Numerical Analysis of Radiation Effect on Heat Flow through Fin of Rectangular Profile

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1 Amrican Journal of Enginring Rsarch (AJER) -ISSN: p-issn : Volum-6, Issu-10, pp Rsarch Papr Opn Accss Numrical Analysis of Radiation Effct on Hat Flow through Fin of Rctangular Profil Tay Stphn Mogaji 1 *, FolashadDsola Owosni 1 1 Dpartmnt of Mchanical Enginring, Fdral Univrsity of Tchnology Akur,School of Enginring and Enginring Tchnology, P.M.B. 704, Ondo Stat, Nigria. ABSTRACT:This papr prsnts rports on numrical analysis of radiation ffcts on hat flow through fin of rctangular profil. Mathmatical modl was dvisd by applying th concpt of nrgy balanc on an lmnt of th fin normal to dirction of hat flow. Galrkin mthod was applid to prss th tmpratur distribution along th fin surfac in trms of finit lmnt formulation. Th influnc of physical paramtrs which includ: lngth, thicknss, fin mtal typ and missivity of th fin on tmpratur distribution ovr th rctangular fin profil surfac with and without considring radiation hat loss ar comparativly studid. Th numrical solution rsults rvald that hat dissipation rat for th fin with thrmal radiation is highr than thos without thrmal radiation. By mploying th valu of tmpratur distribution along th fin longitudinal surfac for th cas considring thrmal radiation ffct (Modl 1), apprciabl nhancmnt of th fin thrmal prformanc was obsrvd for aluminium and coppr matrials compard to th stainlss stl matrial.for th cas of cluding radiation hat loss (Modl 2) in fin dsign, th incurrd rrors obsrvd for both numrical and analytical solution mthods is up to 20% of th total hat loss contributd by radiation. Thrfor, th thrmal radiation ffct has to b considrd in th thrmal analysis of fin mtal typ such as aluminium and coppr as nglcting this ffct will rduc th fins hat dissipation capacity rat and hnc th fin prformanc. Kywords:Galrkin mthod, Hat Dissipation, Physical paramtrs,rctangularfin,thrmal Radiation Dat of Submission: Dat of accptanc: I. INTRODUCTION Th dmand for improvmnt in gomtric dsign of hat changr with a viw to dlivr nrgy fficint toward producing compact and miniaturizd quipmnt in particular in th ara of microlctronics and microcooling has promptd svral rsarchrs and scholars in rcnt yars to rsarch in th ara of hat transfr nhancmnt tchniqus both by primntal and numrical analysis. Findings from various studis [1-5] hav obsrvd that high prformanc surfac will nhanc th hat transfr that taks plac within th hat changr without incurring pnaltis on friction and prssur drop that ar nough to ngat th bnfits of hat transfr augmntation. Mannti t al. [6] also pointd out that nw tchnologis ar giving ris to highr oprating tmpraturs on vn smallr dvics, and thrfor th ncssity of bttr hat transfr prformanc surfac for tnding th srvic lif of lctronic and structural componnts.on of th altrnativs to achiv this objctiv is th us of fin attachmnt. Fins ar th tndd surfacs that ar widly utilizd in many nginring systms to incras th hat transfr ara of th systm, dissipat hat gnratd within th systm and prolong thir functionality [5, 7]. Aziz and Bouaziz [8] also rportd that a longitudinal fin of constant cross-sctional ara (rctangular, circular, lliptic, tc.) is widly usd in practic to nhanc hat dissipation from a hatd primary surfac. In many physical situations, th fin is attachd to on sid of a wall of finit thicknss whil th othr sid of th wall is in contact with a hot fluid from which hat transmittd through th wall is ultimatly rjctd by convction and radiation from th surfac of th fin to th nvironmnt (sink). Du to its asy of manufacturing procss, low cost and simplicity of its dsigning, th rctangular fin typ is widly usd and prfrrd among othr various typs of th fins [9-10]. In principl, hat transfr taks plac through conduction within th fin surfac boundaris whil convction and radiation occur btwn its boundaris and surroundings. It is also intrsting to point out that th arrangmnt of this dvic is most ffctiv whn it oprats in a natural convction nvironmnt whr th convction hat transfr cofficint is low. In this circumstanc, th radiativ componnt of hat loss from th fins is comparabl to th natural convction hat loss. Thrfor, th thr mods of hat w w w. a j r. o r g Pag 36

2 transfr, i.., conduction, convction, and radiation should b considrd in thrmal analysis of fin hat sink. Howvr, th rports from many rsarchrs in th opn litratur as pointd out in th study of [11] hav rvald that most of th isting studis, ithr thortical or primntal, faild to rcogniz th ffcts of thrmal radiation by considring only natural convction in th thrmal prformanc analysis of th fin hat sink. As rportd by Hatami, t al. [12] in th ffort to finding th bst approach to solving problms involving rmoval of css damaging hat from systm componnt many rsarch work ar bing mbarkd upon with various tndd surfac gomtris such as in plain fins, wavy and corrugatd channls, offst-strip fins, louvrd fins and vort gnrators.efficincy of horizontal singl pin fin subjctd to fr convction and radiation hat transfr was rportd by Czs ław t al [13].Highr rsults wr obsrvd by th authors for both masurd and numrical solution rsults in comparison with analytical solution rsults. Studis on optimization of variabl cross-sction convctiv pin fins with variabl hat transfr cofficint and tmpratur dpndnc of th thrmal conductivity wr also invstigatd by Rardon and Razani [14] without considring th ffcts of radiation.considring a convctiv radiativ fin tip and allowing th thrmal conductivity of th fin to vary with tmpratur, Chiu and Chn [9] utilizd Adomian s dcomposition procdur to valuat th hat transfr charactristics of a convcting radiating longitudinal fin of rctangular profil. Aziz and Brs-Grn [15] prsntd rports on prformanc and optimum dsign of convctiv radiativ rctangular fin with convctiv bas hating, wall conduction rsistanc, and contact rsistanc btwn th wall and th fin bas. Similarly, considring tmpratur and humidity ratio diffrncs as th driving forcs for hat and mass transfr, Sharqawy and Zubair [16] carrid out analytical solutions for tmpratur distribution ovr th fin surfac whn th fin is fully wt and concludd that th ovrall fin fficincy is dpndnt on th atmosphric prssur thus yild an incras in ovrall fin fficincy with incrasing th atmosphric prssur. Zhang t al. [17] modlld a fin tub vaporator hat changr for ORC cycl and achivd btwn 60% and 70% wast rcovry fficincy for most of th ngin s oprating rgional in thir study involving hat transfr analysis of a finnd tub vaporator for ngin haust hat rcovry. Rcntly, thrmal analysis of convctiv fin with tmpratur-dpndnt thrmal conductivity and hat gnration was carrid out by Ghasmi t al. [10]. Th authors solvd th nonlinar tmpratur distribution quation in th longitudinal fin with tmpratur dpndnt intrnal hat gnration and thrmal conductivity using Diffrntial Transformation Mthod (DTM). Adopting similar mthod considrd by [10], Torabi t al.[18] in thir study, obsrvd dcras in th fin bas tmpratur with incrasing both radiation and convction ffcts. Rsults from prvious study of [19] also rvald that radiation contributs up to 20% of th total hat dissipation from th fin undr natural convction, a situation that can affct th fin thrmal prformanc additionally [20] rportd that for many practical nginring problms th importanc of th fin's wight initiatd an optimization problm such as to maimiz th fin hat dissipation an appropriat fin dimnsion will hav to b dtrmind for a givn fin volum. Thrfor, in th prsnt study, numrical analyss ofradiation ffct on hat flow through fin of rctangular profil taking into account th influnc of physical paramtrs which includ: thrmal conductivity (K), missivity (ε), lngth (L) and thicknss (t) of th fin on fin thrmal prformanc ar studid. Galrkin mthod was applid to provid th analytical solution in trms of finit lmnt formulation. Galrkin mthod is a numrical analysis tchniqu which mainly dals with prcis mathmatical calculation for dtrmining th approimat solution. It taks simplr lmnts from th main problm, calculats sparatly and thus minimizs th probabilitis of rrors. Th obsrvd hat transfr prformanc intrns of tmpratur distributions ovr th rctangular fin profil surfac with and without considring radiation hat loss ar comparativly rportd. II. METHODOLOGY 2. Problm Dfinition Considr a straight fin of rctangular profil dpictd schmatically in Fig. 1 with cross sctional ara A, lngth, L, constant thrmal conductivity, k and surfac missivity, ε. Th fin is attachd to a primary surfac with a constant tmpratur T and losss hat to th surrounding mdium with tmpratur, T b. Th analysis of th problm is basd on th following simplifid assumptions: Stady stat hat conduction. On-dimnsional hat conduction. Th tmpratur at th bas of th fin is uniform. Thr is no hat transfr from th tip of th fin. Th fin radiats according to th Stfan-Boltzmann law. Thr ar no hat sourcs or sinks in th fin. Constant convctiv hat transfr cofficint. w w w. a j r. o r g Pag 37

3 On th basis of th abov assumptions, and assuming that th rat at which nrgy transfrs to th nvironmnt by compound ffct of convction and radiation from any point on th fin surfac must b balancd by th rat at which nrgy rachs that point du to th conduction in transvrs (y, z), thus for th modl taking radiation ffct and convction into considration and th modl without considring radiation, th consrvation of nrgy balanc, yilds th following rspctiv govrning diffrntial quations that must b satisfid by th fin tmpratur. 2.1 Mathmatical Modls Th gnral hat conduction quation in rctangular coordinats according to Fourir Law of hat conduction is givn by prssion (1): K 2 T T T T 2 y 2 y 2 + q = C t (1) Takn into account th abov statd assumptions, Eq. (1) is rduc to K 2 T 2 Thus th conductiv hat nrgy transfr through th fin is obtaind as: q cond. = K 2 T 2 Th convctiv hat nrgy transfr from th fin to th surrounding is obtaind basd on Nwton law of cooling quation givn by: q conv. = T T (4) Th hat nrgy transfr by radiation from surrounding to th fin du to suppos low convctiv hat nrgy transfr from th fin to th surrounding is modlld as: q rad. = T 4 4 T (5) By linarization Eq. (5) bcoms 3 q rad 4ε T m T T. (6) whr, T m = T i+t 2 Th boundary conditions considrd in this study ar as givn in Eqns. (7) and (8): = 0, T = T b (7) = LT = T L (8) With th boundary conditions givn in Eqns. (7) and (8), th gnral govrning diffrntial quation rsults from an nrgy balanc on an lmnt of th fin normal to dirction of hat flow shown in Fig. 1 is prssd as: = d 2 sf A sf T T α + ε A sf T 4 4 T α (9) Taking into account an infinit log fin, Eq. (9) bcoms: d KA 2 T = 3 3 cs d 2 sf A sf T T + 4ε A sf T m T T + t A t T T + 4ε A t T m T T (10) For as of undrstanding th numrical solution of th thortical quation givn in Eq. (10) is dnotd as Modl 1 for fin taking radiation ffct and convction into considration. KA CS d 2 T Finit Elmnt Solution using Galrkin mthod for Modl 1 In th prsnt study, with a viw to analys numrically th radiation ffct givn in Eq. (10), Galrkin finit lmnt mthod was applid to prssth tmpratur distribution along th fin surfac in th form: N i R s d = 0 (11) whr N i and R s is a shap function and rsiduals rspctivly w w w. a j r. o r g Pag 38

4 Th analysis of th shap function N i with rspct to Fig. 2 is obtaind by using Lagrang mthod of intrpolation as follows: N 1 = 2 = l = l = l l l N 2 = = 0 l 0 = l N 2 1 = 1 2 l N 2 2 = l 2 (14) l 2 (15) N 1 N 2 = 2 l l 2 (16) dn 1 d dn 2 = 1 d l = 1 l d R s = KA 2 T cs d 2 sf P T T d 4ε T 3 m P T T d (19) Thus, application of th Galrkin finit lmnt mthod to Eq. (10) yilds: KA 2 d cs N i d sf P 2 N i T T d 4ε T 3 m P 2 N 1 i T T d = 0 (20) 1 1 In th prsnt study, th tmpratur distribution along th fin obtaind in a matri form by intgrating trm by trm, th stiffnss function, th forcing function and th gradint function for th 2-nod lmnt considrd in Fig. 2 ar givn in Eqns. (21), (22) and (23) rspctivly as follows: dt d 2 d K Modl 1 = KA 1 1 cs sf Pl ε PlT 3 m 21 (21) 6 12 f Modl 1 = sf PlT ε PT m 3 T 1 (22) 2 1 dt f gmodl 1 = KA cs N 2 1 d 1 (23) It is intrsting to point out that th stiffnss matri or conductanc matri govrns all th lmnts cpt th last lmnt as th boundary condition (B.C.) at th tip of fin coms to play in addition. Thus, by imposing th boundary condition at th tip of th fin considring th last lmnt, w hav: K BC = (24) 0 h t + 4ε PT m A t 0 f BC= 3 (25) h t + 4ε PT m T l A t Combining Eqns. (21) to (25) yild gnral matri form of th lmnt as follows: K Modl 1 = KA 1 1 cs sf Pl ε PlT 3 m (26) 0 h t + 4ε PT m A t and f Modl 1 = sf PlT ε PT m 3 T (27) h t + 4ε PT m T l A t Tmpratur distribution at ach nodal point along th fin is obtaind with th approimat solution givn as: n T = v=1 N i T i (28) For a 2- nod lmnt, n=2 2 T = i=1 N i T i = N 1 T 1 + N 2 T 2 (29) dt = dn 1 T d d 1 + dn 2 T d 2 (30) Combining Eqns. (26) to (27) and substitut for T and dt prssd by Eqns. (29) and (30), w hav: 2 dn i d d d KA CS N 1 T 1 + N 2 T 2 d + sf P N i N 1 T 1 + N 2 T 2 d + 4 PT 1 1 M N i N 1 T 1 + N 2 T 2 d 3 sf PT N i d + 4 PT 2 dt M N i d + KA cs N 2 i 1 d d 1 2 = 1 (12) (13) (17) (18) (31) w w w. a j r. o r g Pag 39

5 In ordr to vrify th ffct of th radiation on ffctivnss of fins application for hat dissipation in thrmal systm considrd in this study, gnral form of th lmnt matri quation was also obtaind for th fin without considring radiation hat loss. Thus, th gnral diffrntial quation rsults from nrgy balanc on an lmnt of th fin normal to dirction of hat flow without considring radiation hat loss is prssd as follows: = d 2 sf A sf T Tα (32) Also for as of undrstanding th numrical solution of thortical quation givn in Eq. (32) is dnotd as Modl 2 for th fin without considring radiation hat loss. KA CS d 2 T Finit Elmnt Solution using Galrkin mthod for Modl 2 Application of th Galrkin finit lmnt mthod following thsam procdur givn in Eqns. (28) to (32) yilds: KA 2 d cs N i d sf P 2 N i T T d = (33) 1 1 dt d 2 d Thus, th gnral matri forms of th lmnt for th fin without considring radiation hat loss ar obtaind as: K Modl 2 = KA 1 1 cs sf Pl 21 (34) 6 12 and f Modl 2 = sf PlT 1 (35) 2 1 Equating Eq. (34) to (35) and applying th approimat solution givn in Eq. (28), th tmpratur distribution at ach nodal point along th fin without considring radiation hat loss is stimatd using Eq. (36): KA 2 dn i d CS N 1 d d 1 T 1 + N 2 T 2 d + sf P 2 N i N 1 T 1 + N 2 T 2 d = sf PT 2 N i d (36) 1 1 Th Mathmatical modls dscribd abov wr implmntd in MATLAB softwar to prdict th rspons of hat flow through fin of a rctangular profil considring both Modl 1 and 2. i.th fin with and without considring radiation hat loss. Th softwar input data ar, oprating tmpratur (T b ), ambint tmpratur (T ), convction hat transfr cofficint(h), fin missivity ( ε), Stfan-Boltzmann constant ( ), thrmal conductivity of th matrial (k), lngth of th matrial(l), th primtr of th fin(p) and th sctional ara of fin(ac). It is intrsting to point out that th ffct radiation on hat flow through th fin of th rctangular profil can b valuatd using Galrkin mthod by considring Eqns. (31) and (36) rspctivly, thus th tmpratur distribution along th longitudinal dirction can b achivd with ths modls. In this study, th fin is dividd uniformly into twnty (20) finr lmnts with a total of twnty-on (21) nods in ordr to achiv an accurat numrical solution. Th simulation of both Modl 1 and 2 ar charactrizd by tmpratur distribution taking into considration th influnc of physical paramtrs which includ: lngth, thicknss, fin mtal typ and missivity of th fin on th prformanc of th systms. III. RESULTS AND DISCUSSION In this sction, ovrviw of th fin thrmal prformanc basd on th us of numrical modl dvlopd to valuat th significanc of thrmal radiation ffcts on hat flow through fin of a rctangular profil ar comparativly rportd. Numrical data wr gnratd basd on th influnc of physical paramtrs which includ: lngth (L), thicknss (t), fin mtal typ and missivity (ε) of th fin on tmpratur distribution ovr th rctangular fin profil surfac with and without considring radiation hat loss. For as of th discussion, Modl1 and Modl 2 dnot trms for th fin profil surfac with and without considring radiation hat loss rspctivly. 3.1 Comparativ Analysis Shown in Figurs 3a and 3b is th variation of th fin lngth with th fin bas tmpratur at 100 and 500 o C rspctivly. In ths figurs a fin mtal typ mad of Aluminium was considrd with assumd fin thicknss of m. As can b sn in ths figurs, th modl taking radiation ffct and convction into considration (Modl 1) showd a bttr hat transfr prformanc than th on without thrmal radiation (Modl 2); morovr, it can b notic that by applying th modl with radiation ffct for highr fin bas tmpratur valu (Figur 3b) incrass th hat dissipating rat of th dvic. This bhaviour implis that highr powr dissipation by th dvic on which th fin is attachd.g (lctronic componnts) can b achivd by considring thrmal radiation ffcts and consquntly improving th fficincy of th thrmal systm. In addition, th fin lngth usd for th fin bas tmpratur distribution also play an important rol in th thrmal prformanc of th fin, sinc with incrasing th fin lngth, th fin bas tmpratur also dcrass along th longitudinal dirction of th fin indpndntly of th two modls considrd in this study. Ths bhaviours ar consistnt with thos obsrvd in th study of Safayt-Hossaint al [21]. w w w. a j r. o r g Pag 40

6 Fig3.Tmpratur variation in th fin for various valus of fin Lngth obtaind using Galrkin mthod. Figur 4 prsnts th numrical data for variations of tmpratur distribution along th longitudinal dirction of th fin at a bas tmpratur of 500 o C. Th numrical data wr gnrat for diffrnt fin thicknss valus of 0.01, and (m) rspctivly to valuat th ffcts of fin thicknss on th fin hat dissipation rat bhaviour. It is intrsting to point out that in practical contt; th bas tmpratur is normally rgardd as th oprating tmpratur of lctronic componnts (Khor t al. [11]). From this figur, it is obsrvd that th fin hat dissipation rat virtually incrass with rducing th fin thicknss for th givn bas tmpratur. It can also b noticd that th hat dissipation rat for th cas associatd with considring radiation hat loss (Modl 1) is highr compard to that without considring radiation hat loss (Modl 2). This bhaviour which is th consqunc of th additional accountd radiation loss from th fin surfac apart from th convction loss is obsrvd to b mor pronouncd with th lowst fin thicknss valu considrd in this study. This is concivabl and profitabl as highr powr dissipation ndd for a compact and miniaturizd thrmal quipmnt such as for a rducd fin thicknss obsrvd in this study will giv ris to highr oprating tmpratur and subsquntly improv th safty of th thrmal systm. w w w. a j r. o r g Pag 41

7 Fig 4.Tmpratur variation in th fin for various valus of fin thicknss obtaind using Galrkin mthod. Variations of tmpratur distribution along th longitudinal dirction of th fin considring th ffct of th fin mtal typ on th fin thrmal prformanc bhaviour is displayd in Figurs 5a and 5b. Commonly usd fin mtal typ such as Coppr, Aluminium and Stainlss stl ar considr for comparison in th prsnt study. Numrical primntation wr carrid out for fin bas tmpratur of 100 and 500 o C rspctivly basd on th matrials rspctiv thrmal conductivity valus. It should b highlightd that, thrmal conductivity proprty of a matrial play an important rol on hat transportd from th bas to th tip by conduction insid th fin, concurrnt hat dissipation by convction and radiation to th surroundings taks plac on th fin surfacs. From ths figurs, it can b notic clarly that indpndntly of th fin mtal typ, th thrmal prformanc of fin for th cas considring thrmal radiation ffct (Modl 1) is highr than th cas of cluding thrmal radiation ffct (Modl 2), this bhaviour is obsrvd to b mor pronounc for th fin with highr bas tmpratur valu bing that at high oprating tmpratur, th hat transportd to th hat sink is high and hnc mor hat is dissipatd from th fin to th surroundings and vic vrsa. By mploying th valu of tmpratur distribution along th fin longitudinal surfac for th cas considring thrmal radiation ffct (Modl 1) as th basis for comparison with rspct to th fin typ mtal, it can b obsrvd that whn thrmal radiation is nglctd (Modl 2), th hat dissipation rat with th us of stainlss stl matrial is undrratd with tmpratur distribution (T o C) valu of up to 15.6% at a rducd fin lngth, L= 0.2 m. Similar bhaviour is obsrvd for Aluminium and Coppr matrials at fin lngth of 0.25 m but with undrratd tmpratur distribution (T o C) valu of up to 21.5 and 21.8% rspctivly, th obsrvd diffrncs in th fin thrmal prformanc rsults and as pointd out in th study of Torabit al. [18] is du to th ffcts of thrmal conductivity on hat flow through th rspctiv matrial considrd in this study. It can b infrrd that for a rquird high hat dissipation procss towards th nds of compact and miniaturizd quipmnt in thrmal systm application such as cooling of lctronic dvics, stainlss stl fin matrial could b applicabl providd th matrial cost implication is not put into considration, i. for spcial thrmal application such as in Nuclar ractor systm and othrs. Also, for a rducd production cost with improving thrmal systm fficincy by adopting th cas with thrmal radiation ffct modl proposd in this study, Aluminium fin matrial could b usful. This rsult also justifis on of th rasons why Aluminium fin matrial is commonly usd as fin in practical contt. w w w. a j r. o r g Pag 42

8 Fig5.Tmpratur variation in th fin for diffrnt fin mtal typs obtaind using Galrkin mthod. Th fin thrmal prformanc bhaviour on ffct of incras in th fin matrial missivity is also displayd in Figurs 6 to 8. It is clarly show that fin hat dissipation rat incrass with incrasing missivity valu for various fin mtal typ considrd in this study. Howvr, for th cas with thrmal radiation (Modl 1), th ffct of incrasing missivity valu for stainlss stl matrial is obsrvd to hav ngligibl ffct in incrasing th fin thrmal prformanc compard to that with Aluminium and Coppr matrials which gav apprciabl rspons in nhancing th fin thrmal prformanc as also obsrvd in Fig. 5b, this is mainly du to th fact that stainlss stl is a mtal of high missivity valu in natur. This bhaviour agrd with Khort al.[11] who rportd that th rol of fin bcoms lss notabl spcially whn th missivity of th fin surfac is high, and whn th thrmal radiation hat loss is takn into considration. Fig6.Tmpratur variation in th fin mtal typ (Coppr matrial) for various valus of missivity obtaind using Galrkinmthod. w w w. a j r. o r g Pag 43

9 Fig7.Tmpratur variation in th fin mtal typ (Aluminium matrial) for various valus of missivity obtaind using Galrkinmthod. Fig8.Tmpratur variation in th fin mtal typ (Stainlss stl matrial) for various valus of missivity obtaind using Galrkin mthod. 3.2 Fin thrmal prformanc valuation In ordr to show mor clarly th ffct associatd with cluding radiation hat loss in thrmal prformanc of a rctangular fin profil, analytical calculations ar carrid out to lucidat such ffct in th fin hat dissipation rat capacity by considring a situation whr hat flow through a fin mtal by Aluminum, (Al) taking Fig.1 as a cas study having a thicknss, t = 0.002m, width, w= 0.01m and lngth, L=0.25m is to dissipat hat from th surfac of an lctronic dvic having a spcific oprating tmpratur for a particular cooling application. Th fin is posd to ambint air and its radiating and convcting hat to th nvironmnt at tmpratur, (T ) of 25 ºC, and convction hat transfr cofficint, (h t ) along th lngth and th nd is 20W/m 2 K, th fin missivity,ε =1 and th Stfan-Boltzmann constant, =5.7E-08 W/ m 2 K 4. Dtrmin th hat dissipation prformanc of th solid fin for diffrnt oprating tmpratur, T b ranging from 100 to 500 ºC with and without radiation hat loss. To solv th cas study, th hat dissipation prformancs for th cass without thrmal radiation (Q 1 ) and th cas with thrmal radiation (Q 2 ) wr valuatd using th act solution mthod according to Brgmant al. [22] givn by: Q 1 = P 1 ka Sinm 1L+ 1 mk Cosm 1 L Cosm 1 L+ 1 m1 k Sinm 1L T b T (37) w w w. a j r. o r g Pag 44

10 Q 2 = P 2 ka Sinm 2L+ 2 m2 k Cosm 2L Cosm 1 L+ 2 m2 k Sinm 2L T b T (38) whr T b is th bas surfac tmpratur of th fin, T is th ambint air surrounding tmpratur of th fin, P = 2 w + t is th primtr of th fin, A c = wt is th sctional ara of fin, sf is convctiv hat transfr cofficint of surrounding mdium and K=K Al is th fin mtal typ thrmal conductivity Thus, h 1 = sf + t (38) 3 h 2 = sf + t + 12ε T m (39) m 1 = P 1 KA c (40) m 2 = P 2 (41) KA c By using th valu of Q 2 as th basis for comparison, th rlativ dviation (in prcntag) of th prtinnt paramtrs is quantifid as: δ = Q 1 Q (42) Q 2 In Eq. (42), δcan b rgardd as th rror incurrd by th clusion of thrmal radiation hat loss from th fin surfac. Th analytical solution rsults of th aformntiond cas study using Eqns. (37) and (38) and th rlativ dviation (in prcntag) valu obtaind with th us of Eq. (42) ar prsntd in Tabl1.Th rsults displayd in Tabl 1 clarly show that by considring thrmal radiation ffct in fin dsign, a significant nhancmnt in th fin thrmal prformanc is achivabl. Ths rsults justify th rason for bttr prformanc of th fin for th cas of considring thrmal radiation ffct (Modl 1) compard to th cas of cluding thrmal radiation ffct (Modl 2) as obsrvd with th numrical solution rsults discussd in sction 3.1. Also, it can b sn that undr th sam oprating tmpratur condition of 100 o C, about 20% of th total hat dissipatd might b contributd by radiation, a rsult which is similar to th numrical rsults obtaind in this study and that obsrvd in th study of [19]. Morovr, th rror valu incurrd by th clusion of thrmal radiation hat loss is obsrvd to b mor pronouncd with incrasing th systm oprating tmpratur as shown in Tabl 1and agring with Ra and Wst [23], ths authors whos rsarch work involvd thrmal radiation from finnd hat sinks claimd that, dpnding on th hat sink dsign, oprating tmpratur and ambint nvironmnt, 25% of th total hat dissipatd from th hat sink might b contributd by radiation. Tabl 1.Shows th rsults of th fin prformanc valuation and th analytical solutions approach Tmpratur [ o C] Q 1 Q 2 δ[%] IV. CONCLUSION Numrical analysis of hat flow through fin of a rctangular profil surfac with and without considring radiation hat loss was carrid out in this work. Th ffcts of physical paramtrs which includ: lngth, L, thicknss, t, fin mtal typ and missivity, ε, on th fin thrmal prformanc ar comparativly studid. Th following conclusions can b drawn from th prsnt study: 1. It is obsrvd that hat dissipation rat for th fin with thrmal radiation is highr than thos without thrmal radiation indpndntly of th fin typ mtal considrd in this study. Th cass of cluding thrmal radiation actually undrrat th fin thrmal prformanc in th rang of 15.6%, 21.5% and 21.8% rspctivly for th stainlss stl, th aluminium and th coppr matrials, 2. For th ffct of incrasing th fin matrial missivity subjctd to th cass of considring radiation hat loss, apprciabl nhancmnt of th fin thrmal prformanc was obsrvd for aluminium and coppr matrials compard to stainlss stl matrial. It is argu that this phnomnon is rlatd to th high missivity proprty possss by th stainlss stl which rndr th pctd nhancmnt of th fin matrial missivity for th cas with thrmal radiation lss notabl. 3. By comparing th fin with thrmal radiation and thos without thrmal radiation, it was obsrvd a bttr hat transfr prformanc with incrasing th fin lngth and th fin oprating tmpratur. This implis that highr powr dissipation can b achivd by considring thrmal radiation ffct and consquntly improving th fficincy and safty of th thrmal systms. w w w. a j r. o r g Pag 45

11 4. Enhancmnt of hat dissipation rat with rducing th fin thicknss was obsrvd for high oprating tmpratur rgardlss of th two numrical modls considrd in this study. This bhaviour was obsrvd to b mor pronouncd for th fin with thrmal radiation hat loss considration. ACKNOWLEDGEMENTS Th authors would lik to acknowldg th assistanc of Mr O. Z. Ayodjiof th Dpartmnt ofmchanical Enginring, Fdral Univrsity of Tchnology Akur, Ondo Stat, Nigria and th contribution of Mr N.O. Adwunmiand Mr.O.D.Owosniof th Dpartmnt ofmchanical Enginring, Univrsity of Lagos, Akoka, Yaba, Lagos in supplying th ndd information usd in th prsnt study. REFERENCES [1]. Kiwan S, Effct of radiativ losss on th hat transfr from porous fins, Intrnational Journal of Thrmal Scinc, Vol. 46, pp , [2]. Hatami M, andganji D D, Thrmal prformanc of circular convctiv radiativ porous fins with diffrnt sction shaps and matrials, Enrgy Convrsion and Managmnt,Vol.76, pp , 2013 [3]. Hatami M, HasanpourA&Ganji D.D, Hat transfr study through porous fins (Si3 N4 and AL) with tmpratur-dpndnt hat gnration. Enrgy Convrsion and Managmnt, Vol.74, pp.9 16, [4]. Mogaji T. S., Kanizawa F.T., BandarraFilho E. P. andghrhardt R, Eprimntal study of th ffct of twistd-tap insrts on flow boiling hat transfr nhancmnt and prssur drop pnalty, Intrnational Journal of Rfrigration,Vol.36, pp , 2013 [5]. Hatami M and Ganji D.D, Thrmal and flow analysis of microchannl hat sink (MCHS) coold by Cu watr nanofluid using porous mdia approach and last squar mthod. Enrgy Convrsion and Managmnt, Vol.78,pp , [6]. Mantti L.L, Mogaji T.S, Bck P.A. and Cardoso E.M, Evaluation of th hat transfr nhancmnt during pool boiling using low concntrations of Al 2O 3-watr basd nanofluid, Ep. Thrm. Fluid Sci. (2017), [7]. Hatami M, Hasanpour A, Ganji D.D, Hat transfr study through porous fins (Si3N4 and AL) with tmpratur-dpndnt hat gnration. Enrgy Convrsion Managmnt, Vol.74,pp. 9-16, [8]. Aziz A. and Bouaziz M.N., A last squars mthod for a longitudinal fin with tmpratur dpndnt intrnal hat gnration and thrmal conductivity, Enrgy Convrsion and Managmnt, 52, , [9]. Yu LT, and Chn C.K., Application of Taylor Transformation to Optimiz Rctangular Fins with variabl Thrmal Paramtrs,Applid Mathmatical Modling, Vol.22, pp.11 21,1998. [10]. Ghasmi S E, Hatami M and Ganji DD, Thrmal analysis of convctiv fin with tmpratur-dpndnt thrmal conductivity and hat gnration, Cas Studis in Thrmal Enginring Vol. 4, pp.1 8, [11]. Khor Y.K., Hung Y.M. and Lim B.K, On th rol of radiation viw factor in thrmal prformanc of straight-fin hat sinks, Intrnational Communications in Hat and Mass Transfr, Vol.37, pp , 2010 [12]. Hatami M, GanjiD.D. andgorji-bandpym, Numrical study of finnd typ hat changrs for ICEs haust wast hat rcovry, Cas Studis in Thrmal Enginring, Vol.4,pp , [13]. CzsławO.P.andJanuszW,Efficincy of th Horizontal Singl Pin Fin Subjctd to Fr Convction and Radiation Hat Transfr, JournalHat Transfr Enginring, Vol.28, pp , [14]. Rardon J, and Razani A, Th optimization of variabl cross-sction spins with tmpratur dpndnt thrmal paramtrs. Intrnational Communication in Hat and Mass Transfr, Vol.19, pp , [15]. Aziz A. and Brs-Grn A.B, Prformanc and optimum dsign of convctiv radiativ rctangular fin with convctiv bas hating, wall conduction rsistanc, and contact rsistanc btwn th wall and th fin bas, Enrgy Convrsion and Managmnt, Vol.50,pp , [16]. Sharqawy MH. and Zubair S.M.,Efficincy and optimization of straight fins with combind hat and mass transfr An analytical solution, Applid Thrmal Enginring, Vol.28, pp , [17]. Zhang H.G, Wang E.H and Fan B.Y., Hat transfr analysis of a finnd tub vaporator for ngin haust hat rcovry, Enrgy Convrsion and Managmnt, Vol.65, pp , [18]. Torabi M.Yaghoobi H., andkiani M R. Thrmal Analysis of th Convctiv-Radiativ Fin with a Stp Chang In Thicknss and Tmpratur Dpndnt Thrmal Conductivity, Journal of Thortical and Applid Mchanics, Vol.51, pp , [19]. Rao V.D, Naidu S.V., Rao B.G. and Sharma K.V., Hat transfr from a horizontal fin array by natural convction and radiation - a conjugat analysis, Intrnational Journal of Hat and Mass Transfr, Vol.49, pp , [20]. Razlos P. and Kakatsios X, Optimum dimnsions of convcting radiating fins:part I- longitudinal fins, Applid Thrmal Enginring, Vol.20, pp [21]. SafaytHossain M.D, Raiyan M.F, Sayd S, and Ahamd J.U, Analysis of Thrmal Charactristics of Flard and Rctangular Fin Profils by Using Finit Elmnt Mthod, IOSR Journal of Mchanical and Civil Enginring (IOSR-JMCE),12, [22]. Brgman T. L.,Lavin A. S., Incropra,F. P., DWittD. P., Fundamntals of Hat and Mass, Transfr.7 th Edition, [23]. Ra S.N and Wst, S.E, Thrmal radiation from finnd hat sinks, IEEE Transactions on Parts, Hybrids, and Packaging, Vol.12, pp ,1976. Tay Stphn Mogaji. Numrical Analysis of Radiation Effct on Hat Flow through Fin of Rctangular Profil Amrican Journal of Enginring Rsarch (AJER), vol. 6, no. 10, 2017, pp w w w. a j r. o r g Pag 46