ANALYSIS OF UNSTEADY HEAT CONDUCTION THROUGH SHORT FIN WITH APPLICABILITY OF QUASI THEORY

Transcription

1 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 Rsarch Papr ISSN Vol., No. 1, Jauary IJMERR. All Rights Rsrvd ANALYSIS OF UNSTEADY HEAT CONDUCTION THROUGH SHORT FIN WITH APPLICABILITY OF QUASI THEORY Tjpratap Sigh 1 *, Sajv Shrivastava 1 ad Harbas Sigh Br *Corrspodig Author: Tjpratap Sigh, tjpratap50@yahoo.com Th papr is basd o th aalysis of ustady hat coductio through short fi with applicability of quasi thory. Th ara xposd to th surroudig is frqutly icrasd by th attachmt of protrusios to th surfacs, ad th arragmt provids a mas by which hat trasfr rat ca b substatially improvd. Th protrusios ar calld fis. Th fis ar commoly usd o small powr dvlopig machi as gi usd for motor cycl as wll as small capacity comprssor. Earlir, Wor udr stady stat coductio had b carrid out xtsivly. Ustady hat coductio aalysis for th fis is big do for calculatio of hat trasfr. Ustady Closd form solutios had b drivd arlir by various rsarchrs. Exact solutios ar giv for th ustady tmpratur i flux-bas fis with th mthod of Gr s Fuctios (GF) i th form of ifiit sris for thr diffrt tip coditios. Th tim of covrgc is improvd by rplacig th sris part by closd form solutio. Th prst study supplis a w approach to calculat th thrmal prformac of th short fi. For th short fi cas, xact fi solutio ad a quasi-stady solutio is prstd. Numrical valus ar prstd ad th coditios udr which th quasi-stady solutio is accurat ar dtrmid. Dimsiolss tmpratur distributio is prstd for both quasi stady thory ad xact fi thory. Kywords: Ustady hat, Gr fuctio, Protrusio, Hat trasfr INTRODUCTION Th laws which ar govrig hat trasmissio ar vry importat to th girs i th dsig, costructio, tstig ad opratio of hat xchag apparatus. Hat trasfr is th study of th rat at which rgy is trasfrrd across a surfac of itrst du to tmpratur gradits at th 1 Dpartmt of Mchaical Egirig (Thrmal Egirig), Shri Sharacharya Collg of Egirig ad Tchology (SSCET, Bhilai), Chhattisgarh, Idia. Dpartmt of Mchaical Egirig (Thrmal Egirig), Natioal Istitut of Tchology (NIT, Raipur), Chhattisgarh, Idia. 69

2 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 surfac, ad tmpratur diffrc btw th diffrt surfacs. This variatio i tmpratur is govrd by th pricipl of rgy cosrvatio which wh applid to a cotrol volum or a cotrol mass, stats that th sum of th flow of rgy ad hat across th systm, th wor do o th systm ad th rgy stord ad covrtd withi th systm is zro. Th mchaical gir dal with problms of hat trasfr i th fild of itral combustio gis, stam gratio, rfrigratio ad hatig ad vtilatio. To stimat th cost, th fasibility ad siz of th quipmt cssary to trasfr a spcifid amout of hat i a giv tim, a dtaild hat trasfr aalysis must b mad. Th dimsios of boilrs, hatrs, rfrigrators ad hat xchagrs dpd ot oly o th amout of hat to b trasmittd but rathr o th rat at which hat is to b trasfrrd udr giv coditios. Thrmal systm cotais mattr or substac ad this substac may chag by trasformatio or by xchag of mass with th surroudigs. To prform a thrmal aalysis of a systm, w d to us thrmodyamics, which allows for quatitativ dscriptio of th substac. This is do by dfiig th boudaris of th systm, applyig th cosrvatio pricipls ad xamiig how th systm participats i thrmal rgy xchag ad covrsio. Th ustady rspos of fis is importat i a wid rag of girig dvics icludig hat xchagrs, clutchs, motors ad so o. Hat coductio is icrasigly importat i various aras, amly i th arth scics, ad i may othr volvig aras of thrmal aalysis. A commo xampl of hat coductio is hatig a objct i a ov or furac. Th matrial rmais statioary throughout, glctig thrmal xpasio as th hat diffuss iward to icras its tmpratur. Th importac of such coditios lads to aalyz th tmpratur fild by mployig sophisticatd mathmatical ad advacd umrical tools. Th sctio cosidrs th various solutio mthodologis usd to obtai th tmpratur fild. Th objctiv of coductio aalysis is to dtrmi th tmpratur fild i a body ad how th tmpratur varis withi th portio of th body. Th tmpratur fild usually dpds o boudary coditios, iitial coditio, matrial proprtis ad gomtry of th body. W hy o d to ow tmpratur fild. To comput th hat flux at ay locatio, comput thrmal strss, xpasio dflctio, dsig isulatio thicss, hat tratmt mthod, ths all aalysis lads to ow th tmpratur fild. Th solutio of coductio problm ivolvs th fuctioal dpdc of tmpratur o spac ad tim coordiat. Obtaiig a solutio mas dtrmiig a tmpratur distributio which is cosistt with th coditios o th boudaris ad also cosistt with ay spcifid costraits itral to th rgio. May rsarchrs hav cotributd i ustady hat coductio through fis. Doaldso ad Shouma (197) studid th trasit tmpratur distributio i a covctig straight fi of costat ara for two distict cass, amly, a stp chag i bas tmpratur, ad a stp chag i bas hat flow rat. Th tip of th fi is isulatd. Th authors dvlopd th quatios for th trasit tmpratur distributio ad th hat 70

3 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 flow rat for th two aformtiod cass, ad prst thir rsults graphically. Also icludd is a summary of thir xprimtal wor to vrify thir rsults for th cas of a stp fuctio i hat flow rat. Chapma (1959) who studid th trasit bhaviour of a aular fi of uiform thicss subjctd to a sudd stp chag i th bas tmpratur. His itrst i circular aular fis tmmd from th umrous applicatios of ths typ of fis, spcially o cylidrs of air coold itral combustio gis. Chapma (1959) dvlopd quatios that giv th tmpratur distributio withi th fi, th hat rmovd from th sourc ad th hat dissipatd to th surroudigs, all as fuctios of tim. Ths quatios i graphical form ar vry usful for th dsig girs. Suryaarayaa (1975 ad 1976) also studid th trasit rspos of straight fis of costat cross-sctioal ara. Howvr, rathr tha usig th sparatio of variabls tchiqu followd by Doaldso ad Shouma, h utilizd th Laplac trasforms i ordr to dvlop th solutios for small ad larg valus of tim, wh th bas of th fi is subjctd to a stp chag i tmpratur or hat flux. Th tip of th fi is isulatd i additio, th us of th Laplac trasforms mad it asir for Suryaarayaa to dvlop solutios for th cas of a fi subjctd to a siusoidal tmpratur or hat flux at its bas. Suryaarayaa (1976) has providd a aalysis of th hat trasfr that tas plac from o fluid to aothr sparatd by a solid boudary with fis o o sid. Aziz ad Na (1980) cosidrd th trasit rspos of a smi-ifiit fi of uiform thicss, iitially at th ambit tmpratur, subjctd to a stp chag i tmpratur at its bas, with fi coolig govrd by a powrlaw typ dpdc o tmpratur diffrc. Th choic of a smi-ifiit gomtry abld th trasformatio of th govrig oliar partial diffrtial quatios ito a squc of similarity typ liar prturbatio quatios. Aziz ad Na also discussd th applicability of th rsults to fiit fis. Mao ad Roo (1994) also usd th Laplac trasform mthod to study straight fis with thr diffrt trasits: a stp chag i bas tmpratur; a stp chag i bas hat flux ad a stp chag i fluid tmpratur. Trasit fis of costat crosssctio hav also b studid with th mthod of Gr s fuctios (Bc t al., 199, pp ), a flxibl ad powrful approach that ar applicabl to ay combiatio of d coditios o th fi. Aziz ad Kraus (1995) prst a varity of aalytical rsults for trasit fis, dvlopd by sparatio of variabl ad Laplac trasform tchiqus. Rsults discussd iclud rctagular fis with thr diffrt bas coditios, rctagular fis with powr law covctiv hat loss ad radial fis, alog with svral spcific xampls. Aziz ad Kraus also prst a comprhsiv litratur rviw. Th matrial o trasit fis of costat cross-sctio is also icludd i a boo by Kraus t al. (001, Chap.16). Kim (1976) dvlopd a approximat solutio to th trasit hat trasfr i straight fis of costat cross-sctioal ara ad costat physical ad thrmal proprtis. 71

4 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 Th author utilizd th Katorovich mthod i th variatio formulatio to provid a simpl xprssio of th xact form of th solutio. I som fi applicatios, Nwto s law of coolig is ot applicabl, ad a powr-law typ dpdc of covctiv hat flux o tmpratur bttr dscribs th coolig procss. Such cass iclud coolig of fis du to film boilig, atural covctio, uclat boilig, ad radiatio to spac at absolut zro. Th wor discussd so far has focusd o th trasit rspos of fis of simpl gomtry such as circular aular fis ad straight fis. I additio, svral simplifyig assumptios wr utilizd such as uiform thicss, costat cross-sctioal ara, smi-ifiit lgth, isulatd tip ad small fi thicss-to-lgth ratio to sur o dimsioal hat coductio. Rctly, wor has icludd fis of various shaps ad crosssctios, two ad thr dimsioal hat trasfr, ad practical applicatios of fid hat xchagrs. Campo ad Salazar (1996) xplord th aalogy btw th trasit coductio i a plaar slab for short tims ad th stady stat coductio i a straight fi of uiform crosssctio. Thy mad us of a hybrid computatioal mthod, ow as th Trasvrsal Mthod Of Lis (TMOL), to arriv at approximat aalytical solutios of th ustady-stat hat coductio quatio for short tims i a pla havig a uiform iitial tmpratur ad subjctd to a Drichlt boudary coditio. Th rsultig solutios ar suitabl for obtaiig quality short-tim tmpratur distributios withi th slab wh it is subjctd to a Dirichlt boudary coditio, or a Robi boudary coditio for which th covctiv hat trasfr cofficit is vry larg ad/or th thrmal coductivity of th slab matrial is vry small. I a applicatio typ study, Saha ad Acharya (003) coductd a dtaild paramtric aalysis of th ustady thrdimsioal flow ad hat trasfr i a pi-fi hat xchagr. Th wor was motivatd by th dsir to hac th prform-of compact hat xchagrs, which ar dsigd to provid high hat trasfr surfac ara pr uit volum ad to altr th fluid dyamics to hac mixig. Thr hav b svral umrical studis of trasit fis combid with complicatig factors, such as atural covctio (Hsu ad Ch, 1991; ad Bmadda ad Lacroix, 1996), spatial arrays of fis (Tafti t al., 1999; ad Saha ad Acharya, 004) ad phas chag matrials (Tutar ad Aoca, 004). Thr ar fw publicatios o trasit xprimts for dtrmiig hat trasfr cofficits i fis. Mutlu ad Al-Shmmri (1993) studid a logitudial array of straight fis suddly hatd at th bas. Th istataous hat trasfr cofficit was foud at o poit o th fi as a ratio of th masurd tmpratur to th masur hat flux. Thr ar svral paprs o ivrs tchiqu for dtrmiatio of hat trasfr cofficits from tmpraturs masurd i compact bodis suddly placd i a Covctio viromt (Stolz, 1960; ad Osma ad Bc, 1990). I ths studis, th hat trasfr cofficit is foud from a systmatic compariso btw th trasit data ad a mathmatical modl of th hat coductio i th body of itrst. 7

5 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 EXACT UNSTEADY SOLUTIONS OF FINS FOR THREE DIFFERENT TIP CONDITIONS Cosidr a straight fi iitially i quilibrium with th surroudig fluid viromt at tmpratur T. Th fi has a costat crosssctioal ara, but may b of ay shap (pi, rctagular, tc.). For tim t > 0 a stady hat flux is applid to th bas of th fi. Th tmpratur i th fi satisfis th followig quatios. m T m T T x 1 T x ; 0<x<L...(1) At t = 0, T(x, 0) T = 0...() T x At x = 0, q0...(3) T x At x = L, h T T 0...(4) Quatity m is th fi paramtr giv by hah V. Th boudary coditio at x = L is a gral coditio that rprsts o of thr diffrt tip coditios for th fi. For a tip coditio of th first id, sttig = 0 ad h = 1 rprsts a spcifid d tmpratur (at T = T ). For a tip coditio of th scod id, sttig = ad h = 0 rprsts a isulatd d coditio. For a tip coditio of th third id, sttig = rprsts covctio at x = L. Usually th covctiv cofficit at th d of th fi is ta as sam as that alog th sids of th fi (i.., h = h i gral). Hr, th rsults will b writt out for thr tip coditios. For th tmpraturd coditio (first id), T x, t T q 0 L cos x / L 1 m L 1 xp m L t / L...(5) Whr = ( 1/); For th isulatd d coditio (scod id), T x, t T q m t 1 0 L q0 L cos x / L 1 m L m L 1 xp m L t / L...(6) Whr = ad for covctiv d coditio. T x, t T q 0 L B cos x / L 1 B B m L 1 xp m L t / L...(7) Whr satisfis ta = B Ad whr B = h L/. Each solutio cotais a sris that should b cosidrd i two parts: a trasit part with a xpotial factor ad a stady part with o xpotial factor. Each of th trasit sris cotais a xpotial factor with argumt m L t/l, which dfis th rat of dcay of th ustady. Th dcay rat dpds o fi ffcts (through m L ) ad also o th tip coditio (through ). Th sris solutio for th ustady tmpratur i a flux-bas fi is dvlopd by th mthod of Gr s fuctios. First, a trasformatio (Ozisi, 1993) is usd to rmov th fi trm from th hat coductio quatio. Lt T m t T W...(8) 73

6 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 ad th trasform Equatios (1)-(4) to giv: w x 1 w t ; 0 < x < L...(9) At t = 0, W(x, 0) = 0...(10) At x = 0, W x m t q0...(11) w x At x = L, h W 0...(1) This trasformd problm may b solvd by th mthod of Gr s fuctios i th form (Bc t al., 199, p. 165). W t t 0 0 m t x, t q G x, t x 0, tdt...(13) Th Gr s fuctio associatd with fuctio W is that for a pla wall, giv by Col (008). G x, t x, t 1 x x 0 x N xx x t t / L N...(14) Th first trm (for = 0) is dd oly for a typ (isulatd) boudary at x = L. Eig fuctios X, Eig valus, ad orms N ar dtrmid by th boudary coditios o th fi. For th flux-bas fis of itrst hr, th Eig fuctios ar: X x L cos...(15) Ad th ig valus ad orms ar giv i Tabl 1. Th umbr systm i Tabl 1 for th thr cass listd is XJ whr J = 1,, or 3 to rprst tip coditios of th first id (tmpratur), scod id (isulatd) or third id (covctio), rspctivly. Aftr th tim itgral i Equatio (13) is valuatd, th trasformatio i Equatio (8) ca b rvrsd to fid tmpratur T i th form: T x, t T q m t 1 0 L L q0 L cos x / L 1 N0 m L m L 1 xp m L t / L...(16) Agai, th first trm is oly usd wh th fi tip is isulatd (s Kraus t al., 001, p. 765) for a idpdt drivatio of th isulatd-tip cas). Th abov xprssio, with th ig valus ad orms ar giv i Tabl 1, is limitd to fis with a spcifid hat flux at th bas (x = 0). Howvr, th sam approach could b usd for fis with othr bas coditios with th appropriat pla wall Gr s fuctio. Th pla-wall Gr s fuctios for th tmpratur-bas fi (typ 1 boudary at x = 0) ad th fi with th bas tmpratur applid through a cotact coductac (typ 3 boudary at x = 0) ar availabl lswhr (s Bc t al., 199). Tabl 1: Eig Valus for Thr Diffrt Tip Coditios Cas LN or Eig Coditio X1 1 X ; 0 X3 1; = 0 B B B ta( ) = B Improvmt of Sris Covrgc It has log b ow that classic solutios for th tmpratur i a body hatd o a boudary cotai a slowly covrgig stadystat sris (Ozisi, 1993). I this sctio, th 74

7 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 covrgc of th trasit solutio is improvd by rplacig th stady sris by a fully summd form. Although th stady-fi solutios ar wll ow, a uifid solutio is prstd with th mthod of Gr s fuctios. Th stady tmpratur satisfis th followig quatios. T hah x V T T 0; 0 < x < L...(17) T x At x = 0, q0...(18) T x At x = L, h T T 0...(19) Agai, th boudary coditio at x = L rprsts thr ids of tip coditios. Usig th mthod of Gr s fuctios, th stadyfi tmpratur has th form Col (004). T q xt 0 GX J x, x 0...(0) Th symbol for Gr s fuctio G XJ dots a Cartsia coordiat systm symbol X, boudary of th scod id at x = 0 (symbol ), ad boudary of typ J at x = L (symbol J) for J = 1,, or 3. This umbrig systm is usd to catalog th may GF availabl o th Library wb sit ( Tabl blow shows th ig valus for thr diffrt tip coditios. Gr s fuctio G XJ for th stady-fi is giv by Col (008). G X J, x x R m m L xx m Lx x D xx m x x D Whr D = m (1 R. ml )...(1) Cofficit R is dtrmid by th tip coditio: 1 R 1 ml B ml B Whr B = h L/. typ 1 at x L typ at x L typ 3 at x L Th abov GF may b valuatd at x = 0 ad substitutd ito Th abov tmpratur xprssio, Equatio (0), to giv q L R m Lx 0 x T...() T ml ml R mx Whr cofficit R is giv i th prvious pag. Altratly, stady-fi solutios may b obtaid from computr program TFIN dscribd prviously (Col, 004) that produc aalytical xprssios for th stady tmpratur i fis udr a varity of boudary coditios. Program TFIN is also availabl for dowload at th Gr s Fuctio Library (Col, 008). Nxt th closd-form stady solutios giv abov ar usd i th trasit-fi solutios giv arlir to rplac th slowly covrgig sris by rplacig th sris stady trm with o sris stady trm as obtaid i th Equatio (). Th improvd-covrgc form of th trasit tmpratur i flux-bas fis ar giv by: For th tmpratur tip coditio (first id), T x, t T q0 L mx ml 1 m L ml x 75

8 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 q 0 L cos x / L 1 m L xp m L t / L...(3) Whr = ( 1/) For th isulatd d coditio (scod id), T x, t q0 L T q mx m t q0 L ml m L m Lx ml 1 0 L cos x / L 1 m L xp m L t / L...(4) Whr = ad for covctiv tip coditio (third id) T x, t T q0 L ml B m L ml B ml B ml 1 ml B x ml mx q0 L cos x / L B m L B B 1 xp m L t / L...(5) Whr satisfis ta = B ad whr B = h L/ It is istructiv to xami ths thr tmpratur solutios as a group. Each cotais a stady trm ad ach cotais a additioal trm, a o-sris trasit. Howvr, th isulatd-tip solutio uiquly cotais aothr trm, a o-sris trasit. Quasi Stady Solutio for Short Fis Th short fi is of itrst for our particular applicatio. Th xact tmpratur xprssio for this cas cotais two trms: a sris stady trm ad a sris trasit trm. Th sris cotais a xpotial factor with argumt m L +. By comparig ths argumts, it is clar that as tim icrass th sris trm will dcay mor rapidly. This suggsts that a quasi-stady solutio may b costructd of th form T q (x, t) = T s (x) + T L (t) Hr T s is th stady solutio trm ad T L is th trasit solutio trm. Both th trm cotai sris trm. I quasi stady approach, sris stady trm is big trasformd i to a o sris trm which trasforms th xprssio i to a asily computd algbraic xprssio. Basd o th abov discussio of xpotial argumts, th quasi-stady solutio should b accurat for latr tim. Th umrical rsults giv i th xt sctio ar prstd with th followig dimsiolss variabls: T T X x / L dt t / L q0 L M Bi h v A h Bi A h L v whr = Dimsiolss tmpratur = Dimsiolss locatio = Dimsiolss tim M = Fi paramtr Bi = Biot umbr With ths paramtrs, th dimsiolss quasi-stady tmpratur for short fi is giv by puttig th abov valus i th Equatio (5): 76

9 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 M M M M B M B M M B M B M 1 1 ta cos ta 1 ta M xp M...(6) Ad th dimsiolss xact fi tmpratur for short fi is giv by: 1 1 ta 1 ta ta xp M...(7) 1 Whr satisfis ta = B with dimsio lss paramtr hr = RESULTS AND DISCUSSION Accuracy of Quasi Stady Solutio Th quasi-stady solutio is compard with th xact trasit solutio to dtrmi th coditios udr which th quasi-stady solutio is accurat. Diffrt dimsiolss paramtr wr cosidrd (M, ad locatio) for obtaiig th tmpratur distributio curvs. Whil variatio of o paramtr was cosidrd th othr variabls wr pt costat as idicatd i th graphs. Basd o th aalysis, quasi-stady solutio has b proposd as a accurat solutio at larg dimsiolss tims, which is idpdt of gomtry of th problm. Accuracy of Quasi Stady Solutio for M = 1 at Dimsiolss Locatios 0.0, 0.5 ad 1.0 Figurs 1, ad 3 shows th (dimsiolss) tmpratur vrsus tim at thr diffrt positios o th fi, all for M = 1.0. For all valus of dimsiolss tim th quasi-stady thory stimats th xact valus at x/l = 0 ad x/l = 1.0. For all locatios th agrmt improvs as tim icrass. Figur 1: Tmpratur History i a Fi of Costat Cross Sctio for Both Quasi Stady Thory ad Exact Thory for M = 1 at Locatio x/l = X =

10 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 Figur : Tmpratur History i a Fi of Costat Cross Sctio for Both Quasi Stady Thory ad Exact Thory for M = 1 at Locatio x/l = X = 0.5 Figur 3: Tmpratur History i a Fi of Costat Cross Sctio for Both Quasi Stady Thory ad Exact Thory for M = 1 at Locatio x/l = X = 1.0 Figurs 1, ad 3 shows th (dimsiolss) tmpratur vrsus tim at thr diffrt positios o th fi, all for M = 1. 78

11 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 Th quasi-stady thory stimats th xact valus at x/l = 0.0, 0.5 ad 1.0. For all locatios th agrmt improvs as tim icrass. Accuracy of Quasi Stady Solutio For = X = 0 at Fi Paramtr M = 0., 1.0 ad 5.0 Figurs 4, 5 ad 6 shows tmpratur vrsus tim at X = 0 for M = 0., 1.0 ad 5.0. At M = 5.0 th fi trasit ds quicly so that this fi rachs stady-stat at about = 0.1. As M dcrass th tmpratur distributio tas logr ad logr to rach stady-stat. Fi paramtr M may b itrprtd as a ratio of thrmal rsistacs. Spcifically, M is th thrmal rsistac alog th fi lgth dividd by th covctiv thrmal rsistac from th surfac of th fi. Thus wh M is small, th covctiv thrmal rsistac from th surfac of th fi is larg compard to th thrmal rsistac alog th fi, producig a log, slow trasit. Spcific valus of th dimsiolss tmpratur i th quasi-stady thory for svral valus of dimsiolss tim ad svral valus of fi paramtr M, all at diffrt valus of x/l (dimsiolss locatio). Dimsiolss tmpratur calculatd is th icorporatd ad aalyzd with hlp of graphs for both th thoris. Followigs ar th rsults icorporatd from th graphs. Figurs 4, 5 ad 6 shows tmpratur vrsus tim at X = 0 for M = 0., 1.0 ad 5.0. At M = 5 th fi trasit ds quicly so that this fi rachs stady-stat at about = 0.1. As M dcrass th tmpratur distributio tas logr ad logr to rach stady-stat. Figur 4: Tmpratur History i a Fi of Costat Cross-Sctio for Both Quasi Stady Thory ad Exact Fi Thory at Locatio x/l = X = 0 for M = 0. 79

12 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 Figur 5: Tmpratur History i a Fi of Costat Cross-Sctio for Both Quasi Stady Thory ad Exact Fi Thory at Locatio x/l = X = 0 for M = 1.0 Figur 6: Tmpratur History i a Fi of Costat Cross-Sctio for Both Quasi Stady Thory ad Exact Fi Thory at Locatio x/l = X = 0 for M = 5.0 Fi paramtr M may b itrprtd as a ratio of thrmal rsistacs. Thus wh M is small, th covctiv thrmal rsistac from th surfac of th 80

13 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 fi is larg compard to th thrmal rsistac alog th fi, producig a log, slow ustady stat. CONCLUSION A uifid thory has b prstd for ustady hat trasfr i flux-bas fis for thr tip coditios. Th mthod may b asily xtdd to fis with othr bas coditios. A quasi stady thory has b applid to a cas of straight fi with short lgth tip i th form of a simpl, o-sris xprssio for stady trm. Th quasi-stady thory is simpl ad fficit for computig umrical valus compard to th xact sris solutio. A compariso with a xact sris solutio for th ustady coditio fi shows that th quasi-stady thory is accurat withi dimsiolss tims for all valus of th fi paramtr M. Th rsults show that th quasi-stady fi modl is a simpl way to fid hat trasfr cofficits for largr dimsiolss tims. Complicatd xact ustady solutios ca b simplifid for tmpratur distributio aalysis through Gr s fuctio mthod. A uifid thory is obtaid for ustady hat trasfr i flux-bas fis for thr tip coditios. Basd o th aalysis, th followig coclusios hav b obtaid. For M > 1 th accurat rag xtds to all dimsiolss tims xcpt = 0. Th accuracy icrass for larg dimsiolss tims, whr th ssitivity to hat trasfr cofficit is largst. For M = 0., 1.0 ad 5.0. For = 0 th quasistady thory ovrstimats th xact valus at = X = 0 ad stimats th xact valus at = X = 1.0. For all locatios th agrmt improvs as tim icrass. At M = 5 ad = X = 0 th fi trasit ds quicly so that this fi rachs stady-stat at about = 0.. As M dcrass, th tmpratur distributio tas logr ad logr to rach stady-stat. Wh M is small, th covctiv thrmal rsistac of th fi surfac is larg compard to th thrmal rsistac alog th fi, producig a log, slow trasit. For M 1 ad M > 1. Thr is o rror for dimsiolss tim > 0. th rgio of small rror xtds to arlir tim. Th quasi-stady ad xact tmpraturs agrs closly xcpt at arly tim ( = 0). SCOPE FOR FUTURE WORK It is suggstd that th quasi-stady approach could b succssfully applid to othr fi gomtris with diffrt tip coditios or othr fis for which xact solutios ar difficult to b obtaid. Th rsults show that th quasi-stady fi approach ca b a simpl way to fid hat trasfr cofficit associatd with hat loss. Th hat trasfr cofficits obtaid by this mthod ar itdd for futur us as a xtral boudary coditio for mor laborat thrmal modls. REFERENCES 1. Aziz A ad Na T Y (1980), Trasit Rspos of Fis by Coordiat 81

14 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., 013 Prturbatio Expasio, It. J. Hat ad Mass Trasfr, Vol. 3, pp Aziz A ad Kraus A D (1995), Trasit Hat Trasfr i Extdd Surfacs, Appl. Mch. Rv., Vol. 48, No. 3, pp Bc J V, Col K D, Haji-Shih A ad Litouhi B (199), Hat Coductio Usig Gr s Fuctios Hmisphr, Nw Yor. 4. Bmadda M ad Lacroix M (1996), Trasit Natural Covctio from a Fid Surfac for Thrmal Storag i a Eclosur, Numr. Hat Trasf. A, Vol. 9, No. 1, pp Campo A ad Salazar A (1996), Similarity Btw Ustady-Stat Coductio i a Plaar Slab for Short Tims ad Stady-Stat Coductio i a Uiform, Straight Fi, Hat Mass Trasf., Vol. 31, No. 5, pp Chapma A J (1959), Trasit Hat Coductio i Aular Fis of Uiform Thicss, Chm. Eg. Symp., Vol. 55, No. 9, pp Col K D (004), Computr Programs for Tmpratur i Fis ad Slab Bodis with th Mthod of Gr s Fuctios, Comput. Appl. Eg. Educ., Vol. 1, No. 3, pp Col K D (008), Gr s Fuctio Library, availabl at (Accssd o May 15, 008). 9. Col K D, Tarawh C ad Wilso B (009), Aalysis of Flux Bas Fis for Estimatio of Hat Trasfr Cofficit, It. J. Hat ad Mass Trasfr. 10. Doaldso A B ad Shouma A R (197), Ustady-Stat Tmpratur Distributio i a Covctig Fi of Costat Ara, Appl. Sci. Rs., Vol. 6, Nos. 1-, pp Hsu T H ad Ch C K (1991), Trasit Aalysis of Combid Forcd ad Fr- Covctio Coductio Alog a Vrtical Circular Fi i Micro Polar Fluids, Numr. Hat Trasf. A, Vol. 19, No., pp Kim R H (1976), Th Katorovich Mthod i th Variatioal Formulatio to a Ustady Hat Coductio Ltt., Hat Mass Trasf., Vol. 3, No. 1, pp Kraus A D, Aziz A ad Wlty J (001), Extdd Surfac Hat Trasfr, Wily, Nw Yor. 14. Mao J ad Roo S (1994), Trasit Aalysis of Extdd Surfacs with Covctiv Tip, It. Commu. Hat Mass Trasf., Vol. 1, pp Mutlu I ad Al-Shmmri T T (1993), Stady-Stat ad Trasit Prformac of a Shroudd Logitu-Dial Fi Array, It. Commu. Hat Mass Trasf., Vol. 0, pp Osma A M ad Bc J V (1990), Ivstigatio of Trasit Hat Cofficits i Quchig Exprimt, J. Hat Trasf., Vol. 11, pp Ozisi M N (1993), Hat Coductio, p. 89, Wily, Nw Yor. 18. Saha A K ad Acharya S (003), Paramtric Study of Ustady Flow ad Hat Trasfr i a Pi-Fi Hat Exchagr, It. J. Hat Mass Trasf., Vol. 46, No. 0, pp

15 It. J. Mch. Eg. & Rob. Rs. 013 Tjpratap Sigh t al., Saha A K ad Acharya S (004), Ustady Simulatio of Turbult Flow ad Hat Trasfr i a Chal with Priodic Array of Cubic Pi-Fis, Numr. Hat Trasf. A, Vol. 46, No. 8, pp Stolz G Jr. (1960), Numrical Solutio to a Ivrs Problm of Hat Coductio for Simpl Shaps, J. Hat Trasf., Vol. 8, pp Suryaarayaa N V (1975), Trasit Rspos of Straight Fis, J. Hat Trasf., Vol. 97, pp Suryaarayaa N V (1976), Trasit Rspos of Straight Fis Part II, J. Hat Trasf., Vol. 98, pp Tafti D K, Zhag L W ad Wag G (1999), Tim-Dpdt Calculatio Procdur for Fully Dvlopd ad Dvlopig Flow ad Hat Trasfr i Louvrd Fi Gomtris, Numr. Hat Trasf. A, Vol. 35, No. 3, pp Tutar M ad Aoca A (004), Numrical Aalysis of Fluid Flow ad Hat Trasfr Charactristics i Thr-Dimsioal Plat Fi-ad-Tub Hat Exchagrs, Numr. Hat Trasf. A, Vol. 46, No. 3, pp APPENDIX Nomclaturs Gr A h Surfac ara of fi for covctio (m ) Thrmal diffusivity (m s 1 ) Bi Biot umbr, h i (V/A h )/K Eig valu [Equatio (14)] B Biot umbr, hl/ Dimsiolss tmpratur G Gr s fuctio, X Dimsiolss x-coordiat h Hat trasfr cofficit (W m K 1 ), dt Dimsiolss tim Thrmal coductivity (W m 1 K 1 ) Suprscripts L Lgth of fi (m) q Quasi-stady N Norm [Equatio (14)] (m) s Stady stat m Fi paramtr, (m 1 ) M Dimsiolss fi paramtr = ml q 0 Hat flux (W m ) Q Iput hat (W) T Tmpratur (K) t Tim (s) V Fi volum (m 3 ) W Trasformd tmpratur [Equatio (13)] 83