduciviis in h dircions x y and. h marix [D] in h discussd cas is nrd as follows kx 0 0 D 0 ky k () By diffrniaing marix [N] (7) wih rspc o x y and w form marix [B] h invrs marix of which is nrd as follows B a yh b xh b xa y a yh b xh b xa y a yh b xh b xa y a yh b xh b xa y a y b x b xa y a y b x b x a y a y b x b xa y a y b x b xa y () Whil calculaing marics of h fini lmn l us assum ha a b h d y dx (3) a b 0 h fini lmn hrmal conduciviy marix [K] (8) has wo pars: h firs ingral dscribs h hrmal conduciviy of h fini lmn drmind by h hrmal conduciviy cofficins of h marial; h scond ingral dscribs h hrmal conduciviy of h fini lmn drmind by h convcion ha xchang wih nvironmn via h laral surfacs and h nds of h fini lmn. h hrmal conduciviy marix of h lmn always has h firs componn whil h scond has o b valuad only whn h laral surfacs and h nds of h fini lmn ar opn i.. whn hy hav a conac wih h nvironmn. Whil calculaing h firs ingral of h hrmal conduciviy marix of h lmn (8) h valus of marics [B] [B] and [D] ar insrd in is xprssion. I can b sn ha pos-ingral xprssions ar rahr complx. ha is why i is difficul o calcula hm manually; bsids i is asy o mak misaks. ha is why mahmaical ransformaions ar mad by using compur algbra sysms APLE and AHEAICA.

3 05 Having ingrad h firs ingral of h hrmal conduciviy marix of h connciv fini lmn w g B DBd 8 () m m m3 m m m m m 3 m3 m m m m m3 m m m5 m6 m7 m8 m6 m5 m8 m 7 m7 m8 m5 m6 m8 m7 m6 m5 m 3a k a h k 0b k b h k 3b k m 3a k a h k 8b k b h k 3b k 3 m 3a k a h k 0b k b h k 3b k 3 3 m 3a k a h k 8b k b h k 3b k m 6a k a h k 0b k b h k 6b k 5 m 6a k a h k 6b k b h k 6b k 6 3 m 6a k a h k 0b k b h k 6b k 7 3 m 6a k a h k 6b k b h k 6b k 8 k kx 3k y k 3k x k y k3 kx 3k y k 3k x k y S h valu of h scond componn N N ds (5) of h hrmal conduciviy marix of h connciv fini lmn (8) dpnds on h surfac of h lmn (laral or nd) h ha ransfr wih convcion aks plac. h valu of his ingral dos no dpnd on h hrmal conduciviy cofficins ha is why by mploying h oucoms of [] w can calcula h valu of ingral (5) for h surfac.g. S : x b y aa 0 h ha S S 0 0 S hr 8 S S S h valus of ingral (5) ar wrin accordingly for surfacs S S 3 S S 5 S 6 S 7 S 8 h valus of h lmns of marix [ S i ] dpnd on h numbrs of h nods blonging o laral or nd surfacs. h marix of h hrmal capaciy of h connciv fini lmn (9) according o [3] has h following form chs * * C C C * hr 6 * C C C Bcaus h ingrals of h hrmal load vcor (0) do no dpnd on h hrmal conduciviy cofficins on h basis of [] h valu of h firs ingral is Q N QhS d 8 h valus of h ohr wo ingrals of h hrmal load vcor (0) dpnd on h numbrs of h nods of h laral or nd surfacs of h fini lmn. For insanc for surfac S qn d S S 0 qha S 0 ha 0 N ds h ransformaion formula of h coordina nods of h connciv fini lmn W shall now analys h bas (quadrangl) of h connciv fini lmn (Fig. ) wih nodal poins a middl sid poins numbrd counr-clockwis from h firs frly chosn nod. W shall ransfr h middl sid nods of h bas x y x y 3 x y x y o h cornr nods AxA ya A BxB yb B C C C Dx y D D D C x y i.. w shall xprss h coordinas of nods A B C D by h coordinas of nodal poins 3. o solv his

4 06 problm h quaions of lins going via givn poins known from analyical gomry ar going o b usd. D 3 C C x x y y D x x y y A Fig. Numbring of h nods of h bas of h lmn Bcaus h branchd srucur of a complx shap is analysd h connciv fini lmn can b in various posiions in is fini lmns schm. ha is why i is ncssary o analys a fw cass of h posiions of h nods of h connciv fini lmns in spac.. c cons x x3 x x y y3 y y In his cas h coordinas of h cornr poins ar nrd as follows a a A c b b B c c c C c D c x x x y y y x x x y y y y y a x x y x x x y x x3 x a y x y y y x y y 3 3 y y b x x y x x x y x x3 x b y x y y y x y y 3 3 y y c x x y x x x y x x x x3 c y x y y y y y y y d x x y x x x y x x3 x d y x y y y x y y For h poins no o b in on lin a ncssary y x condiiondiscriminan d 0. y x3. = c = cons x x3 y y ar as follows 3 In his cas h coordinas of h cornr nods y c Bx y c C x y c Dx y c A x 3 B h coordinas of h cornr nods ar calculad as follows A x x y y B x x y y. 3 h coordinas of h cornr poins ar nrd as follows A x x y y 3 3 B x x y y 3 3 C x x y y D x x y y cons x x x x y y y y x3 y3 3 h rank of marix has o x y b qual. inors of scond ordr ar mad of h lmns of marix y x y x x y which hav o saisfy h condiion 3 0. Considr h following cass: If 0 hn a3 a b 3 b A x x B x x c3 c d3 d C x3 x 3 D x3 x. a y y y y y y y y a y y y y x b x y x y x b y x c y y y y y y y y y c y x y

5 d y y y y y y y y d y y If 0 hn a33 a b33 b A y y B y y c33 c d33 d C y3 y3 D y3 y a x x x x x x a x x x x b y x x x x x b x x x x c x x x x x x c x x x x d x x x x x x d x x x bas of h fini lmn ar found as follows K R K R C R C R 5. Numrical rsuls. W shall analys h mpraur fild in a srucur of an anisoropic marial in h cas of a ransin ha ransfr procss. h algorihm was wrin by FORRAN. A coold branchd srucur is givn []. h air moving across h uppr surfac has a mpraur of 0ºC convcion cofficin is 8. W/(m C). h lowr surfac is coold by a liquid of h mpraur of 96ºC convcion cofficin is 359 W/(m C). h soluion schma of h fini lmns of h srucur wih a quadrangular connciv fini lmn is shown in Fig If 0 hn 3 0 bcaus.. his conradics h condiion 3 0. h ransformaion formula of h mpraur valus of h connciv fini lmn In h cas whn h schm of h fini lmns of h srucur is composd only from connciv quadrangular fini lmns i is no ncssary o ransla h mpraur valus of h cornr nods o h middl sid nods of bass. Ohrwis whn connciv fini lmns in h srucur join plan fini lmns i is ncssary o ransla h mpraur valus of h cornr nods o h middl sid nods of bass. Having valuad prviously formulad condiions (7) w g h following ransformaion marix of mpraur valus R R R 0 0 R hus h hrmal conduciviy marix and hrmal capaciy marix of h connciv fini lmn o calcula mpraur valus in h middl sid nods of h Fig. 3 h grid of h fini lmns of h srucur o analys h mpraur convrgnc wo fini lmn schms wr adapd: wih and wihou a connciv quadrangular fini lmn. h convrgnc of h rsuls is illusrad by obsrving h chang of mpraur of a crain poin of h calculad schm in im.g. 3 (Fig. 3). h viw of h mpraur convrgnc of nod 3 of h wo calculad schms is prsnd in Fig.. Fig. h mpraur in im of nod 3 of h wo calculad schms h corrcnss of h formula of h ransfor-

6 08 maion of nod coordinas and mpraurs valus of h connciv fini lmn is sablishd by calculaing mpraur valus in h connciv nods 6 and 6 (Fig. 3). h rsuls of h calculaion ar prsnd in h abl blow. h comparison of h mpraur valus Numbrs of h nods (Fig. 3) 6. Conclusions Obaind mpraur valus ºC Conrol mpraur valus [] ºC h numrical soluion of h problm of a ransin mpraur fild convrgs.. In cass of h firs and scond sampling of h juncion on h curvs of h mpraur valus of nod 3 diffr. According o h calculad schm wih a connciv fini lmn h calculad mpraur valus (a nod 3) ar closr o h conrol valus han h valus obaind according o h schm wihou a connciv fini lmn. 3. h obaind ransformaion formula of h nod coordinas and mpraur valus allow making a fini lmn schm of h analysd srucur boh wih cornr and middl sid nods of a bas of a connciv fini lmn. his allows a mor fficin modlling of joining a fw plas wih diffrn normals o a horional surfac. Rfrncs. Barauskas R.; Kasparaiis A.; Kaušinis S.; Ladinas R. 0. mpraur filds xchangs anformaions of a prcis lngh comparaor microscop chanika 7(3): hp://dx.doi.org/0.5755/j0.mch Lwis R.W.; Nihiarasu P.; Sharamu K.N. 00. Fundamnals of h Fini Elmn hod for Ha and Fluid Flow. Wily 356p. hp://dx.doi.org/0.00/ arnlyė S.; aciulvičius D riangular prismaic fini lmn for approximaion of juncions in mpraur disribuion problms of hin-walld branchd and complx srucurs Liuvos mchanikos rinkinys 9: (in Russian).. arnlyė-urskinė S Discriaion of pla juncion ons in hrmal problms chanika 3(6): urskinė S h modlling by fini lmns of plan juncion ons in ha ransfr problms Liuvos mamaikos rinkinys 0: 36-. S. urskinė PLONASIENIŲ IŠSIŠAKOJANČIŲ KONSRUKCIJŲ BAIGINIŲ ELEENŲ ODELIAI ŠILUOS PERNEŠIO UŽDAINIUOSE R i u m ė Sraipsnyj nagrinėjamas plonasinės išsišakojančios konsrukcijos pagaminos iš anioropinės mdžiagos plokšlių sandūros onos nsacionaraus mpraūros lauko analiės uždavinys. okių konsrukcijų plokšlių sandūros onoj vyksa didli mpraūrų svyravimai kuri uri įakos konsrukcijos įmpo ir dformuoo būvio analii. Sandūros onai diskriuoi naudojamas jungiamasis rdvinis baiginis lmnas kurio pagrindas yra kurkampis. Gaua jungiamojo baiginio lmno šilumos laidumo marica kai mdžiaga yra anioropinė. Išvsos koordinačių ransformacijos ir mpraūros vrčių ransformacijos formulės kurios įgalina sudaryi nagrinėjamo objko baiginių lmnų inkllį ik su jungiamojo lmno pagrindo kampiniais ik su pagrindo krašinių vidurių magais. ai lidžia fkyviau modliuoi kai jungiamos klios plokšlės urinčios skiringas horionalaus paviršiaus normals. Išvsos formulės paikrinos sprndžian šaldomos plokšlės mpraūros lauko uždavinį. S. urskinė HE FINIE ELEEN ODELS OF HIN-WALLED BRANCHED SRUCURES IN HEA RANSFER PROBLES S u m m a r y h papr dals wih h problm of analysis of h ransin mpraur fild in h pla juncion on of a hin-walld branchd srucur of an anisoropic marial. In h juncion on of h plas of such srucurs gra mpraur oscillaions ak plac which hav an influnc on h analysis of h srss-srain sa of h srucur. h juncion on of h plas was modlld by a connciv fini lmn wih a quadrangular bas. h hrmal conduciviy marix of h connciv fini lmn was obaind in h cas of an anisoropic marial. h coordina ransformaion and mpraur valu ransformaion formula wr drivd which allowd composing a grid of h fini lmns of h xamind objc boh wih h cornr and middl sid nods of h connciv fini lmn. his allows a mor fficin modlling of joining a fw plas wih diffrn normals o a horional surfac. h drivd formula wr chckd by h soluion of a problm of h mpraur fild of a coold pla. Kywords: hin-walld branchd srucur connciv fini lmn ha ransfr. Rcivd arch 0 0 Accpd arch 9 0