Influence Of Geometric Parameters Of An Annular Fin Of Different Cross-Sections On Its Efficiency

Transcription

1 Inlunc O Gomtrc Paramtrs O An Annular Fn O Drnt Cross-Sctons On Its Ecncy Nabla Labs and Youb Khald Bnkahla Unrsté ds Scncs t d la Tchnolog Houar Boumdn USTHB, Faculté d Gén Mécanqu t d Gén ds Procédés. B.P. 3, El-Ala Bab-Ezzouar, 6 Algr, ALGEIE. -mal: nablalabs@yahoo.r, youbnkahla@yahoo.r Fns ar usd n arous thrmal dcs such as ndustral changrs, aporators, tc. Th objct o thr us s to ncras th hat transr suracs and thror, th hat chang. To contrbut to th comprhnson o hat transr through ths dcs, a numrcal study basd on nt olum mthod s undrtakn. Th purpos o th prsnt study s to amn, n th rst stp, th nlunc o gomtrc paramtrs o an annular n o arous cross-sctons (rctangular, hyprbolc, trangular and parabolc) on ts cncy. Ths paramtrs ar: ts rducd lngth and ts charactrstc paramtr m. In th scond stp, th prormancs o th our cross-sctons ar compard. Th aldaton o th computr cod s conrmd by th comparson btn th rsults obtand usng th prsnt modl and thos aalabl n th ltratur, that s to say th analytcal soluton o Gardnr and th numrcal on o Ullman and Kalman. Ky ords: annular n, cncy, nt olum mthod.. Introducton Fns, n thr arous orms, ar tndd suracs usd n ordr to ncras th total surac o chang and thror, to mpro hat transr. Gardnr (945) count among th rst ho studd arous orms o ns. H prsntd th cncy o longtudnal n, annular on o rctangular and hyprbolc cross-sctons and nally n orm o spns. In a prospct or optmzaton o th prol and th dmnsons o th annular ns, Ullman and Kalman (989) prsntd abacuss gng th cncy accordng to th charactrstc paramtr o annular ns o arous cross-sctons. In th prsnt study, annular n o arous cross-sctons (rctangular, hyprbolc, trangular and parabolc) s consdrd. Ths n s sat o an unstady hat transr ollong th radal drcton. Th purpos o ths ork s to dtrmn th nlunc o th charactrstc paramtr as ll as th rducd lngth o ths our orms o ns, on th cncy thn to compar thr thrmal prormancs.

2 . Formulaton O Th Problm A thrmal balanc on a drntal lmnt o lngth dr s carrd out. Th lattr s takn thn an annular n o lngth L and arabl thcknss (r) astnd to a cylndr o radus r and subjct to a unorm tmpratur T o at ts bas. Consdrng som smplyng assumptons (T and T unorms, ry lo Bot numbr and nglgbl hat lu acuatd at th trmty o th n) and by ntroducng dmnsonlss arabls,, θ, τ, k a, as ll as th charactrstc paramtr m o th n and nally, by adoptng th ollong las to dscrb th araton accordng to th tmpratur o th thrmal conductty gn by Zubar t al. (996) as ll as th hat transr cocnt ctd by Laor and Kalman (996): k() = k ( ) () h h = () lad to th ollong quaton o nrgy: ( ) / a d d r 4 m k = τ (3) Th spatal and tmporal boundary condtons ar gn blo: or = (,τ) = (4) or = ) (, = τ (5) or τ = (,τ) = (6) By consdrng all ths arabls constants and unorms and that only and τ consttut th ndpndnt arabls, quaton (3) can b sold numrcally by mans o nt olum mthod. By ntgratng th quaton accordng to tm and takng a lnar prol o btn th ponts and o th olum o control, on obtans: ( ) ( ) Δτ Δτ = / a a d d r 4 m k k (7) In act, th our orms o th studd n dr by thr dmnsonlss thcknss (Tabl ). Th lattr s gn by th ollong prsson:

3 n = (8) Tabl : laton btn n, and th orm o th cross-scton o th annular n. Gomtrcal orms n ctangular hyprbolc - trangular parabolc A n s otn charactrzd by ts cncy hch rprsnts th quantty o hat lang th bas o th n normalzd by that acuatd all th surac o th n as mantand at T : η = m ( ) ( ) = n n ( a ) n 4 r ( a ) / d (9) It should b notd that th rsults prsntd on ths papr do not tak nto account th arablty accordng to th tmpratur o thrmal conductty and hat transr cocnt. Actually, th paramtr and th ponnt ll b takn qual to zro. 3. sults And Dscusson 3. Valdaton o th computr cod To aldat our computng program, a comparson btn th prsnt rsults and thos o th ltratur as mad; t s about th numrcal soluton o Ullman and Kalman (989). Th lattr study consdrs an annular n o a rctangular cross-scton or hch all th physcal proprts as ll as tmpraturs T and T ar constant and unorm. Th hat lo s undrctonal and taks plac n stady stat. Bsds, th n hos trmty s solatd, dos not gnrat ntrnal hat. Th comparson shos a good agrmnt btn our rsults and thos o th ltratur. Fgur shos ndd an almost prct concordanc btn th numrcal soluton o Ullman and Kalman (989) and th prsnt modl, cpt that a lght sht or th lo alus o th paramtr m, to hch a drnc o 4% s notd.

4 ,,,9 Prsnt study Ullman and Kalman (989),8,7 Ecncy, η,6,5,4,3,,,,,5,,5,,5 3, 3,5 4, 4,5 5, 5,5 m = L ( h / k ) / Fgur : Comparson btn th cncy rsultng rom th prsnt study and thos o Ullman and Kalman. ctangular n, ( /r ) =. and =.. Ect o th charactrstc paramtr and th rducd lngth o th studd ns on th cncy Fgur shos th oluton o th cncy or th our n s prols accordng to thr charactrstc paramtr m, or =, 3, 4 and 5 hn th stady rgm s rachd (τ =.8). Th curs prsnt th sam orm: a dcrasng monotonous uncton o th paramtr m rangng btn th alus and, tndng asymptotcally toards a plat hn m tnds to 5. Th mamum alus o th cncy ar at th lo alus o m. W notc that hatr th alu o m, th cncy dcrass hn ncrass. Ths dcras s ry pronouncd or th lo alus o m, thn, mor and mor slo or th grat ons. Bcaus o th lo lngths o th n (or, k and h gn) and/or th grat alus o th thrmal conductty o th matral (or L, and h gn), th ral hat lu dsspatd by th n bcoms ndd qual to th mamum hat lu hch could b dsspatd by th n.

5 , =,8 = 3,6 = 4,,8,4, n = = 5,6, Ecncy, η,4,,,8,,6,4,,,8,,6,4, n = n = - n =,,,5,,5,,5 3, 3,5 4, 4,5 5, m = L ( h / k ) / Fgur : Eoluton o th cncy o an annular n o arous cross-sctons accordng to th charactrstc paramtr m, or arous alus o.. τ = Comparson btn th thrmal prormancs o th studd ns Fgurs 3-a and 3-b sho th araton o cncy o th studd annular ns accordng to th charactrstc paramtr m atr th rach o th stady rgm (τ =.8) and or qual to and 5, rspctly. Th comparson btn th ns n trm o thrmal prormancs, shos that th rctangular n posssss, hatr th alu o m, th gratst cncy. Ths obsraton s rasd by sral authors.,,8 = n = n = n = - n =,,8 = 5 n = n = n = - n = Ecncy, η,6,4 Ecncy, η,6,4,,,,,5,,5,,5 3, 3,5 4, 4,5 5, m = L ( h / k ) /,,,5,,5,,5 3, 3,5 4, 4,5 5, m = L ( h / k ) / Fgur 3: Eoluton o th cncy accordng to th charactrstc paramtr m or arous cross-sctons o th studd n τ =.8. (a) =, (b) = 5.

6 It s also ntrstng to not that or = 5 th hyprbolc and parabolc prols mrg or alus o m hghr than.. Smlarly, or =, th hyprbolc and trangular prols mrg almost or m. On th othr hand, hl gong rom = to = 5, th trangular n bcoms mor cnt than th hyprbolc on. 4. Concluson Th purpos o th prsnt study as to dtrmn th nlunc o th gomtrcal paramtrs (m and ) o an annular n o arous cross-sctons on th cncy by consdrng constant physcal proprts. Th prormanc o th computng program as conrmd by comparng th prsnt rsults th th data o th ltratur. W not that th mamum alus o th cncy ar obtand or th lo alus o th charactrstc paramtr o th n and or th lo alus o. In addton, hatr th condtons, th annular n o rctangular cross-scton (n = ) posssss th bst alus o cncy n comparson to th thr othr gomtrcal orms. 5. rncs Gardnr K.A., 945, Ecncy o tndd suracs, ASME J. Hat Transr, 67. Laor K. and H. Kalman, 996, Prormanc and optmum dmnsons o drnt coolng ns th a tmpratur dpndnt hat transr cocnt, Int. J. Hat and Mass Transr, 39, pp Ullman A. and H. Kalman, 989, Ecncy and optmzd dmnson o annular ns o drnt cross-scton shaps, Int. J. Hat and Mass Transr, 3, pp 5-. Zubar S.M., A.Z. Al-Gan and J.S. Nzam, 996, Th optmal dmnsons o crcular ns th arabl prol and tmpratur-dpndnt thrmal conductty, Int. J. Hat and Mass Transr, 39, pp Nomnclatur C p spcc hat o th n matral h constant trm n th cocnt o hat chang h cocnt o concton hat chang k thrmal conductty o th matral k thrmal conductty takn at C k a dmnsonlss thrmal conductty, = k(t) / k L lngth o th annular n m charactrstc paramtr o th n, = L ( h / k ) / n paramtr hch dscrbs th araton o th prol o th n r radal poston r radus o th cylndrcal tub r radal poston o th trmty o th n, = r L dmnsonlssal tror radus, = r / r paramtr hch dpnds on th orm o th n prol t tm T tmpratur T tmpratur at th bas o th n T surroundng tmpratur dmnsonlss radal coordnat, = r / r Grk symbols: (r) thcknss o th n at a radal dstanc r dmnsonlss thcknss, = (r)/ η cncy o th annular n ρ dnsty o th n dmnsonlss tmpratur, = (T - T )/(T - T ) τ dmnsonlss tm, = k t /ρ С p r Subscrpts, control olums acs at ast and st sds poston o th nod