Songklanakarin Journal of Science and Technology SJST belhocine

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1 Exact solutio of boudary valu problm dscribig th covctiv hat trasfr i fully- dvlopd lamiar flow through a circular coduit Joural: Sogklaakari Joural of Scic ad Tchology Mauscript ID SJST-- Mauscript Typ: Origial Articl Dat Submittd by th Author: -Mar- omplt List of Authors: blhoci, ali; Mchaical Egirig,, Mchaical Egirig, Wa Omar, Wa Zaidi ; Mchaical Egirig, Mchaical Egirig Kyword: Gratz problm, Sturm-Liouvill problm, Partial diffrtial quatio, dimsiolss variabl

2 Pag of Rsarch Articl Exact solutio of boudary valu problm dscribig th covctiv hat trasfr i fully- dvlopd lamiar flow through a circular coduit Ali Blhoci * ad Wa Zaidi Wa Omar aculty of Mchaical Egirig, Uivrsity of Scics ad th Tchology of Ora, L.P El - MNAOUER, USTO ORAN Algria aculty of Mchaical Egirig, Uivrsiti Tkologi Malaysia, UTM Skudai, Malaysia * orrspodig author, addrss: blhoci@gmx.fr Abstract This papr proposs a xact solutio of th classical Gratz problm i trms of a ifiit sris rprstd by a oliar partial diffrtial quatio cosidrig two spac variabls, two boudary coditios ad o iitial coditio. Th mathmatical drivatio is basd o th mthod of sparatio of variabls whos svral stags wr illustratd to rach th solutio of th Gratz problm. A MATLAB cod was usd to comput th igvalus of th diffrtial quatio as wll as th cofficit sris. Howvr, th xprssio of th Nusslt umbr as ifiit sris solutio ad th Gratz umbr was dfid basd o th hat trasfr cofficit ad th hat flux from th wall to th fluid. I additio, th aalytical solutio was compard to th umrical valus obtaid by th sam author usig ORTRAN program show that th orthogoal collocatio mthod givig bttr rsults. It is importat to ot that th aalytical solutio is i good agrmt with publishd umrical data.

3 Pag of Kywords: Gratz problm, Sturm-Liouvill problm, Partial diffrtial quatio, dimsiolss variabl Nomclatur a :paramtr of coflut hyprgomtric fuctio b : paramtr of coflut hyprgomtric fuctio c p :hat capacity : cofficit of solutio dfid i Eq. a;b;x : stadard coflut hyprgomtric fuctio K : thrmal coductivity L : lgth of th circular tub N G Nu P R r T T T ω υ max Grk lttrs ζ θ : Gratz umbr : Nusslt umbr : Pclt umbr : radial dirctio of th cylidrical coordiats : radius of th circular tub : tmpratur of th fluid isid a circular tub : tmpratur of th fluid trig th tub : tmpratur of th fluid at th wall of th tub : maximum axial vlocity of th fluid : igvalus : dimsiolss axial dirctio : dimsiolss tmpratur

4 Pag of : dimsiolss radial dirctio ρ : dsity of th fluid µ : viscosity of th fluid. Itroductio Th solutios of o or mor partial diffrtial quatios PDEs, which ar subjctd to rlativly simpl limits, ca b tackld ithr by aalytical or umrical approach. Thr ar two commo tchiqus availabl to solv PDEs aalytically, amly th variabl sparatio ad combiatio of variabls. Th Gratz problm dscribs th tmpratur or coctratio fild i fully dvlopd lamiar flow i a circular tub whr th wall tmpratur or coctratio profil is a stp-fuctio Shah & Lodo,. Th simpl vrsio of th Gratz problm was iitially glctig axial diffusio, cosidrig simpl wall hatig coditios isothrmal ad isoflux, usig simpl gomtric cross-sctio ithr paralll plats or circular chals, ad also glctig fluid flow hatig ffcts, which ca b grally dotd as lassical Gratz Problm Braga, d Barros, & Sphair,. Mi, Yoo, ad hoi prstd a xact solutio for a Gratz problm with axial diffusio ad flow hatig ffcts i a smi-ifiit domai with a giv ilt coditio. Latr, th Gratz sris solutio was furthr improvd by Brow. Hsu studid a Gratz problm with axial diffusio i a circular tub, usig a smi- ifiit domai formulatio with a spcifid ilt coditio. Ou ad hg mployd th sparatio of variabls mthod to study th Gratz problm with fiit viscous dissipatio. Thy obtaid th solutio i th form of a sris whos igvalus ad igfuctios of which satisfy th Sturm Liouvill systm. Th solutio tchiqu follows th sam approach as that applicabl to th classical Gratz

5 Pag of problm ad thrfor suffrs from th sam wakss of poor covrgc bhavior ar th trac. Th sam tchiqus hav b usd by othr authors to driv aalytical solutios, ivolvig th sam spcial fuctios hrbach, D Gouray, Pirr, & Plouraboué, ; Plourabou & Pirr, ; Plourabou & Pirr,. Basu ad Roy aalyzd th Gratz problm for Nwtoia fluid by takig accout of viscous dissipatio but glctig th ffct of axial coductio. Papoutsakis, Damkrisha, ad Lim prstd a aalytical solutio to th xtdd Gratz problm with fiit ad ifiit rgy or mass xchag sctios ad prscribd wall rgy or mass fluxs, with a arbitrary umbr of discotiuitis. olho, Piho, ad Olivira studid th trac thrmal flow problm for th cas of a fluid obyig th Pha-Thiad Tar PTT costitutiv quatio. This appars to b th first study of th Gratz problm with a viscolastic fluid. Th solutio was obtaid with th mthod of sparatio of variabls ad th suig Sturm Liouvill systm was solvd for th igvalus by mas of a frly availabl solvr, whil th ordiary diffrtial quatios for th igfuctios ad thir drivativs wr calculatd umrically with a Rug Kutta mthod. I th work of Bilir, a umrical profil basd o basd o th fiit diffrc mthod was dvlopd by usig th xact solutio of th o-dimsioal problm to rprst th tmpratur chag i th flow dirctio. Ebadia ad Zhag aalyzd th covctiv hat trasfr proprtis of a hydrodyamically, fully dvlopd viscous flow i a circular tub. Lahjomri ad Oubarra ivstigatd a w mthod of aalysis ad improvd solutio for th xtdd Gratz problm of hat trasfr i a coduit. A xtsiv list of cotributios rlatd to this problm may b foud i th paprs of Papoutsakis, Ramkrisha, Hry,

6 Pag of ad Lim ad Liou ad Wag. I additio, th aalytical solutio proposd fficitly rsolvs th sigularity ad this mthodology allows xtsio to othr problms such as th Hartma flow Lahjomri, Oubarra, & Almay,, cojugatd problms ith & Aad, ad othr boudary coditios. Trasit hat trasfr for lamiar pip or chal flow was aalyzd by may authors. Ats, Darici, ad Bilir ivstigatd th trasit cojugatd hat trasfr i thick walld pips for thrmally dvlopig lamiar flow ivolvig two-dimsioal wall ad axial fluid coductio. Th problm was solvd umrically by a fiitdiffrc mthod for hydrodyamically dvlopd flow i a two-rgioal pip, iitially isothrmal i which th upstram rgio is isulatd ad th dowstram rgio is subjctd to a suddly applid uiform hat flux. Darici, Bilir, ad Ats i thir work solvd a problm i thick walld pips by cosidrig axial coductio i th wall. Thy hadld a trasit cojugatd hat trasfr i simultaously dvlopig lamiar pip flow. Th umrical stratgy usd i this work is basd o a fiit diffrc mthod i a thick walld smi-ifiit pip which is iitially isothrmal, with hydrodyamically ad thrmally dvlopig flow ad with a sudd chag i th ambit tmpratur. Darici ad S umrically ivstigatd a trasit cojugat hat trasfr problm i micro chals with th ffcts of rarfactio ad viscous dissipatio. Thy also xamid th ffcts of othr paramtrs o hat trasfr such as th Pclt umbr, th Kuds umbr, th Brikma umbr ad th wall thickss ratio. Rctly, Blhoci dvlopd a mathmatical modl to solv th classic problm of Gratz usig two umrical approachs, th orthogoal collocatio mthod ad th mthod of rak-nicholso.

7 Pag of I this papr, th Gratz problm that cosists of two diffrtial partial quatios will b solvd usig sparatio of variabls mthod. Th Kummr quatio is mployd to idtify th coflut hyprgomtric fuctios ad its proprtis i ordr to dtrmi th igvalus of th ifiit sris which appars i th proposd aalytical solutio. Also, thortical xprssios of th Nusslt umbr as a fuctio of Gratz umbr wr obtaid. I additio, th xact aalytical solutio prstd i this work was validatd by th umrical valu data prviously publishd by th sam author rprstd by th orthogoal collocatio mthod which givig bttr rsults.. Backgroud of th Problm As a good modl problm, w cosidr stady stat hat trasfr to fluid i a fully dvlopd lamiar flow through a circular pip ig.. Th fluid trs at z at a tmpratur of T ad th pip walls ar maitaid at a costat tmpratur of T ω. W will writ th diffrtial quatio for th tmpratur distributio as a fuctio of r ad z, ad th xprss this i a dimsiolss form ad idtify th importat dimsiolss paramtrs. Hat gratio i th pip du to th viscous dissipatio is glctd, ad a Nwtoia fluid is assumd. Also, w glct th chags i viscosity i tmpratur variatio. A sktch of th systm is show blow. ig.... Th hat quatio i cylidrical coordiats Th gral quatio for hat trasfr i cylidrical coordiats dvlopd by Bird, Stwart, ad Lightfoot is as follows; ρ T p + u r t T u + r r θ T + u θ Z u Z T z k ρ p T k r T + z r r r T T T + z + r θ

8 Pag of u r uθ µ + + u r r θ r uz + z u + µ z u Z u Z + + r θ r µ u r u + r r θ r r θ θ u r + z + osidrig that th flow is stady, lamiar ad fully dvlopd flow R, ad if th thrmal quilibrium had alrady b stablishd i th flow, tht. Th t dissipatio of rgy would also b gligibl. Othr physical proprtis would also b costat ad would ot vary with tmpratur such as ρ, µ, p,k. This assumptio also T implis icomprssibl Nwtoia flow. Axisymmtric tmpratur fild, θ whr w ar usig th symbol θ for th polar agl. By applyig th abov assumptios, Eq. ca b writt as follows: u Z T z k ρ p r T r r r Giv that th flow is fully dvlopd lamiar flow Poisuill flow, th th vlocity profil would hav followd th parabolic distributio across th pip sctio, rprstd by r u Z u R Whr u is th Maximum vlocity xistig at th ctrli. By rplacig th spd trm i Eq., w gt: r u T k R z ρ r T p r r r Solvig th quatios rquir th boudary coditios as st i ig., thus

9 Pag of z, T T r, T r r R, T Tω It is mor practical to study th problm with stadardizd variabls from to. To do this, w variabls without dimsio kow as adimsioal ar itroducd, dfid as Tω θ T, Tω T Eq. givs x r ad R y z. Th substitutio of th adimsioal variabls i L x R T Tω T T T T u θ k ω θ ω θ + R L y xr R x R x ρcp Aftr makig th cssary arragmts ad simplificatios, th followig simplifid quatio is obtaid. whr th trm u Rρ k kl x θ θ θ + y ρp ur x x x p is th adimsioal umbr kow as Pclt umbr P, which i fact is Ryolds umbr dividd by Pradtl umbr. I stady stat coditio, th partial diffrtial quatio rsultig from this, i th adimsioal form ca b writt as follows: x θ L y RP x θ x x x This quatio, if subjctd to th w boudary coditios, would b trasformd to th followigs: z, r, T T y, θ, T r x, θ θ, y x,

10 Pag of r R, T Tω x, θ θ, y. It is hrby proposd that th sparatio of variabls mthod could b applid, to solv Eq... Aalytical Solutio usig Sparatio of Variabls Mthod I both qualitativ ad umrical mthods, th dpdc of solutios o th paramtrs plays a importat rol, ad thr ar always mor difficultis wh thr ar mor paramtrs. W dscrib a tchiqu that chags variabls so that th w variabls ar dimsiolss. This tchiqu will lad to a simpl form of th quatio with fwr paramtrs. Lt th Gratz problm is giv by th followig govrig quatio θ L θ θ x + y PR x x x whr, P is th Pclt umbr, L is th tub lgth ad R is th tub radius. With th followig iitial coditios: I : y, θ B : x, θ x B : x, θ Itroducig dimsiolss variabls as dmostratd Huag, Matloz, W, & William, as follows: x r R ζ kz ρc v p max R y z L

11 Pag of By substitutig Eq. ito Eq. th it bcoms: Kowig that Thrfor, v max u Notic that th trm k y L ζ ρ c p v max R k y L y L ζ u ρ c p R u ρ c p R. R k uρc k Thus, Eq. ca b writt as p R i Eq. is similar to th Pclt umbr, P. yl ζ PR Basd o Eqs. -, os ca writ th followig xprssios; θ θ x θ θ x θ ζ θ L θ. y y ζ PR ζ Now, by rplacig Eqs. - ito Eq., th govrig quatio bcoms: Eq. will b rducd to: L PR θ L θ + ζ PR θ Th right trm i Eq. ca b simplifid as follows: θ θ θ + ζ θ θ θ +

12 Pag of ially, th quatio that charactrizs th Gratz problm ca b writt i th form of: θ ζ θ Now, usig a rgy balac mthod i th cylidrical coordiats, Eq. ca b dcomposd ito two ordiary diffrtial quatios. This is do by assumig costat physical proprtis of a fluid ad glctig axial coductio ad i stady stat. Th associatd boudary coditios for a costat-wall-tmpratur cas for Eq. ar as follows: at trac ζ, θ at wall, θ at ctr, θ ad dimsiolss variabls ar dfid by: θ T T ω ω T T, r ad r ζ kz ρ c p vmax r whil th sparatio of variabls mthod is giv by ially, Eq. ca b xprssd as follows: ad θ Z ζ R dz dζ Z d R dr + + R d d whr is a positiv ral umbr ad rprsts th itrisic valu of th systm. Th solutio of Eq. ca b giv as: ζ Z c

13 Pag of whr c is a arbitrary costat. I ordr to solv Eq., trasformatios of dpdt ad idpdt variabls d to b mad by takig: Ι v v Π R v S v Thus, Eq. is ow giv by; d S ds ν + ν S dν dν Eq. is also calld as coflut hyprgomtric as citd i Slatr, ad it is commoly kow as Kummr quatio... Thorm of uchs A homogous liar diffrtial quatio of th scod ordr is giv by '' ' y + P Z y + Q Z y If PZ ad QZ admit a pol at poit ZZ, it is possibl to fid a solutio dvlopd i th whol sris providd that th limits o limz Z Z Z P ad lim Z Z Z Z Q xist. Z Th mthod of robius sks a solutio i th form of Z y Z Z λ a Z whr, λ is a cofficit to b dtrmid whilst proprtis of th hyprgomtric fuctios ar dfid by ; α α α + Z α α + α + L α + Z α,, Z + Z + + L+ + L+ +! + + L θ +! α,, γ covrg Z Usig drivatio agaist Z, th fuctio is ow bcom

14 Pag of d dz [ α,, Z ] rom Eq., os will gt ; α α + α + α + Z ++ Z ! α α + α + L α + Z + L+ + + L α +! d,, q α α +, +, Z Thus, th solutio of Eq. ca b obtaid by: whr,,... ad igvalus + L+,, R c,,,, ar th roots of Eq.. Ths ca b radily computd i MATLAB sic it has a built-i hyprgomtric fuctio calculator. Th solutios of our quatio ar th igfuctios to th Gratz problm. It ca b show by a sris xpasio that th igfuctis ar: G,, whr is th coflut hyprgomtric fuctio or th Kummr fuctio. Ths fuctios ar powr sris i similar to a xpotial fuctio Abramowitz & Stgu,. Th fuctios rprstd abov hav th symmtry proprty sic thy ar v fuctios. Hc th boudary coditio at is satisfid. Sic th systm is liar, th gral solutio ca b dtrmid usig suprpositio approach: θ ζ..,,

15 Pag of Th costats i Eq. ca b sought usig orthogoality proprty of th Sturm- Liouvill systms aftr th iitial coditio is big applid as statd blow; c.,,.,, d Th itgral i th domiator of Eq. ca b valuatd usig umrical itgratio. or th Gratz problm, it is oticd that: θ θ ζ whr is th fuctio of th wight / igvalus B., θ θ B., I ζ, θ θ ζ ζ G,, G,, is th fuctio of th wight Sturm-Liouvill problm. I ζ, θ d dg G d d dg for, G for d

16 ,, ζ θ Rlatio of orthogoality, j i x Y x Y x W j i,,,, d m m m, m +,,,, θ ζ ζ,, ζ θ ζ By cosidrig,,, Eq. ad owards ca b giv as; θ θ ζ θ d θ ζ θ d +,,,,,, d ζ ζ ζ,,,, d ζ ζ By combiig Eqs., ad, th quatio ca b rducd to; Pag of

17 d,,,, Lt s multiply Eq. by Eq. ad th itgrat Eq.,,, m m m d d m m m m m m,,.,,,, Th outcoms of multiplicatio ad itgratio procss will produc th followig: i If m th rsult is qual to zro ii If m th rsult is d,, d,, Substitutig Eq. ito Eq., th quatio bcoms;,, d,, Ad th costats ca b obtaid by; d,,,, Pag of

18 Pag of. Rsults ad discussio.. Evaluatio of th first four igvalus ad th costat A fw valus of th sris cofficits ar giv i Tabl togthr with th corrspodig igvalus. Th rsults of calculatd valus of th ctr tmpratur as a fuctio of th axial coordiat ζ ar also summarizd i Tabl. Tabls -. Th ladig trm i th solutio for th ctr tmpratur is thr for:. ζ θ ζ,. G.. Graphical rprstatio of th xact solutio of th Gratz problm Th ctr tmpratur profil is show i ig. usig fiv trms to sum th sris. As s i this figur, th valu of dimsiolss tmpratur θ dcrass with icrasig valus of dimsiolss axial positio ζ. Not that th fiv-trm sris solutio is ot accurat for ζ<. Mor trms dd hr for th sris to covrg. ig.... ompariso btw th aalytical modl ad th prvious modl simulatio rsults I ordr to compar th prvious umrical rsults carrid out prviously by Blhoci with th aalytical modl of our hat trasfr problm, w chos to prst th rsults of umrical distributio of tmpratur with th mthod of orthogoal collocatio which givs th bst rsults. ig. plots th compariso rsults. It is clar from ig. that thr is a good agrmt btw umrical rsults ad ctr aalytical solutio of th Gratz problm. ig..

19 Pag of.. Hat trasfr cofficit corrlatio Th hat flux from th wall to th fluid q ω z is a fuctio of axial positio. It ca b calculatd dirctly by usig th rsult: T qω z k R, z r but as w otd arlir, it is customary to dfi a hat trasfr cofficit h z via qω z h z Tω T b whr th bulk or cup-mixig avrag tmpratur T b is itroducd. Th way to xprimtally dtrmi th bulk avrag tmpratur is to collct th fluid comig out of th systm at a giv axial locatio, mix it compltly, ad masur its tmpratur. Th mathmatical dfiitio of th bulk avrag tmpratur was giv i a arlir sctio. T R b R π rv r T r, z dr π rv r dr whr th vlocity fild v r v r. You ca s from th dfiitio of th R hat trasfr cofficit that it is rlatd to th tmpratur gradit at th tub wall i a simpl mar: T k R, z h z r T ω W ca dfi a dimsiolss hat trasfr cofficit, which is kow as th Nusslt umbr. T b

20 Pag of Whr b θ ζ, hr Nu ζ k θ ζ θ is th dimsiolss bulk avrag tmpratur. By substitutig from th ifiit sris solutio for both th umrator ad th domiator, th Nusslt umbr ca b writt as follows. hr Nu ζ k ζ ζ b G G d Th domiator ca b simplifid by usig th govrig diffrtial quatio for G, alog with th boudary coditios, to fially yild th followig rsult. Nu ζ ζ G ζ G W ca s that for larg ζ, oly th first trm i th ifiit sris i th umrator, ad likwis th first trm i th ifiit sris i th domiator is importat. Thrfor, as; ζ, Nu.. rom our quatio, it ca b xprssd: whr N Gr N N Gr,,. π NGr Nu, a π + NGr π,,. πζc pvmaxr is th Gratz umbr. kl ig. is th plot showig Nusslt umbr alog th odimsioal lgth of a tub with uiform hat flux. As xpctd, th graphs show that th Nusslt umbr is vry high at th bgiig of th trac rgio of th tub ad thraftr dcrass

21 Pag of xpotially to th fully dvlopd Nusslt umbr. ig. idicats that th Nusslt umbr dcrasd i th try rgio ad rapidly rachd a costat valu i th fully dvlopd rgio ig... oclusio I this papr, a xact solutio of th Gratz problm is succssfully obtaid usig th mthod of sparatio of variabls. Th hyprgomtric fuctios ar mployd i ordr to dtrmi th igvalus ad costats, ad latr to a fid solutio for th Gratz problm. Th mathmatical mthod prformd i this study ca b applid to th prdictio of th tmpratur distributio i stady stat thrmally lamiar hat trasfr basd o th fully dvlopd vlocity for fluid flow through a circular tub. I futur work xtsios, w rcommd prformig th Gratz solutio by sparatio of variabls i a varity of ways of accommodatig o-nwtoia flow, turbult flow, ad othr gomtris bsids a circular tub. It is importat to ot that th prst aalytical solutios of th Gratz problm ar i good agrmt with prviously publishd umrical data of th author. It will b also itrstig to solv th quatio of th Gratz problm usig xprimtal data for compariso purposs with th proposd xact solutio. Rfrcs Abramowitz, M., & Stgu, I. Hadbook of Mathmatical uctios, Dovr, Nw York. Ats, A., Darıcı, S., & Bilir, S.. Ustady cojugatd hat trasfr i thick walld pips ivolvig two-dimsioal wall ad axial fluid coductio with

22 Pag of uiform hat flux boudary coditio. Itratioal Joural of Hat ad Mass Trasfr,,. Basu, T., & Roy, D. N.. Lamiar hat trasfr i a tub with viscous dissipatio. Itratioal Joural of Hat ad Mass Trasfr,,. Blhoci, A.. Numrical study of hat trasfr i fully dvlopd lamiar flow isid a circular tub. Itratioal Joural of Advacd Maufacturig Tchology, -. Bilir, S.. Numrical solutio of Gratz problm with axial coductio. Numrical Hat Trasfr Part A-Applicatios,, -. Bird, R. B., Stwart, W. E., & Lightfoot, E. N.. Trasport Phoma, Joh Wily ad Sos, Nw York. Braga, N. R., d Barros, L. S., & Sphair, L. A.. Gralizd Itgral Trasform Solutio of Extdd Gratz Problms with Axial Diffusio IM -th July, ambridg, Eglad, pp.-. Brow, G. M..Hat or mass trasfr i a fluid i lamiar flow i a circular or flat coduit.aihe Joural,,. olho, P. M., Piho,. T., & Olivira, P. J.. Thrmal try flow for a viscolastic fluid: th Gratz problm for th PTT modl. Itratioal Joural of Hat ad Mass Trasfr,,. Darıcı, S., Bilir, S., & Ats, A.. Trasit cojugatd hat trasfr for simultaously dvlopig lamiar flow i thick walld pips ad miipips. Itratioal Joural of Hat ad Mass Trasfr,,.

23 Pag of Ebadia, M. A., & Zhag, H. Y.. A xact solutio of xtdd Gratz problm with axial hat coductio. Itratioal Joural of Hat ad Mass Trasfr,, -. hrbach, J., D Gouray,., Pirr,., & Plouraboué,.. Th Gralizd Gratz problm i fiit domais. SIAM Joural o Applid Mathmatics,,. ith, R. M & Aad, N. K.. iit Elmt Aalysis of ojugat Hat Trasfr i Axisymmtric Pip lows. Numrical Hat Trasfr,, -. Gratz, L.. Ubr di Wärmlitugsfähigkit vo lüssigkit. Aal dr Physik,,. doi:./adp. Hsu,. J.. Exact solutio to try-rgio lamiar hat trasfr with axial coductio ad th boudary coditio of th third kid. hmical Egirig Scic,,. Huag,. R., Matloz, M., W, D. P., & William, S.. Hat Trasfr to a Lamiar low i a ircular Tub. AIhE Joural,,. Lahjomri, J., & Oubarra, A.. Aalytical Solutio of th Gratz Problm with Axial oductio. Joural of Hat Trasfr,, -. Lahjomri, J., Oubarra, A., & Almay, A.. Hat trasfr by lamiar Hartma flow i thrmal trac rgio with a stp chag i wall tmpraturs: Th Gratz problm xtdd. Itratioal Joural of Hat ad Mass Trasfr,, -. Liou,. T., & Wag,. S.. A omputatio for th Boudary Valu Problm of a Doubl Tub Hat Exchagr. Numrical Hat Trasfr Part A,, -.

24 Pag of Mi, T., Yoo, J. Y., & hoi, H.. Lamiar covctiv hat trasfr of a bigham plastic i a circular pipi. Aalytical approach thrmally fully dvlopd flow ad thrmally dvlopig flow th Gratz problm xtdd. Itratioal Joural of Hat ad Mass Trasfr,,. Ou, J. W., & hg, K... Viscous dissipatio ffcts i th trac rgio hat trasfr i pips with uiform hat flux. Applid Scitific Rsarch,,. Papoutsakis, E., Damkrisha, D., & Lim, H... Th Extdd Gratz Problm with Dirichlt Wall Boudary oditios. Applid Scitific Rsarch,, -. Papoutsakis, E., Ramkrisha, D., & Lim, H... Th Extdd Gratz Problm with Prscribd Wall lux. AlhE Joural,, -. Pirr,., & Plouraboué,.. Statioary covctio diffusio btw two coaxial cylidrs. Itratioal Joural of Hat ad Mass Trasfr, -,. Pirr,., & Plouraboué,.. Numrical aalysis of a w mixd-formulatio for igvalu covctio-diffusio problms. SIAM Joural o Applid Mathmatics,,. S, S., & Darici, S..Trasit cojugat hat trasfr i a circular microchal ivolvig rarfactio viscous dissipatio ad axial coductio ffcts. Applid Thrmal Egirig,,. Shah, R. K., & Lodo, A. L.. Lamiar low orcd ovctio i Ducts. Rtrivd from Slatr, L. J.. oflut Hyprgomtric uctios. ambridg Uivrsity Prss.

25 Pag of Rsarch Articl Exact solutio of boudary valu problm dscribig th covctiv hat trasfr i fully- dvlopd lamiar flow through a circular coduit Ali Blhoci * ad Wa Zaidi Wa Omar aculty of Mchaical Egirig, Uivrsity of Scics ad th Tchology of Ora, L.P El - MNAOUER, USTO ORAN Algria aculty of Mchaical Egirig, Uivrsiti Tkologi Malaysia, UTM Skudai, Malaysia * orrspodig author, addrss: blhoci@gmx.fr List of figurs ig.. Schmatics of th classical Gratz problm ad th coordiat systm ig.. Variatio of dimsiolss tmpratur profil θ with dimsiolss axial distac ζ ig.. A compariso btw th prst aalyticalcal rsults with th umrical Orthogoal collocatio data of Blhoci ig.. Nusslt umbr vrsus dimsiolss axial coordiat

26 Pag of z R luid at v r z Dimsiolss Tmpratur θ T,,,,,,,,,, r TR, z T ω ig.. d m o d m o d m o d m o Approximatio Solutio θζ, d m o tr d Aalytical m o Solutio d m o with d trms m oθ d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o, d m o d m o d m o d m o,,,,,,,,,,,,, Dimsiolss Axial Positio ζ ig..

27 Pag of Nusslt Numbr Nu Dimsiolss Tmpratur θ,,,,,,,,,,, d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o tr Aalytical Solutio d m o d m o Orthogoal d m o ollocatio d m o Solutio d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o, d m o d m o d m o d m o,,,,,,,,,,,,, Logitudial oordiat ζ ig.. d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o Nud m o, d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o d m o,,,,,, Dimsiolss Axial Positio ζ ig..

28 Pag of Rsarch Articl Exact solutio of boudary valu problm dscribig th covctiv hat trasfr i fully- dvlopd lamiar flow through a circular coduit Ali Blhoci * ad Wa Zaidi Wa Omar aculty of Mchaical Egirig, Uivrsity of Scics ad th Tchology of Ora, L.P El - MNAOUER, USTO ORAN Algria aculty of Mchaical Egirig, Uivrsiti Tkologi Malaysia, UTM Skudai, Malaysia * orrspodig author, addrss: blhoci@gmx.fr List of tabls Tabl. Eigvalus ad costats for Gratz s problm. Tabl. Rsults of th ctr tmpratur fuctios θ ζ

29 Pag of Tabl. Eigvalus officit G Tabl. ζ Tmpratur θ θ ζ,..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,