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1 39 Enopy 003, 5, Enopy ISSN Enopy Analysis in Pipe Flow Sbjeed o Eenal Heaing Iyad Talal Al-Zahanah Mehanial Engineeing Depamen, KFUPM, Dhahan, Sadi Aabia; iyadz@fpm.ed.sa; Tel: ; Fa: Reeived: 8 Oobe 00 / Aeped: 5 May 003 / Pblished: 3 Deembe 003. Absa In he pesen sdy, hea ansfe and enopy analysis fo flow hogh a pipe sysem is onsideed. The Reynolds nmbe and he pipe wall empeae effes on enopy disibion and oal enopy geneaion in he pipe ae invesigaed. Nmeial sheme employing a onol volme appoah is inoded when solving he govening eqaions. Seel is seleed as pipe maeial, while wae is sed as flid. I is fond ha ineasing pipe wall empeae and Reynolds nmbe ineases he enopy podion ae, in whih ase, enopy geneaion de o hea ansfe dominaes ove ha oesponding o flid fiion. Keywods: pipe, flow, hea ansfe, enopy Inodion Flow hogh pipes finds wide appliaions in indsy. The flow and hea ansfe haaeisis an be fhe impoved hogh sdying ievesibiliies assoiaed in he flow sysem. The flow hogh pipes in hemal envionmen sffes fom fiional and hea ansfe losses; in whih ase, fiion edes he pesse; while hea ansfe lowes he hemal effiieny of he piping sysem. Fom he indsial appliaion poin of view, flow hogh piping sysem is blen whih eqies appopiae modeling o pedi he flow popeies. Consideably eseah sdies wee aied o o invesigae he flow and hea ansfe haaeisis of pipe flow. Nmeial simlaion of ansiional flow and hea ansfe in smooh pipe was aied o by Hien and Songling []. They indiaed ha in he flly developed egion, flow and hea ansfe wee no affeed by inle blene inensiies, and he ageemen beween he pediions and daa fo he fiion oeffiien was good. The behavio of fiion and hea ansfe oeffiiens of wae flowing blenly in a elaively long pipe was invesigaed by Choi and Cho []. They poposed a new blen hea ansfe oelaion fo he loal Nssel nmbe. Pesse dop and hea ansfe in blen d flow wee sdied by Ghaiban e al [3] sing a wo paamee vaiaional mehod. They showed ha he analysis lead o developmen of a Geen s fnion; whih was sefl in solving a vaiey of onjgae hea ansfe poblems. An

2 39 Enopy 003, 5, epeimenal sdy of blen wae flow hogh abp onaions was aied o by Bllen e al [4]. They showed ha he pesse loss oeffiien was Reynolds nmbe dependen, whih ineased wih deeasing Reynolds nmbe. Hea ansfe in blen flow was invesigaed by Thae and Joshi [5]. They inoded diffeen vesions of low Reynolds nmbe - blene models. They indiaed ha he - models pefomed elaively bee han he Reynolds sess models fo pediing he mean aial empeae and he Nssel nmbe. The eensive sdy fo blen pipe and hannel flows was o by Wosni e al [6]. They indiaed ha fo pipe and hannel flows he inflene of bonday laye was impoan; howeve, he Reynolds nmbe had less inflene on he fiional losses as Reynolds nmbe ineased fhe. Ievesibiliy assoiaed wih pesse dop and hea ansfe gives insigh ino he enegy desed in he hemal sysems. Consideable eseah sdies wee aied o o invesigae he enopy geneaion in he flow and hemal sysems. Bejan [7] applied seond law analysis o he hemal enegy soage sysem. His appoah was based on minimizing he desion of hemodynami availabiliy as apposed o maimizing he oal amon of hemal enegy soage. The ievesible eension of he Cano yle of maimm mehanial wo deliveed fom a sysem of finie esoes was invesigaed by Sieniyz [8]. He showed ha bonds of he lassial availabiliy shold be eplaed by songe bonds fo finie ime poess. Demiel and Sandle [9] sdied linea-noneqilibim hemodynami model fo opled hea and mass ansfe poess. They pesened he phenomenologial eqaions wih esisane oeffiiens, whih wee apable of efleing he eend of he ineaions beween hea and mass flows. Enopy geneaion ding lase heaing poess was invesigaed by Yilbas [0]. He indiaed ha he enopy geneaion nmbe was inflened by he lase plse lengh. Enopy geneaion in pipe flow wih inenal esisane was invesigaed by Yilbas e al []. They showed ha enopy geneaion de o hea ansfe dominaes ove is onepa oesponding o flid fiion. Sine he enopy analysis povides insigh ino he ievesibiliy assoiaed in a hemal sysem, he pesen sdy in onded o invesigae he enopy geneaion in he pipe flow. Moeove, in a pevios sdy [], he Reynolds nmbe was ep onsan as well as pipe wall heaing fl was fied. Conseqenly, invesigaion ino inflene of he Reynolds nmbe and pipe wall empeae on he enopy geneaion is neessay. In he analysis, diffeen pipe wall empeaes and flow Reynolds nmbes ae onsideed o eamine ompehensively he affeing paamees fo he enopy geneaion. Mahemaial Modeling The flow siaion in he pesen sdy is involved wih an inompessible flow hogh a pipe, whih is eenally heaed a diffeen empeaes. The pipe is shown shemaially in Fige. The flow field is assmed o be aisymmei and a nifom oe wall empeae is assmed along he pipe lengh. The eqaions govening he flow field ae simplified afe he onsideaion of Bossinesqe appoimaions, in whih ase - blene model an be sed o aon fo he blene haaeisis. In ylindial pola oodinaes he onsevaion eqaions ae wien as: Coniniy:

3 393 Enopy 003, 5, () 0 (v) Momenm: ( ) ( ) ( ) ( ) d dp ρ vρ Enegy: ( ) ( ) () 3 T P P ρt vtρ whee P and P ae lamina and blen Pandl nmbes espeively. In ode o deemine he blen visosiy and he Pandl nmbe, he - blene model is sed. The onsiive eqaions fo he blen visosiy ae as follows: ( ) 4 ρ d whee and ae he blen inei enegy geneaion and he dissipaion vaiables espeively. The anspo eqaion fo is: ( ) ( ) ( ) ( ) 5 D ρ P ρ vρ whee ( ) 6 ) ρ)( ( D The anspo eqaion fo is: ( ) ( ) ( ) 7 w ρ ρ P ρ vρ The geneaion of blene inei enegy, and is dissipaion a he inne wall of he pipe ( i ) is zeo. The Pandl nmbes in anspo eqaions of inei enegy geneaion and dissipaion ae P and P, espeively. The Pandl nmbe vaies wih Reynolds nmbe [3]. The vales in Table ae employed ding he simlaions, sine eah simlaion is aied o fo a fied Reynolds nmbe.

4 394 Enopy 003, 5, In ode o minimize ompe soage and n imes, he dependen vaiables a he walls wee lined o hose a he fis gid fom he wall by eqaions, whih ae onsisen wih he logaihmi law of he wall. Conseqenly, he eslan veloiy paallel o he wall in qesion and a a disane y (whee y 30) fom i oesponding o he fis gid node was assmed o be epesened by he law of he wall eqaions [4], i.e.: V τ d w /ρ / κ ln[e( d ) / y ρ ] () 8 whee, κ is a nivesal von-kaman onsan and e is a nivesal blene paamees and hei vales ae κ 0.47 and e 9.37, fom whih he wall shea sesses wee obained in solving he momenm eqaions. The onsans sed in he anspo eqaions ae [4]: P ; d ;.44 ;.9; Re ( ρ) ; P.0 A adial sep lengh of appoimaely wo visos sb-laye hiness (y 5), whee y y [(τ w /ρ) / /(/ρ)] is employed. In he aial and adial dieions, he gid onains nodes in he flid egion and nodes in he solid egion, employed o obain he gid independen solion. The gid nodes ae disibed o give a high onenaion lines nea he wall povided ha he wall adjaen nodes ae posiioned a y 5. The gid independeny ess wee aied o by sing he diffeen gid nodes and he gid disibion ess wee also onded and based on he findings, he gids giving opimm solion is ensed as onsisen wih he ealy wo [3]. The Nssel nmbe (N) pedied fom he pesen sdy and he esls of he pevios sdy [3] ae shown wih Reynolds nmbe in able. I an be obseved fom he able ha he pesen pediions fo N agee well wih he pevios esls. Howeve, small disepanies ae obseved beween boh esls a high Reynolds nmbes. This may be de o slighly ove pediing he blen inei enegy geneaion a high Reynolds nmbe by he blen model (- model) inoded in he pesen sdy. Howeve, his diffeene is small. Sine he flid flow in he pipe and he eenal heaing of he pipe ae plae a seadysae, he ondion in he solid is: and The bonday ondiions: T T ( ) 0 (9) The elevan bonday ondiions fo he onsevaive eqaions of flow and solid ae: T ) A pipe ais ( 0): 0 and 0 ) A inne solid wall ( i ): No-slip ondiions ae onsideed, i.e.: 0 and v 0 w w

5 395 Enopy 003, 5, ) A pipe inle (0): Unifom flow and nifom empeae wee assmed. 4) A pipe ole (L): All he gadiens of he vaiables wee se o zeo, i.e.: ϕ η 0 whee φ is he flid popey and η is any abiay dieion. 5) A oe sfae of he pipe ( 0 ): Unifom sfae empeae is assmed, i.e.: T T 0 (K) 6) A solid-flid inefae ( i ), i.e.: i and 0 L Ts Tf s f and T s Tf Enopy Analysis The ievesibiliy involved in he hemal sysem de o momenm and enegy anspo, esls in oninos enopy podion in he sysem. The loal enopy geneaion pe ni volme fo an inompessible Newonian flow sysem is [5]: S& f ( T) T T Φ ( 0) whee he fis em epesens he enopy geneaion de o hea ansfe while he seond em is he enopy geneaed de o visos dissipaion in he flow sysem. In pola oodinaes he visos dissipaion (Φ) an be wien as: v Φ v whee is he aial veloiy and v is he adial veloiy. Toal enopy geneaion ae ove he volme is: v ( ) & Sgen ( ) S& dv and he ae o ievesibiliy is: & I T S & o gen ( 3) Eq. (0) is sed o deemine he volmei enopy geneaion ae while Eq.(3) is sed fo he ievesibiliy ae.

6 396 Enopy 003, 5, Resls and Disssion Flow hogh a heaed pipe and enopy geneaion in he flow sysem is pesened fo hee oe wall empeaes (500 K, 750 K and 000K) and hee Reynolds nmbes (0000, and 50000). The pipe sed in he sdy is 4 mee long. Fo ompe soage easons, i is diffil o se longe pipes. To geneae enogh enopy in he pipe and o sdy he inflene of empeae disibion in he pipe on enopy disibion, lage vales of pipe oe wall empeaes (in he ange fom 500 K o 000 K) ae sed in he sdy. I shold also be noed ha he wae empeae seleed was eessive, whih eqied pessized wae. The pipe diamee is aen as 0.08 mee and he pipe hiness as 0.04 mee. In his sdy, he empeae disibion in he pipe is pesened in a non-dimensional fom (T * ), whih is allaed as: T - T T - T whee T inle is he pipe inle empeae and T mean is he mean empeae of all gid poins. Fige shows dimensionless empeae (T * ) onos in he flid fo wo pipe wall empeaes and Reynolds nmbe of Dimensionless empeae (T * ) onos show simila behavio fo all he pipe wall empeaes employed (500 K and 000 K). In geneal, a pipe inle whee is less, dimensionless empeae aains low vales. In his ase, he flid eneing ino he pipe does no have enogh daion o gain hemal enegy fom he pipe wall. Conseqenly, onsan empeae ono eends fhe inside he flid in his egion. As he pipe lengh eends, high empeae onos ae developed in he egion lose o he pipe wall. This is bease of he onveive heaing of he flid in he viiniy of he pipe wall. In he ase of enopy podion, as seen fom Fige 3, enopy onos do no follow he empeae onos. This is bease of he enopy is popoional o he empeae gadien ahe han empeae. When ompaing he enopy onos obained fo 500 K and 000K, he onos diffe signifianly in he wo ases. This is moe pononed in he egion lose o he pipe wall. This is bease of he losses de o flid fiion in he wall egion as well as high flid empeae gadien de o elevaed wall empeae, whih enhanes hea ansfe aes fom he wall o flid. Fige 4 shows dimensionless empeae (T * ) onos fo hee pipe wall empeaes and Reynolds nmbe of 30000, while Fige 5 shows oesponding enopy onos. Tempeae onos behave simila o hose obained fo Reynolds nmbe 0000, povided ha he magnide of empeae onos in he sfae egion ineases. This os bease of he high onveive hea ansfe aes, whih enhanes wih ineasing Reynolds nmbe. Enopy onos hange fo he Reynolds nmbe as ompaed o hei oesponding o This is moe pononed fo pipe wall empeae 000 K. I shold be noed ha he fiional losses de o visos dissipaion edes as Reynolds nmbe ineases fo he blen flow. Despie his fa, enopy onos aain high vales fo Reynolds nmbe of This is bease of he enopy geneaion de o hea ansfe aes, whih impove onsideably a high Reynolds nmbe. Fige 6 shows dimensionless empeae (T * ) onos fo hee oe wall empeaes and Reynolds nmbes of 50000, while Fige 7 shows oesponding enopy onos. T * inle mean ( 4)

7 397 Enopy 003, 5, Tempeae onos in he viiniy of he pipe wall ineases fhe as ompaed o hose shown in Fige 4 bease of he high onveive hea ansfe aes fom he pipe wall o flid. Enopy onos diffe signifianly fom hose pesened in Figes 3 and 5, pailaly fo empeae 000 K. Conseqenly, ineasing Reynolds nmbe and pipe wall empeae enhanes he hea ansfe aes, whih in n esls in high ae of enopy podion. Fige 8 shows oal enopy ae poded in he flid sysem wih wall empeae as Reynolds nmbe is vaiable. Enopy podion ae ineases onsideably wih ineasing pipe wall empeae and Reynolds nmbe. This siaion indiaes ha enopy geneaion de o hea ansfe dominaes ove is onepa oesponding o flid fiion. This is bease of ineasing Reynolds nmbe in blen flow lowes he fiional losses; heefoe, he enopy podion ae is loweed. Conseqenly, ineasing enopy podion ae a high Reynolds nmbe is bease of hea ansfe poess only. I an be obseved fom Fige 8 ha enopy geneaion an be minimized by eding he pipe wall empeae, while ineasing he Reynolds nmbe. Conlsions Hea ansfe and enopy analysis fo flow hogh pipe sysem ae aied o. A nmeial mehod sing a onol volme appoah is employed in he ompaions. Seel pipe is seleed as pipe maeial, while wae is onsideed as flid. The simlaions ae epeaed fo hee Reynolds nmbes and hee pipe wall empeaes. I is fond ha enopy onos do no follow he empeae pofiles. Enopy podion ae is low a he pipe inle and as he pipe lengh ineases i signifies. Moeove, enopy podion ae is high in he egion lose o he pipe wall, whih is moe pononed fo high wall empeae. This is bease of high empeae gadien in his egion. Ineasing Reynolds nmbe enhanes he enopy podion ae. This is bease of he enhaned hea ansfe aes, i.e. onveive heaing of he flid enhanes a high Reynolds nmbe. This indiaes ha enopy podion de o hea ansfe dominaes ove he enopy podion de o flid fiion, sine fiional losses ede wih ineasing Reynolds nmbe in he pipe flow. ACKNOWLEDGEMENT The aho hans Pofesso Bei Yilbas ( bsyilbas@fpm.ed.sa) fo his nmeos sefl ommens and sppo. The aho also wold lie o han King Fahd Univesiy of Peolem and Mineals fo is sppo in his wo. Nomenlae L P T v V lengh of he pipe pesse adial oodinae empeae a a gid poin flid aial veloiy adial veloiy eslan adial veloiy aial oodinae

8 398 Enopy 003, 5, P Re T Panl nmbe Reynolds nmbe empeae hemal ondiviy Gee Symbols e onsan in law of wall κ onsan in law of wall τ shea sess flid dynami visosiy ρ flid densiy φ any abiay vaiable blen dissipaion vaiable Sbsips f s i o w flid solid inne oe wall blen Spesips * dimensionless Lamina Reynolds Nmbe (Re) Tblen Pandl Nmbes Nssel Nmbe (N) P () P () Refeene No. (3) Pesen Sdy Abs. % Diff Table. The blen Pandl nmbes and he esling Nssel nmbes fo he validaion and he pesen sdy ases.

9 399 Enopy 003, 5, L nifom wall empeae, T o o i Fige. Shemai diagam of he pipe and oodinaes /i /I (a) (b) Fige. Dimensionless empeae onos fo he ases of Reynolds nmbe0000, fo oe wall empeaes (a) 500 K and (b) 000 K /i /i (a) (b) Fige 3. Enopy onos fo he ases of Reynolds nmbe0000, oe wall empeaes (a) 500 K and (b) 000 K

10 400 Enopy 003, 5, /i /i (a) (b) Fige 4. Dimensionless empeae onos fo he ases of Reynolds nmbe30000, wall empeaes (a) 500 K and (b) 000 K /i R/Ri (a) (b) Fige 5. Enopy onos fo he ases of Reynolds nmbe30000 fo oe wall empeaes (a) 500 K and (b) 000 K.

11 40 Enopy 003, 5, /i /i (a) (b) Fige 6. Dimensionless empeae onos fo he ases of Reynolds nmbe50000 fo oe wall empeaes (a) 500 K and (b) 000 K /i /i (a) (b) Fige 7. Enopy onos fo he ases of Reynolds nmbe fo oe wall empeaes (a) 500 K and (b) 000 K.

12 40 Enopy 003, 5, Toal enopy/0^6 (W/m^3 K) Re 0000 Re Re Oe wall empeae, To ( K) Fige 8. The oal enopy geneaion fo diffeen oe wall empeaes and Reynolds nmbes. Refeenes. Z. Hien, and L. Songling, Nmeial Simlaion of Tadiional Flow and Hea Tansfe in A Smooh Pipe, Inenaional Jonal of Hea Mass Tansfe, Vol.34, pp , 99.. E. Choi,E. and Y.I. Cho, Loal Fiion and Hea Tansfe Behavio of Wae in Tblen Pipe Flow wih Lage Hea Fl a The Wall, Jonal of Hea Tansfe, Vol. 7, pp.83-88, N. Ghaiban, A. Haji-Sheih, and S.M. Yo, Pesse Dop and Hea Tansfe in Tblen D Flow: a Two - Paamee Vaiaional Mehod, Jonal of Hea Tansfe, Vol. 7, pp , P.R. Bllen, D.J. Cheeseman, and L.A. Hssain, A Sdy of Tblen Flow in Pipe Conaions, Jonal of Poess Mehanial Engineeing, Vol. 0, pp.70-80, S. Thae and J. Joshi, CFD Modeling of Hea Tansfe in Tblen Pipe Flows, AIChE Jonal, Vol. 46, No.9, pp , M. Wosni, L. Casillo and W. Geoge, A Theoy fo Tblen Pipe and Channel Flows, Jonal of Flid Mehanias, Vol. 4, pp. 5-45, A. Bejan, A Sdy of Enopy Geneaion in Fndamenal Conveive Hea Tansfe ASME, Jonal of Hea Tansfe, Vol. 0, pp , S. Sieniyz, Cano Poblem of Maimm Wo fom a Finie Resoe Ineaing wih Enviomen in a Finie Time Physia A, Vol. 64, pp , Y. Demiel, and S.I. Sandle, Linea-Noneqilibim Themodynamis Theoy fo Copled Hea and Mass Tanspo Inenaional Jonal of Hea and Mass Tansfe, Vol. 44, pp , B.S. Yilbas, Thee-Dimensional Lase Heaing Model and Enopy Geneaion Consideaion, ASME, Jonal of Enegy Resoes Tehnology, Vol., pp. 7-4, B.S. Yilbas, S.Z. Shja and M.O. Bdai Seond Law Analysis of Swiling Flow in a Cila D wih Resiion Inenaional Jonal of Hea Mass Tansfe, Vol. 4, pp , 999.

13 403 Enopy 003, 5, I. Al-Zahanah, B.S. Yilbas and M.S. Hashmi Enopy Geneaion in Pipe flow De o Diffeen Solid o Flid Condiviy Raios Poeedings of he Twelfh Inenaional Symposim on Tanspo Phenomena, ISTP-. 3. W.M. Kays, and M.E. Cawfod, Conveive Hea and Mass ansfe, Mgaw-Hill, In., A.D. Gosman, E.E. Khalil, and H.J. Whielaw, The Callaion of Two -Dimensional Tblen Reilaing Flows Pblished in Tblen shea flows (Seleed papes fom he Fis Inenaional Symposim on Tblen Shea Flows, he Pennsylvania Sae Univesiy, Univesiy Pa, Pennsylvania, USA, Apil 8-0, 977), Spinge-Velag Belin Heidelbeg, New Yo, A. Bejan, Enopy Geneaion Minimizaion, CRC Pess, New Yo (995) (C) 003 by MDPI (hp:// Repodion fo nonommeial pposes pemied.

Chapter Finite Difference Method for Ordinary Differential Equations

Chapter Finite Difference Method for Ordinary Differential Equations Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence