Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI a*, Zaman Zabkhsh GANJI b, and Hosan Domr GANJI c a Department of Mechancal Engneerng, Mazandaran Unversty, Babol, Iran b IOR/EOR Research Insttute, Natonal Iranan Ol Company (N.I.O.C.), Tehran, Iran c Islamc Azad Unversty, Scence and Research Branch, Tehran, Iran Orgnal scentfc paper UDC: 536.241:517.957 DOI: 1.2298/TSCI11S1111G In ths paper, homotopy perturbaton method has been used to evaluate the temperature dstrbuton of annular fn wth temperature-dependent thermal conductvty and to determne the temperature dstrbuton wthn the fn. Ths method s useful and practcal for solvng the nonlnear heat transfer equaton, whch s assocated wth varable thermal conductvty condton. The homotopy perturbaton method provdes an approxmate analytcal soluton n the form of an nfnte power seres. The annular fn heat transfer rate wth temperaturedependent thermal conductvty has been obtaned as a functon of thermogeometrc fn parameter and the thermal conductvty parameter descrbng the varaton of the thermal conductvty. Key words: homotopy perturbaton method, numercal method, annular fn, thermal conductvty n heat transfer Introducton Most scentfc problems and phenomena requre hgh performance heat transfer components wth progressvely smaller weghts, volumes, costs. So, one of the most sgnfcant mportance s the optmzaton of the desgn of fns for hgh performance, lght weght and compact heat transfer components. Extended surfaces are wdely utlzed n varous ndustral applcatons. Kern et al. [1] have presented an extensve revew on ths topc. In addton, large amount of papers exst on the problem of convectve fns. It has provded the optmum dmensons of straght fns, crcular fns and spnes of dfferent profles wth several numercal examples by Azz [2]. Constant thermal conductvty was consdered n prevous study. Addtonally, a consderable amount of research has been conducted nto the varable thermal parameters. It has utlzed the regular perturbaton method and a numercal soluton method to calculate a closed form soluton for a straght convectng fn wth temperature dependent thermal conductvty by Azz et al. [3]. *ncorrespondng author; e-mal: ddg_davood@yahoo.com
S112 Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for There are few phenomena n dfferent felds of scence occurrng lnearly. Most scentfc problems such as heat transfer are nherently nonlnear. We know that except a lmted number of these problems, most of them do not have analytcal soluton. Therefore, these nonlnear equatons should be solved usng other methods. Some of them are solved usng numercal technques. In the numercal method, stablty and convergence should be consdered so as to avod dvergence or napproprate results. In the analytcal perturbaton method, we should exert the small parameter n the equaton. Therefore, fndng the small parameter and exertng t nto the equaton are dffcultes of ths method. Snce there are some lmtatons wth the common perturbaton method, and also because the bass of the common perturbaton method s upon the exstence of a small parameter, developng the method for dfferent applcatons s very dffcult. Homotopy perturbaton method (HPM) whch was recently developed by J-Huan He [4, 5] s one of the most successful and effcent methods n solvng nonlnear equatons. In contrast to prevously ntroduced analytc methods, HPM s ndependent of any small or large parameter. In the works of prevous authors Ganj et al. [6, 7] and others [8, 9] have successfully appled HPM n solvng dfferent types of nonlnear problems.e. coupled, decoupled, homogeneous and non-homogeneous equatons arsng n dfferent physcal problems such as heat transfer, flud flow, oscllatory systems and etc. The am of ths paper s to gve the analytc soluton of the nonlnear equaton of the annular fns wth temperature dependent thermal conductvty and compare the HPM results wth numercal results gven by Arslanturk [1]. Fgure 1. Geometry of a straght fn Problem descrpton An annular fn wth temperaturedependent thermal conductvty as shown n fg. 1 s consdered n predctng the fn geometry. The fn of thckness t, base radus r, and tp radus r o s exposed to a convectve envronment at the constant ambent temperature T and heat transfer coeffcent h. The base temperature T b of the fn s constant, and the fn tp nsulated. Snce the fn s assumed to be thn, the temperature dstrbuton wthn the fn does not depend on axal ecton. The one-dmensonal energy balance equaton s gven: d dt t k( T ) r 2 hr( T T ) (1) dt T Tb at r r, at r r The thermal conductvty of the fn materal s assumed to be a lnear functon of temperature accordng to k(t) = k [1 + k(t T b )] where k b s the thermal conductvty at the ambent base temperature of the fn and l s the parameter descrbng the varaton of the thermal conductvty. Employng the followng dmensonless parameters: o (2)
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for T T hr r r r t T T k r r r o, B, Tb T,,, b S113 (3) the formulaton of the problem reduces to: 2 1 2B, 1 (4) (1 ) (1 ) 1 at (5) at 1. (6) Applcaton of homotopy perturbaton method to fn temperature dstrbuton eq. (4) as: We set up the homotopy perturbaton method formulaton [9] by rewrtng p 2 (1 pl ) ( ) 1 2B (1 ) (1 ) where p s the homotopy parameter, and L s the lnear operator as L º d 2 /dx 2.We seek a perturbaton soluton for q n the form of a power seres n p as under: ( ) ( ) p ( ) p ( ) p ( ) (8) 2 1 2 Assumng that the seres (2) converge for p = 1, the fnal soluton for q s gven by: ( ) ( ) The HPM s an analytcal method that has been used to solve effectvely, easly, and accurately a large class of lnear and nonlnear, ordnary or partal, determnstc or stochastc dfferental equatons wth approxmate whch converge rapdly to accurate solutons. Ths technque wll be used to solve the evoluton equatons derved here as follows. We begn wth ntal approxmatons: ( ) 1 (1) Substtutng for q from eq. (8) nto eq. (7), and equatng lke powers of p on both sdes, we obtan, for n 1: n 1 1 1 2B L ( ) (11) n m n m 1 m n m 1 m m n 1 n 1 n 1 m 1 1 wth boundary condtons: n, n 1 (12) As wth the ntal approxmaton, eq. (12) must be frst solved for q n. Below we present frst few terms of the expansons (13): (7) (9)
S114 Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for 1 B 2 2B( 1) ( ), (13) 2 2 B 4 2B (1 ) 3 B (1 2 ) 2 2( ) 2 2 2 6 3 Accordng to eq. (9) the assumpton p 1, we get: (14) ( ) ( ) 1( ) 2( ) (15) Concluson Fgure 2. Temperature dstrbuton by HAM for varous b when l = 2, d = 1/3, and B -=.1 In ths study, temperature dstrbuton of annular fn wth temperature-dependent thermal conductvty was analyzed usng HPM. When compared wth other numercal methods, t s clear that HPM provdes hghly accurate analytc solutons for nonlnear problems. Fgures 2 and 3 show q(x) that are obtaned by usng homotopy perturbaton method for varous values of b and d when l = 2. Fnally, as shown n fg. 4, t has been attempted to show the accuracy, capabltes and wde-range applcatons of the homotopy perturbaton method n comparson wth the numercal soluton of nonlnear temperature dstrbuton of annular fn wth temperature-dependent thermal conductvty. Fgure 3. Temperature dstrbuton by HAM for varous B when l = 2, d = 1.3 and b =.3 Fgure 4. The comparson between HPM and numercal soluton for q(x) for l = 2 d = 1/3 and B =.1 Acknowledgement The authors are grateful to the Research and Technology Drectorate of Natonal Iranan Ol Company, Tehran, Iran for fnancal support.
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S115 References [1] Kern, D. Q., Kraus, D. A., Extended Surface Heat Transfer, McGraw-Hll, New York, USA, 1972 [2] Azz, A., Na, T. Y., Perturbaton Methods n Heat Transfer, Hemsphere Publshng Corporaton, Washngton, DC, USA, 1984 [3] Azz, A., Enamul Hug, S. M., Perturbaton Soluton for Convectng Fn wth Varable Thermal Conductvty, Journal of Heat Transfer ASME, 97 (1975), pp. 3-31 [4] He, J.-H., The Homotopy Perturbaton Method for Nonlnear Oscllators wth Dscontnutes, Appled Mathematcs and Computaton, 151 (24), 1, pp. 287-292 [5] He, J.-H., Homotopy Perturbaton Technque, Computer Methods n Appled Mechancs and Engneerng, 17 (1999), 8, pp. 257-262 [6] Ganj, Z. Z., Ganj, D. D., Approxmate Solutons of Thermal Boundary-Layer Problems n a Sem- Infnte Flat Plate by Usng He s Homotopy Perturbaton Method, Internatonal Journal of Nonlnear Scences and Numercal Smulaton, 9 (28), 4, pp. 415-422 [7] Ganj, Z. Z., Ganj, D. D., Esmaelpour, M., Study on Nonlnear Jeffery-Hamel Flow by He s Sem- Analytcal Methods, Computers and Mathematcs wth Applcatons, 58 (29), 11-12, pp. 217-2116 [8] Ylm, A., Homotopy Perturbaton Method for the Mxed Volterra-Fredholm Integral Equatons, Chaos, Soltons & Fractals, 42 (29), 5, pp. 276-2764 [9] Donald Arel, P., Homotopy Perturbaton Method and the Natural Convecton Flow of a Thrd Grade Flud through a Crcular Tube, Nonlnear Scence Letters A, 1 (21), 3, pp. 43-52 [1] Arslanturk, C., Correlaton Equatons for Ooptmum Desgn of Annular Fns wth Temperature Dependent Thermal Conductvty, Heat and Mass Transfer, 45 (29), 4, pp. 519-525 Paper submtted: July 3, 21 Paper revsed: September 2, 21 Paper accepted: November 11, 21