EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS

25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut of Scinc & Tchnology, Opp. IIM, Indor Rau- Pithampur Road, Rau, Indor *Author for Corrspondnc ABSTRACT Ovr th yars fficincy of a fin has bn usd as a masur of its prformanc. Howvr, if hat transfr through fins is givn mor importanc, fin ffctivnss is mor rlvant for this purpos. Variation of ths two paramtrs is of opposit natur with rspct to fin gomtry paramtrs. In this work a comparison of ths two prformanc paramtrs is mad for diffrnt gomtrical proportions of rctangular circumfrntial fins. Hat transfr rat pr unit volum of fin is idntifid as a nw paramtr for assssmnt of fin prformanc and dsign. Importanc of this paramtr is stablishd through study of variation of its variation with fin gomtry paramtrs. Th mthod of analysis prsntd in this work may b usd to nsur dsign adquacy of fins. Th ovrall conclusion of this study is that dsignd should air at thin and short fins for bst rsults. Kywords: Fins; Rctangular Circumfrntial Fin; Fin Efficincy; Fin Effctivnss Notations h Convctiv hat transfr cofficint at fin surfac k Thrmal conductivity of fin matrial 2t m k q Hat transfr rat r Radius at arbitrary locatd point on th fin r Bas radius of fin r Outr radius of fin t Tmpratur at a point locatd at radius r t Tmpratur at root (bas radius) of fin t s Ambint tmpratur E Fin ffctivnss I, I Bssl s function of first kind and its drivativ, rspctivly K, K Bssl s function of scond kind and its drivativ, rspctivly Fin thicknss t - t s t - t s r /r Fin fficincy INTRODUCTION Hat transfr through rctangular circumfrntial fins is dtrmind by taking into account th conduction through th matrial followd by convctiv hat transfr to th surroundings. Th quations thus obtaind yild a solution involving Bssl s functions (Krn, 972). Boundary layr mthod has also bn usd in arly rsarch to obtain altrnativ solutions in polynomial form (Bharani t al., 992). Both fficincy and ffctivnss of a fin hav bn idntifid as th guiding paramtrs for assssmnt of adquacy of its dsign (Bharani t al., 992; Kraus t al., 2). But th variation of ths two paramtrs is of opposit natur for diffrnt paramtrs dfining fin gomtry lik fin volum, fin lngth, and fin Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 37

25 Vol. 3 () January-March, pp.37-5/tripathi thicknss. Hnc, th dsignr has to compromis ovr final valus of ths two paramtrs. Exprimntal tchniqus hav also bn usd in th past to obtain th optimum solution to this problm (Rizzi and Catton, 23). Howvr, application orintd approach clarly indicats that mor mphasis should b givn to ffctivnss in dsign of finnd surfacs for hat transfr (Bharani t al., 992; Razlos, 23; Pawar t al., 25). Th prsnt work xamins fficincy and ffctivnss, and also introducs hat transfr rat pr unit volum of fin as a nw paramtr for assssing adquacy of fin dsign. Th mthod of analysis prsntd in this work may b usd to nsur dsign adquacy of fins. Th ovrall conclusion of this study is that dsignd should air at thin and short fins for bst rsults. Analysis of Rctangular Circumfrntial Fin This sction prsnts th mathmatical foundation for th analysis of rctangular circumfrntial fins, followd by sampl calculation. Mathmatical Modl Figur shows th rctangular circumfrntial fin. Th hat transfr through such fin is dscribd by th following diffrntial quation 4, 2 2 d d 2 2 r m r () 2 dr dr Th gnral solution of this quation is, = C I (mr) + C 2 K (mr) (2) Assuming no hat transfr through th dg of th fin, th solution is, K( ) I( mr) I( ) K( mr) (3) I ( mr ) K ( mr ) I ( mr ) K ( mr ) Th hat transfr rat is givn by, I ( ) K( mr ) K( ) I( mr ) q 2 k r m (4) I( mr ) K( ) I( ) K( mr ) Effctivnss is givn by, km K( mr ) I E h I( mr ) K ( ) ( mr ) K( ) I I ( mr ) K ( mr ) ( mr ) (5) Efficincy is givn by, 2 r km 2 2 2 ( r r ) h I ( mr ) K( ) I ( mr ) I ( mr ) K ( mr ) ( mr ) (6) I ( ) K I ( mr ) K Hnc, 2 ( ) I K K I ( ) ( ) ( ) ( ) I K I K ( ) ( ) ( ) ( ) (7) Whr, Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 38

25 Vol. 3 () January-March, pp.37-5/tripathi 2 2h r ( ) (8) k Sampl Calculations Th following paramtr valus ar usd for th sampl calculations prsntd in this sction, h=2 W/m 2 K, k=4 W/mK, r i =.25m, δ =.2 m, volum of fin 6 cc. 2 V 2 Outr radius of fin, r ri = r =.56 m V Hat dissipation ara of fin = =.8 m 2 r Charactristic lngth L = L i = 2.5, L o = 28 h Biot numbr B i =.5 2k 2h Also, m =22.366 k Th radius ratio is, =.4464 And, r 2h =.6932 k Now hat transfrrd can b calculatd by, I ( ) K( mr ) K( ) I( mr ) q 2 k r m I( mr ) K( ) I( ) K( mr ) For, θ o =, q = 25.89 W Thn fficincy is, I 2 K K I ( ) ( ) ( ) ( ) =.8685 ( ) I K I K ( ) ( ) ( ) ( ) Effctivnss is, km K ( mr ) I( ) K( ) I( mr ) E = 4.4 h I( mr ) K( ) I( ) K( mr ) RESULTS AND DISCUSSION In th prsnt study th fin thicknss was varid from 2 mm to mm in intrvals of 2 mm. Th prformanc paramtrs usd in th study ar fin fficincy and fin ffctivnss. Th matrial volum rang was varid from 8 cc to 8 cc in qual stps. Effct of Outr Radius of th Fin on Prformanc Paramtrs Th variation in prformanc paramtrs for constant fin thicknss and bas radius is shown in Tabl. Figur 2 shows th variation of prformanc paramtrs with rspct to th outr radius of fin, for a constant fin thicknss of 2 mm and a constant bas radius of 25 mm. Th fficincy dcrass from 8% Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 39

25 Vol. 3 () January-March, pp.37-5/tripathi to 3% whil th ffctivnss incrass from 4 to 74. Figur 3 shows variations for fin thicknss of 4 mm with bas radius of 25 mm. Hr th fficincy initially incrass from 8% to 9% and thn dcrass to 67%, whil th ffctivnss incrass stadily from 39 to 74. Figur 4 shows th variations for fin thicknss of 6 mm and bas radius of 25 mm. Hr th dcras in fficincy is from 99% to 85% accompanid by an incras in th ffctivnss from 7 to 25. Figur 5 corrsponds to fin thicknss of 8 mm and bas radius of 25 mm. Th fficincy is sn to dcras from 99% to 92% and ffctivnss is sn to incras from 4 to 5. Figur 6 shows variations for fin thicknss of mm and bas radius of 25 mm. Th fficincy in this cas dcrass from 99.5% to 96% whil th ffctivnss incrass from 3 to. Ths rsults clarly show that th fficincy and th ffctivnss vary in opposit mannr whn th outr radius of th fin (and hnc th matrial volum) is incrasd. It can b notd that th hat transfr rat incrass continuously with incras in outr radius (and hnc th matrial volum), for constant fin thicknss and constant bas radius. Thus, in ordr to gt maximum hat transfr rat, th outr radius should b maximum. Howvr, as indicatd by th variation of th ffctivnss, both th curvs of fficincy and ffctivnss bcom flattr with incrasing valu of outr radius or th matrial volum. Thus th rlativ advantag of largr outr radius of fin (and largr matrial volum) diminishs continuously with incras in outr radius. Also, as vidnt from th graph, th fficincy shows a continuous dcras with incrasing outr radius. Sinc th basic purpos of a fin is to maximiz th hat transfr from th parnt body, it is obvious that a largr valu of ffctivnss is mor rlvant in assssmnt of fin prformanc. It is also quit obvious from ths graphs that th ffctivnss is high for highr outr radii (largr matrial volum) of th fin, for which th fficincy is lowr. Similarly, for lowr valus of outr fin radii (smallr matrial volum), th fficincy is highr but th ffctivnss is low. Effct of Fin Volum on Prformanc Paramtrs Th ability of hat transfr through fin is rflctd by its ffctivnss. Furthr, th cost of manufacturing, among othr factors, mainly dpnds upon th quantity of th matrial. Thrfor dsign adquacy of a fin gomtry, both in trms of hat transfr rat and cost ffctivnss, can b ascrtaind through study of variation in prformanc paramtrs with variation in fin thicknss, kping th matrial volum constant. Th prformanc paramtrs wr dtrmind for fin thicknss rang of 2 mm to mm, for constant bas radius of 25 mm, and ar prsntd in Tabl 2. Figur 7 shows variation of prformanc paramtrs for constant matrial volum of 6 cc, Figur 8 for 32 cc, Figur 9 for 48 cc, Figur for 64 cc and Figur for 8 cc. Hr again it is obsrvd that ffctivnss and fficincy vary in opposing mannr as th fin thicknss is incrasd. Also, it is obsrvd that a largr fin thicknss givs highr fficincy but lowr ffctivnss. Kping in mind that ths variations ar for a constant volum, and thrfor a constant matrial and manufacturing cost in gnral, it is obvious that bst ffctivnss can b obtaind by kping th fin thicknss at th minimum. Th practical limits on minimum fin thicknss will b imposd by limitations of th manufacturing procss, which is mostly prssur di-casting ths days. Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 4

25 Vol. 3 () January-March, pp.37-5/tripathi Tabl : Gomtrical & Prformanc Paramtrs (Bas Radius 25 mm, Biot No. 5x 4 ) Fin Thicknss, Outr Radius, Ara, Volum, Hat Effctivnss Efficincy mm cc Transfr mm mm 2 Rat, W/m 2 2.56 8 6 25.82 4. 8.69.76 6 32 37.78 6.6 77.39.9 24 48 42.93 68.36 44.72.4 32 64 45.34 72.2 35.42.6 4 8 46.55 74.3 29. 4.44 4 6 5.42 2.27 8.69.56 8 32 28.54 22.72 89.7.67 2 48 38.97 3.4 8.22.76 6 64 47. 37.5 73.59.84 2 8 53.35 42.48 66.69 6.38 26.67 6.54 5.59 98.78.48 53.33 32 2.48.87 96.2.56 8. 48 29.59 5.7 92.48.63 6.67 64 37.79 2.6 88.56.7 33.33 8 45.8 23.93 84.52 8.36 2 6 7.96 3.7 99.45.44 4 32 5.7 6.25 98.4.5 6 48 23.2 9.2 96.34.56 8 64 3.6 2. 94.24.62 8 36.77 4.64 9.94.34 6 6 6.38 2.3 99.7.4 32 32 2.67 4.4 98.98.46 48 48 8.8 6. 97.96.52 64 64 24.76 7.89 96.72.56 8 8 3.5 9.72 95.33 Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 4

25 Vol. 3 () January-March, pp.37-5/tripathi Tabl 2: Prformanc Paramtrs (Bas Radius 25 mm) Fin Volum, cc Fin Thicknss, Outr Radius, Hat Transfr Effctivnss Efficincy % mm mm Rat 6 2 56 25.8 4. 8.68 4 44 5.4 2.27 96.35 6 38.53 5.59 98.78 8 36 7.95 3.6 99.45 34 6.38 2.32 99.79 32 2 76 37.78 6.6 59.3 4 56 28.53 22.72 89.7 6 48 2.48.87 96.2 8 44 5.7 6.25 98.4 4 2.67 4.4 98.98 48 2 9 42.93 68.36 44.72 4 67 38.99 3.4 8.22 6 56 29.59 5.7 92.48 8 5 23.2 9.2 96.34 46 8.8 5.99 97.96 64 2 4 45.34 72.9 35.42 4 76 47. 37.5 73.59 6 63 37.79 2.6 88.56 8 56 3.6 2. 94.24 52 24.76 7.87 96.72 8 2 6 46.55 74.3 29. 4 84 53.35 42.48 66.69 6 7 45.8 23.93 84.52 8 62 36.77 4.64 9.94 56 3.5 9.72 95.33 Tabl 3: Fin Effctivnss for Diffrnt Fin Volums and Thicknsss (Bas Radius 25 mm) Fin Thicknss, mm Volum 6 cc Volum 32 cc Volum 48 cc Volum 64 cc Volum 8 cc 2 4. 6.6 68.36 72.9 74.3 4 2.27 22.8 3.4 37.5 42.48 6 5.59.87 5.7 2.6 23.93 8 3.7 6.25 9.2 2. 4.64 2.3 4.4 5.99 7.89 9.72 Tabl 4: Hat Transfr Rat pr unit Volum for Diffrnt Fin Volums and Thicknsss (Bas Radius 25 mm) Fin Thicknss Volum Volum Volum Volum Volum mm 6 cc 32 cc 48 cc 64 cc 2.6.8.89.7.58 4.96.89.8.74.67 6.66.64.62.59.56 8.5.49.48.47.46 8 cc.4.4.39.39.38 Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 42

25 Vol. 3 () January-March, pp.37-5/tripathi Figur : Rctangular Circumfrntial Fin Figur 2: Variation of prformanc paramtrs for fin thicknss 2 mm Figur 3: Variation of prformanc paramtrs for fin thicknss 4 mm Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 43

25 Vol. 3 () January-March, pp.37-5/tripathi Figur 4: Variation of prformanc paramtrs for fin thicknss 6 mm Figur 5: Variation of prformanc paramtrs for fin thicknss 8 mm Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 44

25 Vol. 3 () January-March, pp.37-5/tripathi Figur 6: Variation of prformanc paramtrs for fin thicknss mm Figur 7: Variation of prformanc paramtrs for matrial volum 6 cc Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 45

25 Vol. 3 () January-March, pp.37-5/tripathi Figur 8: Variation of prformanc paramtrs for matrial volum 32 cc Figur 9: Variation of prformanc paramtrs for matrial volum 48 cc Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 46

25 Vol. 3 () January-March, pp.37-5/tripathi Figur : Variation of prformanc paramtrs for matrial volum 64 cc Figur : Variation of prformanc paramtrs for matrial volum 8 cc Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 47

25 Vol. 3 () January-March, pp.37-5/tripathi Figur 2: Variation of Effctivnss of constant volum fins for diffrnt fin thicknss Figur 3: Hat Transfr pr unit Volum of Fin Variation in Effctivnss with Fin Thicknss and Fin Volum In this sction ffctivnss is xamind from th point of viw of gomtrical paramtrs involvd in fin dsign. Th bas radius of fin is gnrally pr-spcifid in th form of outr diamtr of pip. Thrfor fin gomtry can b dfind in trms of fin thicknss and outr radius. Also, ffct of matrial quantity on prformanc of fin could b xamind mor clarly if volum of fin is usd in plac of outr radius. Tabl 3 prsnts th data from th study and Figur 2 shows variation of ffctivnss of constant volum fins Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 48

25 Vol. 3 () January-March, pp.37-5/tripathi for diffrnt fin thicknss. It is clar that smallr fin thicknss lads to highr ffctivnss, as indicatd in th prvious sctions. It may b obsrvd from Tabl 3 that for fin thicknss of 2 mm, an incras in volum from 6 cc to 32 cc (an incras of 6 cc) rsults in incras in ffctivnss from bout 4. to 6.6 (an incras by about 9). But, incrasing th volum from 32 cc to 48 cc (again an incras of 6 cc) rsults in incras in ffctivnss from 6.6 to 68.36 (an incras by about 8). Thus, it is obsrvd that incras in matrial quantity rsults in lss than proportionat improvmnt in ffctivnss. Figur 2 indicats that ffctivnss dtriorats rapidly with incras in fin thicknss. Thrfor, th dsignr must giv du considration to this fact and considr altrnativ manufacturing procsss in ordr to achiv low fin thicknss. Th family of curvs shown in Figur 2 may b usd by th dsignr in idntifying suitabl fin thicknss, basd upon considrations of cost of manufacturing and availabl manufacturing procsss. Thn dpnding upon th dsirabl har transfr rat, th volum can b dtrmind basd upon ffctivnss. Variation in Hat Transfr Rat pr unit Volum with Fin Thicknss and Fin Volum Hat transfr rat pr unit volum of fin should b considrd an important paramtr for assssmnt of fin prformanc. This paramtr may b usd as an indicator of ovrall wight of th systm. Tabl 4 shows th valus for this paramtr for diffrnt combinations of fin thicknsss and fin volums. Ths rsults ar plottd in Figur 3. It is obsrvd that this paramtr is vry snsitiv to both fin thicknss and fin volum. Also, unlik th othr prformanc paramtrs lik fin ffctivnss and fin fficincy its bhavior is not monotonous. Figur 3 shows that maximum hat transfr rat pr unit volum of fin may b achivd by slcting a combination of small fin thicknss and small fin volum. A small fin volum, for a givn fin thicknss and bas radius, would lad to a small outr radius of fin. Thus, in ordr to achiv a high hat transfr rat pr unit volum, th dsignr should aim at small fin thicknss combind with small fin hight. This may lad to incrasing th numbr of fins to achiv dsird ovrall hat transfr rat, but would nsur optimum dsign from th point of viw of matrial utilization as wll as wight of th systm. Th curv corrsponding to fin volum of 8 cc clarly indicats that xcssivly larg volum may lad to rducd hat transfr pr unit volum of fin, if th fin thicknss is not slctd judiciously. In this mannr, analysis of this paramtr is usful in avoiding such pit-falls in dsign of finnd surfacs. CONCLUSION Analysis of rctangular circumfrntial fins was prformd in ordr to study th ffct of fin gomtry on fin prformanc paramtrs. Th following ar th main conclusions basd on this work:. Variations of ffctivnss and fficincy ar of opposit natur for diffrnt paramtrs of fin gomtry. Fin ffctivnss should b givn priority in fin dsign from th point of viw of hat transfr rat through th fin. Thrfor th objctiv of fin dsign should b to maximiz fin ffctivnss. 2. For constant fin thicknss and bas radius highr fin ffctivnss may b achivd through largr outr radius of fin, although th valu of fficincy rducs substantially in this cas. 3. It was also obsrvd that for constant fin volum (matrial quantity) and constant bas radius highr fin ffctivnss may b achivd by kping th valu of fin thicknss as low as practicabl. Furthr, th ffct of fin thicknss on ffctivnss was found to b mor prominnt compard to fin volum. 4. Hat transfr rat pr unit fin volum was idntifid as a critical paramtr in fin dsign, lading to optimum matrial utilization and ovrall wight of th systm. Study of this paramtr indicats that th dsignr should aim for fins of small thicknss as wll as small hight. Hnc th logical approach for fin dsign should b to slct a small fin thicknss, basd on manufacturing and cost considrations, followd by dtrmination of th fin volum to achiv dsird fin ffctivnss and hat transfr rat. This approach should rsult in optimum fin gomtry both from hat transfr and cost point of viw. REFERENCES Bharani S, Tripathi K and Garg VN (992). Prformanc of Rctangular Circumfrntial Fins. Journal of Institution of Enginrs (India) 73 49-52. Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 49

25 Vol. 3 () January-March, pp.37-5/tripathi Janna WS and Hochstin JI (22). A Boundary Layr Mthod of Obtaining an Approximat Solution to th Infinit Fin Problm. ASME Intrnational Mchanical Enginring Congrss & Exposition, Nw Orlans. Krn DQ (972). Extndd Surfac Hat Transfr (McGraw Hill). Kraus A, Aziz A and Wlty J (2). Extndd Surfac Hat Transfr (John Wily). Pawar CB, Tripathi S, Aharwar KR and Tripathi K (25). Comparison of diffrnt prformanc paramtrs using diffrnt fin proportions of rctangular circumfrntial fin. National Confrnc on Emrging Trnds in Thrmal Enginring, Comptitiv Manufacturing and Managmnt (TECM), Jan 25, Chnnai India. Razlos P (23). A Critical Rviw of Extndd Surfac Hat Transfr. Hat Transfr Enginring 24(6). Rizzi M and Catton I (23). An Exprimntal Study of Pin Fin Hat Sinks and Dtrmination of End Wall Hat Transfr. ASME Summr Hat Transfr Confrnc, July 23, Las Vgas. Copyright 24 Cntr for Info Bio Tchnology (CIBTch) 5