Heat flow in composite rods an old problem reconsidered

Transcription

1 Ha flow in copoi ro an ol probl rconir. Kranjc a Dparn of Phyic an chnology Faculy of Eucaion Univriy of jubljana Karljva ploca 6 jubljana Slovnia an J. Prnlj Faculy of Civil an Goic Enginring Univriy of jubljana Jaova jubljana Slovnia h inrfac praur of wo ro wih qual cro cion join a on n an wih iffrn iniial praur iniially alway acquir h valu characriic for wo iinfini ro. hi valu which i hown o b a conqunc of nrgy conrvaion i in gnral iffrn fro h hral quilibriu praur in fini ro. o illura hi rul wo paricular ca ar icu. I. INRODUCION In hi brif papr w wih o ain crain apc of h wll known probl of linar flow of ha in copoi ro. h uual ran of h probl giv a ahaical oluion bu lack a clar phyical picur. I i our goal o clarify a crain poin concrning h i pnnc of inrfac praur an h approach o hral quilibriu which a la o our knowlg ha no bn ar an clarifi in h pa. W giv a clar planaion oghr wih a clan ahaical oluion ha confir i. Conir wo i-infini ro of qual an unifor cro cion join a on n an inula on h i. If h iniial praur ar an > h corrponing praur iribuion in h ro for > i wll known o b for h bnfi of h rar a brif rivaion i prn in h ppni an k k rf < a rf > b whr k an k ar hral conucivii of h lf < an h righ > ro an an ar h corrponing hral iffuivii. Fro h abov quaion i follow li li > >

2 which i aying ha h inrfac praur i conan an qual o h final quilibriu praur of h ro. On h ohr han if h ro ar of fini an qual lngh h quilibriu praur obain by an lnary calculaion uing h conrvaion of nrgy i ρ c ρ c / / q 3 ρ c ρ c whr ρ i an c i ar h niy an h pcific ha of h rpciv ro. hi rul which o no pn on h lngh of h ro o b a o wih h corrponing rul for i-infini ro givn by q.. II. EMPERURE DISRIUION IN COMPOSIE RODS OF FINIE ENGH Conir wo ro of lngh an an of qual cro-cion align wih h -ai an join a. h lf ro wih h iniial praur covr h inrval < < an h righ ro wih h iniial praur > covr h inrval < <. uing ha h ro ar inula on h i ipli ha. Conqunly h praur iribuion i rin by h ha iffuion quaion of h for i i i i 4 oghr wih h iniial coniion 5a h bounary coniion a h n 5b an h bounary coniion a h inrfac > > 5c k k. 5 Uing h aplac ranforaion 3 h ha iffuion quaion in r of h aplac ranfor i h following orinary inhoognou con orr iffrnial quaion.

3 3 i i i i i. 6 h corrponing iniial an bounary coniion ar 7a 7b 7c k k. 7 h oluion of 6 aifying h iniial an bounary coniion 7 i coh inh inh coh inh coh < < 8a coh inh inh coh inh coh < <. 8b In paricular aking h lii w g li 9a li. 9b h lii rprn h aplac ranfor of a an b rpcivly. pplying h final valu hor for aplac ranfor 3

4 4 li li o h q. 9 w rcovr h rul. Whn w apply hi hor o q. 8 w obain li li ' q / / / / which ruc o 3 for an o for / /. o fin h invr aplac ranfor of 8a an 8b w u h riu hor 3 i.. riu of whr h u run ovr all pol of. σ an i β. Fro h prion 8a an 8b w uc ha an ar ingl-valu funcion of wih ipl pol a an β whr ±β 3... ar h roo of h quaion co β inσβ in β coσβ which i obain by quaing noinaor of 8a or 8b o zro. In wha follow w will conir wo ca only. Ca whr w an Ca whr w choo / /. Ca : pplying h riu hor in hi ca noing ha in hi ca σ / i follow an q q β β β / inσβ co β / [ σ co β coσβ σ in β inσβ ] β / in β coσβ / [ σ co β coσβ σ in β inσβ ] whr q i fin by 3. Morovr h q. can b rwrin a co β coσβ a b in β inσβ. 3 h roo of ar hu n o b h roo of quaion

5 5 co β coσβ 4a which ar all ral an ipl 4 oghr wih h coon roo of in β inσβ. 4b Howvr h coon roo if hy i o no conribu o h u a an b bcau of h prnc of inβ an inσβ facor in h nuraor of h prion. hrfor h u in run only ovr h roo of q. 4a. Morovr w can alway approia σ a a raio of wo ingr naly C r/p. In hi ca h roo of 4a can b wrin a β β n β npπ n... 5 wih β rprning h roo of 4a on h inrval fro o pπ. aking ino accoun 5 an uing 4a w can wri h praur iribuion in h ro in h final for a q in σβ σ in β n in β co β in in β in σβ in σβ σ in β σβ n β npπ β npπ co[ β npπ / ] β npπ in[ β npπ / ] β npπ 6a an q inσβ coσβ in σ in σβ σ in β β n σβ nrπ co[ σβ nrπ / ] σβ nrπ in σβ in σ in σβ σ in β n β σβ nrπ in[ σβ nrπ / ]. σβ nrπ 6b Ca : Sinc / / h rul 8 iplify o h following prion coh 7a coh

6 6 coh. 7b coh h corrponing invr aplac ranfor can b foun in Rf. 3 p. 3 for apl an ar n π / in π n n n n π / in π n n n π 8a π. 8b III. NUMERIC RESUS ND DISCUSSION Fro q. an 8 i follow ha h inrfac praur i conan in h ca of i-infini ro an ro wih hir lngh raio / / an i qual o h valu givn by q.. On h ohr han h inrfac praur for h ro of qual lngh i for apl β npπ in β co β in σβ q. 9 β npπ in σβ σ in β n I i plo in Fig. a a funcion of /. W obrv howvr ha vn in hi ca h inrfac praur who iniial valu i again givn by q. rain approialy conan for a rlaivly larg fracion of i / which i h i characriic for h ro o rach hral quilibriu. hi iniial an apparnly gnral bhavior of inrfac praur i unranabl bcau for i uch ha << h ha flow i inpnn of h lngh of h ro. For i-infini ro hi coniion i clarly aifi a all i an conqunly h inrfac praur i hu rigorouly conan. For fini ro wih arbirary raio of hir lngh h inrfac praur of cour ar o chang wih incraing i an vnually approach h quilibriu valu givn by q.. hi i illura in Fig. howing h praur iribuion along h ro of qual lngh for variou valu of i. For coparion h praur iribuion in i-infini ro i alo hown in Fig. 3 an for fini ro wih h lngh raio / / in Fig. 4.

7 7 Fig.. praur a h conac for h ca of a coppr iniial praur ºC an aluinu ro iniial praur ºC boh of lngh a a funcion of /. h uppr horizonal lin i h iniial conac praur > > 65. ºC givn by Eq. h lowr horizonal lin i h final quilibriu praur q ºC givn by Eq. 3. Fig.. Spaial iribuion of praur in a coppr an an aluinu ro boh of lngh in conac a for variou i: 5 5 an. h iniial praur a h conac i 65. ºC h final quilibriu praur q ºC.

8 8 Fig. 3. Spaial iribuion of praur in a coppr an aluinu ro of i-infini lngh in conac a for variou i: 5 an. h praur a h conac > i 65. ºC a all i. Fig. 4. Spaial iribuion of praur in a coppr an aluinu ro of lngh raio / / in conac a for variou i: 5 an. h praur a h conac o no chang an i a all i 65. ºC which i alo h final quilibriu praur of h ro. On h bai of h rul w conclu ha h inrfac praur i givn iniially in all ca by h valu obain fro q.. Morovr w hav hown ha hi valu

9 9 corrpon o h hral quilibriu valu q of wo fini ro who lngh raio i / /. hai i / / / / ρc ρ c q ρ c ρ c in agrn wih q.. h coniion ha h inrfac praur rain approialy conan i << <<. Whn hi coniion ca o b aifi h inrfac praur ar o chang cp in h ca of i-infini ro whr hi coniion i nvr viola an in h paricular ca of fini ro wih hir lngh raio aju in uch a way ha h quilibriu praur q coinci wih h iniial inrfac praur. h quilibriu praur for i-infini ro u hrfor b copar o h quilibriu praur of wo fini ro of lngh an wih hir raio qual o / /. h raniion o h lii rin by h ha conucion proc ilf i uch ha hi raio i kp fi. hu whn w join wo ro or wo boi wih iffrn praur all rgion on ach i of h inrfac who iz ar roughly of h orr an rpcivly alo iialy rach h quilibriu praur. h rpciv porion of h boy wih iniial praur > cool by h aoun q a whil h rpciv porion of h colr boy wih iniial praur ha up by h aoun q b k kρc whr. pracical apl illuraing hi rul i h following k kρc wll-known iuaion. Whn w p wih bar f on a col urfac our f will cool off by an aoun givn by a which i h largr h largr i. Conqunly ho urfac for which a fin abov i larg will b prciv a colr han urfac wih all vn hough hir praur ar h a. a Elcronic ail: oaz.kranjc@pf.uni-lj.i H. S. Carlaw an J.C. Jagr Conucion of Ha in Soli n Eiion Ofor U. P. Ofor 959 p V. uikov nalyical Ha Diffuion hory caic Pr Nw York 968 p M. R. Spigl aplac ranfor Schau Oulin Sri McGraw-Hill Nw York Rfrnc p Rfrnc p. 89.

10 PPENDIX In h ca of praur iffuion in i-infini ro align wih h -ai h only quaniy wih inion of lngh in aiion of cour o h coorina i. Conqunly h only inionl quaniy on which h praur iribuion ay pn i /. Sinc h iffuion quaion i linar an hoognou wih rpc o praur i uni can b aign arbirarily. If i o praur characriic o h probl a han w ay wri f f whr w hav inclu a facor of for lar convninc. hi ranforaion i originally u o olzann 5. Inring hi funcion ino iffuion quaion 4 w obain 4 f f. uing ha wo ro wih iniial praur lf ro an righ ro ar join a an ingraing h abov quaion w wri h oluion in h for < / rf π 3 > rf / π 4 whr rf i h rror funcion fin a rf π rf rf rf. Ipoing h iniial an bounary coniion 5 nabl u o rin h ingraion conan wih h rul. Inring h prion ino 3 an 4 w obain h rul quo in h inroucion.