Analytical investigations of thermo-geometrical parameters effects on heat transfer and performance of longitudinal radiating fin

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1 Internatinal Jurnal f Energy and Pwer Engineering 4; 3(3): Published nline June, 4 ( di: 48/j.ijepe Analytical investigatins f therm-gemetrical parameters effects n heat transfer and perfrmance f lngitudinal radiating fin Bay Yemisi Ogunmla, Gbeminiyi Sbamw Department f Mechanical Engineering, Faculty f Engineering, University f Lags, Akka, Lags, Nigeria address: bayemi@yah.cm (B. Y. Ogunmla), mikegbeminiyiprf@yah.cm (G. Sbamw) T cite this article: Bay Yemisi Ogunmla, Gbeminiyi Sbamw. Analytical Investigatins f Therm-Gemetrical Parameters Effects n Heat Transfer and Perfrmance f Lngitudinal Radiating Fin. Internatinal Jurnal f Energy and Pwer Engineering. Vl. 3, N. 3, 4, pp di: 48/j.ijepe Abstract: In this wrk, analytical mdels are develped fr determining temperature distributin, heat transfer and the efficiency f radiating fin. The develped mdels are used t investigate the effects f varius therm-gemetric prperties n the temperature distributin and cnsequently n the perfrmance f the radiating fin. The effects f fin tip cnditins n the thermal perfrmance f the fin are als examined. Frm the results, it is shwn that the variatin in the thermgemetric parameters (fin tip Bit number, mdified Bit number and efficiency) f the fin has prnunced effects n the temperature distributin and cnsequently n the perfrmance f the fin especially when a large temperature difference exists between the prime surface and the envirnment. The results btained in this study serve as basis fr determining the level f accuracy f any ther apprximatin methd use in the analysis f the prblem and als, it culd be used t imprve the design f radiating fin in heat transfer equipment. Keywrds: Analytical Slutin, Temperature Distributin, Heat Transfer, Therm-gemetric Parameters, Radiating Fin. Intrductin The ever grwing demands fr high-perfrmance heat transfer cmpnents with prgressively smaller weights, vlume, csts r accmmdating shapes have greatly increased the use f extended surfaces/fins t enhance heat dissipatin lss frm ht primary surfaces. The Pwer plants in the autmbiles are becming highly pwer-packed with increasing pwer t weight and/r vlume rati. The designs f radiatr at present fr the high-perfrmance heat transfer cmpnents depend n the empirical methds, wherein existing experimental data are used as the rules f thumb fr the design prcess. Als, the drive n autmbile manufacturers fr develping cmpact and energy efficient cars warrants a thrugh ptimizatin prcess in the design f all engine cmpnents, including radiatrs. Radiatrs are installed in autmbiles t remve heat frm the under hd which include engine cling and heat remval during aircnditining prcess. The use f higher utput engines with tightly packed under hd packaging demands a better understanding f the cmplex cling fluid flw characteristics and resulting thermal perfrmance f the radiatr. Mrever, in the recent years, interest in fin with radiatin heat transfer has been simulated in space explratin since radiatin is almst the main mechanism by which waste heat frm electrnic part r ther heat generating equipment in satellite can be dissipated. The enrmus applicatins f this type f fin have arused interest f varius wrkers as there have been extensive wrks n the heat transfer characteristics, which attempt t imprve the design and prvide the ptimized mass f the fins. In the attempts f studying the functins, investigating the heat transfer characteristics and ptimizing the perfrmance f the fin, Cckfield [] discussed the rle f radiatrs as a structural element in spacecraft applicatins while Wilkins Jr. [] develped expressins fr heat analysis f triangular fins t space at abslute zer temperature Few years later, Schnurr [3] ptimized the design f lngitudinal and triangular radiating fins with respect t weight. Meanwhile, wrkers such as Gpinath and Ashk [4] studied the prblem f a unifrm area radiating fin fr large values f the radiatin-cnductin interactin parameter and develped fr imprving the previusly knwn asympttic slutin t make it applicable fr relatively smaller values f radiatin cnductin interactin parameter. Besides, Lve and Francis [5] analyzed lngitudinal radiating fins equally spaced

2 4 Bay Yemisi Ogunmla and Gbeminiyi Sbamw: Analytical Investigatins f Therm-Gemetrical Parameters Effects n Heat Transfer and Perfrmance f Lngitudinal Radiating Fin arund a cylindrical heat surce, while Eslinger and Chung [6] presented a finite element slutin fr the heat transfer frm a radiating and cnvecting fin r fin arrays. Schnurr et al [7] ptimized the radiating fin arrays with respect t weight. Here, a nn linear ptimizatin apprach was used t determine the minimum weight design fr radiating fin arrays used in space applicatins and the analysis were brught ut. In the fllwing year, Trung [8] investigated the perfrmance f radiating annular fins f varius shapes while Karam and Eby [9] presented the linearised slutin f cnducting radiating fin. On the ther hand, Chang [] develped a cmputer mdel f a heat pipe radiatr based n tw crrelatins fr the rectangular radiating fin efficiency. In the same vein, Zaulichnyi [] analyzed the prblem f heat transfer by means f radiatrs with finned heat pipes with nedimensinal apprximatin and prpsed an algrithm t design radiatrs f a specified capacity with cnstraints n their weight and dimensin. Mrever, Sunil Kumar and Venkateshan [] ptimized tubular radiatr with annular fins n a nnisthermal base. Few years later, Krikkis and Razels [3] presented the crrelatins fr ptimum dimensins f lngitudinal rectangular and triangular radiating fins with mutual irradiatin. Neumann [4] has investigated analytically the thermal design f heat pipe/fin type space radiatrs fr the case f unifrmly tapered fins as well as fr flat fins with cnstant thermal cnductivity assumptin. Equally, Cihat [5] evaluated the temperature distributin alng a radiating fin by the analytical ADM (Admian Decmpsitin Methd) methd while Hsseini et al [6] applied the HPM (Hmtpy Perturbatin Methd) methd fr a variable thermal cnductivity. In furtherance f this research, Bazdidi and Kamrava [7] prvided a numerical calculatin f temperature distributin analysis using finite vlume technique fr linear functin f temperature-dependent thermal cnductivity. The heat transfer in radiating fin ccurs nnlinearly and as such it is difficult t find the exact analytical slutins fr such prblems. Therefre, frm the reviewed literatures, apprximate analytical slutins were used amng which hmtpy perturbatin methd (HPM), Admain decmpsitin methd (ADM). The study (using either analytical r numerical methds) f therm-gemetrical parameter effects and the material thermal prperties effects f a radiating fin and its heat transfer characteristics is limited in pen literatures. Therefre, this wrk presents analytical slutins fr the study f the effects f therm-gemetrical parameters, material thermal prperties and the heat transfer characteristics n the perfrmance f radiating fin. 3. The temperature f the medium surrunding the fin is unifrm. 4. The fin thickness is small, cmpared with its height and length, s that temperature gradients acrss the fin thickness and heat transfer frm the edges f the fin may be neglected. 5. The fin base temperature is unifrm. 6. There is n cntact resistance where the base f the fin jins the prime surface. 7. There are n heat surces within the fin itself. 8. Heat transfer t r frm the fin is prprtinal t the temperature excess between the fin and the surrunding medium. Under these assumptins, the general differential equatin f radiating fin is given by 4 4 ( ) dt σεlt T d T Ldf x ( ) Lf ( x) + = () dx dx dx k The prfile functin f(x) fr lngitudinal fins usually will take the frm f ( x) δ x = b ( n )/( n).. Lngitudinal Fin f Rectangular Prfile Fr the lngitudinal fin f rectangular prfile displayed with its terminlgy and crdinate system in Fig.., the expnent n the general fin prfile f Eq. () satisfies the gemetry when n =. The prfile functin fr this fin then becmes f δ () ( x) = (3) ( ) df x Because δb = δ and = dx On substituting Eq. (3) int Eq. (), the gverning differential equatin becmes 4 4 ( T T ) d T σε = (4) dx kδ Which is an rdinary nnlinear secnd-rder differential equatin with cnstant cefficients... The Physical Mdel. Mdel Assumptins and Develpment The fllwing assumptins are made in the develpment f the gverning equatin. The heat flw in the fin and its temperatures remain cnstant with time. i.e. a steady state heat transfer.. The cnvective heat transfer cefficient n the faces f the fin is cnstant and unifrm ver the entire surface f the fin. Fig.. Schematic diagram f lngitudinal radiating fin

3 Internatinal Jurnal f Energy and Pwer Engineering 4; 3(3): Develpment f the Gverning Equatin Frm the cnservatin f energy equatin based n the assumptins 4 4 ( T T ) d dt σε k( T) = dx dx δ Since the thermal cnductivity is assumed cnstant Which culd als be written as, 4 4 ( T T ) (5) d T σε = (6) dx kδ d T σε 3 3 ( T T )( T T T TT T ) = dx kδ The bundary cnditins T x = L, k = hlt ( T ), x x =, T = T b, (7) In the reviewed literatures, numerical methds r apprximate analytical methds were used in finding slutins t the kind f nn-linear differential equatin Eq. (6). Since, ne f ur majr aims in this wrk is t generate a clsed frm slutin fr this equatin, the fllwing dimensinless parameters were used. hl x T T Bi = X = θ = k L T T Bi m σl ε 3 3 = ( T + T T + T T + T kδ b, (8) The bundary cnditins are d θ Bi θ = m dx X =, θ = θ X =, = Biθ X On using the dimensinless parameters, the gverning equatins and bundary cnditins transfrm t; On slving Eq. 9 with the bundary cnditin f Eq. Bicsh(( Bim )( L)) + Bim sinh(( Bim )( L)) sinh(( Bim )( X ) θ = csh(( Bim ) X) (( Bim )csh(( Bim ) L) + Bisinh(( Bim( L)) (3) (9) () The Bit number as shwn in the Eq. (9) gives a simple index f the rati f the internal resistance f a bdy t cnductive heat t its external resistance t cnvective heat. The mdified Bit number in Eq. () gives a simple index f the rati f the internal resistance f a bdy t cnductive heat t its external resistance t radiative heat. This means that a small mdified Bit number represent a small resistance t heat cnductin, and thus small temperature gradients r rate f heat transfer within the fin. As part f ur analysis, the use f different values f the Bit numbers (calculated using different values f thermal cnductivity) represents different fin materials. If negligible heat lss at the tip f the fin is cnsidered, then bundary cnditins are 8 x =, T = T b, T x = L, = x () Using the same dimensinless parameter as given in Eq. X =, θ = θ X =, = X () Bicsh(( Bi )( )) sinh(( )( )). m L + Bim Bim L ( Bim )csh(( Bim ) Q = ( Bim)sinh(( Bim )) (( Bim )csh(( Bim ) L) + Bisinh(( Bim ( L)) Bicsh(( Bim )( L)) + Bim sinh(( Bim )( L)) (/ Bim )(csh(( Bim ) ) η = (/ Bim )sinh(( Bim )) (( Bim )csh(( Bim ) L) + Bisinh(( Bim ( L)) (4) (5) On slving equ. with the bundary cnditins f Eq.. { } θ = csh(( Bi X)) tanh(( Bi )( L) sinh(( Bi )( X) (6) m m m { } Q = ( Bi )sinh(( Bi )) tanh(( Bi )( L) ( Bi )csh(( Bi )( X) (7) m m m m m

4 4 Bay Yemisi Ogunmla and Gbeminiyi Sbamw: Analytical Investigatins f Therm-Gemetrical Parameters Effects n Heat Transfer and Perfrmance f Lngitudinal Radiating Fin {[ tanh(( Bi )( L) ]((/ Bi ))(csh(( Bi ) ) } = m m m m m ( /( Bi ))sinh(( Bi )) η (8) Where, Q. ql = KAT ( T ). b a 4. Results and Discussin Since a high thermal cnductivity implies that energy can be cnducted mre easily, the results in the figures-6 are as expected. The dimensinless temperature distributin falls mntnically alng fin length fr all varius mdified Bit number. Fr larger values f the mdified Bit number Bi m, the mre heat radiated frm the fin thrugh its length and the mre thermal energy is 5 efficiently transferred int envirnment thrugh the fin length. In the situatin f negligible heat lss frm the fin tip t the envirnment, the fin temperature decreases alng the fin length als, and the temperature decreasing rate is the same arund fin base area fr all values f tip Bit numbers. This is due t heat which is majrly dispsed t the surrunding thrugh the fin tip. Als fr high values f tip Bit number, the tip tends tward being isthermal and the thermal energy is efficiently transferred int the surrunding thrugh the fluid-slid interface, hence, a lwer temperature cntur is expected as cmparing t thse with small values f tip Bit number Bim =. Bim = Bim = Bim = Bim =. θ (Dim ensinless Tem perature) Bim =. Bim = Bim = Bim = Bim = Fig.. Dimensinless Temp. Against Dimensinless length fr Bi=.. Fig. 3. Dimensinless Temp. Against Dimensinless length fr Bi=. 5 θ (Dim ensinless Temperature) Bim =. Bim = Bim = Bim = Bim =. θ (Dim ensinless Tem perature) Bim =. Bim = Bim = Bim = Bim =..... Fig. 4. Dimensinless Temp. Against Dimensinless length fr Bi=.4 Fig. 5. Dimensinless Temp. Against Dimensinless length fr M=Bi=.6

5 Internatinal Jurnal f Energy and Pwer Engineering 4; 3(3): Bim =. Bim = Bim = Bim = Bim =. Bim =. Bim = Bim = Bim = Bim =... Fig. 6. Dimensinless Temp. Against Dimensinless length fr Bi=.8.. Fig. 7. Dimensinless Temp. Against Dimensinless length fr Bi=. Bim =. Bim = Bim = Bim = Bim =... Bim =. Bim = Bim = Bim = Bim =... Fig. 8. Dimensinless Temp. Against Dimensinless length fr Bi=.. Fig. 9. Dimensinless Temp. Against Dimensinless length fr Bi=.5. Bim =. Bim =. Bim =.4 Bim =.6 Bim =.8 Bim =.. Bim =. Bim =. Bim =.4 Bim =.6 Bim = Fig.. Dimensinless Temp. Against Dimensinless length fr Bi=.75 Fig.. Dimensinless Temp. Against Dimensinless length fr Bi=. Fig. shws the variatin f dimensinless temperature with dimensinless length and als the effect f the Mdified Bit number n the straight fin with a negligible heat lss frm the fin tip(bi=). Frm the figure, as the Mdified Bit number increases, the rate f heat transfer thrugh the fin increases as the temperature in the fin drps faster as depicted in the figure. The prfile has steepest temperature gradient at Bi m =., but its much higher value gtten frm the lwer value f thermal cnductivity than the ther values f Bi m in the prfiles prduces a lwer heat-transfer rate. This shws that the perfrmance r efficiency f the fin is favured at lw values f Mdified

6 44 Bay Yemisi Ogunmla and Gbeminiyi Sbamw: Analytical Investigatins f Therm-Gemetrical Parameters Effects n Heat Transfer and Perfrmance f Lngitudinal Radiating Fin Bit number since ur aim (high effective use f the fin) is t minimize the temperature decrease alng the fin length, where the best pssible scenari is when T=T b everywhere. Figures 3- shw the variatin f dimensinless temperature with dimensinless length in lngitudinal radiating fin with cnvective tip. The effects f Bit number and Mdified Bit number n the dimensinless temperature distributin and in cnsequent n the rate f heat transfer are shwn. Frm the figures, it is shwn that as the as the Bit number (Bi) and the Mdified Bit number (Bi m ) increase, the rate f heat transfer thrugh the fin increases. This is because as the fin radiative heat transfer increases and cnvective heat transfer increases at the tip f the fin, mre heat is transferred by cnductin thrugh the fin thereby increases the temperature distributin in the fin and cnsequently the rate f heat transfer. Therefre, high effectiveness f fin culd be achieved by using small values f Bit numbers, which culd be realised using a fin f small length r by using a material f better thermal cnductivity. Mrever, the results depict that care must be taken in the chice f length Bi =. Bi =. Bi =.4 Bi =.6 Bi =.8 Bi =. f fin used during applicatins. This is because, as Bi m (which increases as the fin length increases) tends t infinity, the fin efficiency tends t zer. The fin t a large extent f its length will remain at ambient! This cnsequently results in weak cnductin limit. The extended area is largely useless in the heat transfer prcess and hence inefficient. Therefre, very lng fins are t be avided in practice. A cmprmise is reached fr nedimensinal analysis f fins < Bi <.. When the Bit number is greater than., tw dimensinal analysis f the fin is recmmended as ne-dimensinal analysis predicts unreliable results fr such limit. Figures -5 shws the effects f fin tip cnditins n the dimensinless temperature distributin. Frm the figures, it culd be seen that the at the value where the Mdified Bit number equals as shwn in Fig., there was prnunced effects f fin tip cnditin n the temperature distributin alng the fin length. As the value f the Mdified Bit number increases, the effects f tip end cnditin vanishes as there are verlapping temperature distributin prfiles as shwn in fig Bi =. Bi =. Bi =.4 Bi =.6 Bi =.8 Bi =... Fig.. Dimensinless Temp. Against Dimensinless fr Bim=.. Fig. 3. Dimensinless Temp. Against Dimensinless fr Bim=.. Bi =. Bi =. Bi =.4 Bi =.6 Bi =.8 Bi =.. Bi =. Bi =. Bi =.4 Bi =.6 Bi = Fig. 4. Dimensinless Temp. Against Dimensinless length fr Bim=.5 Fig. 5. Dimensinless Temp. Against Dimensinless length fr Bim=.

7 Internatinal Jurnal f Energy and Pwer Engineering 4; 3(3): Bim =. Bim =. Bim =.4 Bim =.6 Bim =.8.. Bi=. Bi =. Bi =.4 Bi =.6 Bi = Bi (Bit number) Fig. 6. Dimensinless Temp. Against Bit number Bim (Mdified Bit number) Fig. 7. Dimensinless Temp. Against Mdified Bit number Q (Dimensinless Heat lss rate) Bie =. Bie =. Bie =.4 Bie =.6 Bie =.8 Bie =. Fin efficiency Bi =. Bi =. Bi =.4 Bi = Bim (Mdified Bit number) Bim (Mdified Bit number) Fig. 8. Dimensinless Heat lss Against Mdified Bit number Fig. 9. Fin efficiency Against Mdified Bit number The effects f Bit number n the dimensinless temperature distributin fr different Mdified Bit number is shwn in Fig. 6 while Fig. 7 shws the variatin f dimensinless temperature distributin with Mdified Bit number fr different values f Bit number. In the limit Bi, the fin heat transfer tends t zer even thugh the fin is entirely at the base temperature. This ccurs when the heat transfer cefficient arund the fin is relatively small as cmpared t the thermal cnductivity f the fin and the fin length is als nt t large. The effect f fin tip end cnditin n the rate f heat lss frm the fin tip is shwn Fig. 8. As it culd be seen frm the figure, as the Bit number increases, mre heat is being cnvected away frm the fin tip. Als, frm Fig. 9 the fin efficiency decreases mntnically (fr different Bit number) with increasing mdified Bit number. The fin efficiency is unity in the limit Bim. In this limit, the actual heat transfer rate frm the fin is zer! This fin parameter plays a very imprtant rle in determining the amunt f heat transfer frm the fin as it accunts fr the effects f decrease in temperature n the heat transfer frm the fin. Since, the fin temperature drps alng the fin length, the fin efficiency decreases with increase in fin length. Therefre, in practice required fin length shuld be prperly determined because the fin length that causes the fin efficiency t drp belw 6% usually cannt be justified ecnmically and shuld be avided. 5. Cnclusin In this wrk, temperature distributin in a straight fin having cnstant thermal cnductivity with cnvecting and insulated tip has been slved analytically. The clsed frm slutin was used t investigate the effects f the fin tip Bit number, Mdified Bit number and tip end cnditins n the temperature distributin, rate f heat transfer and efficiency f the straight. Frm the results, it is shwn that the variatin in the thermgemetric parameters (fin tip Bit number, Mdified Bit number, and efficiency) f the fin has prnunced effects n the temperature distributin and cnsequently n the perfrmance f the fin especially when a large temperature difference exists between the prime surface and the envirnment. Therefre, the ttal heat transfer, the fin effectiveness, and the fin efficiency are all enhanced when using a fin material with a high thermal cnductivity. This depicts that the higher the thermal cnductivity f the fin material, the better! Fr increase in the efficiency, the length f the fin culd be decreased.

8 46 Bay Yemisi Ogunmla and Gbeminiyi Sbamw: Analytical Investigatins f Therm-Gemetrical Parameters Effects n Heat Transfer and Perfrmance f Lngitudinal Radiating Fin This will decrease the ttal heat transfer area and the ttal energy transfer f a single fin, but multiple fins culd be used t cunter this effect, and still keep the efficiency high. These results serve as basis fr cmparisn f any ther methd f analysis f the prblem and they als prvide platfrm fr imprvement in the design f fin in heat transfer equipments such as air land space vehicles and their pwer surces, in chemical, refrigeratin, and crygenic prcesses, in electrical and electrnic equipment, in cnventinal furnaces and gas turbines, in the design f firebx fr the generatin f steam pwer frm fssil fuels.in prcess heat dissipatrs and waste heat bilers, and in nuclear-fuel mdules, steam pwer plants, autmbiles radiatrs etc. Nmenclature A crss sectinal area f the fins, m Bi Bit number Bi m Mdified Bit number h heat transfer cefficient, Wm - k - K thermal cnductivity f the fin Wm - k - L Lenght f the fin, m M dimensinless fin parameter m fin parameter m - P perimeter f the fin, m T Temperature, K T a ambient temperature, K x axial distance, m X dimensinless lenght Q dimensinless heat transfer η efficiency f the fin Greek Symbls δ thickness f the fin, m ε emissivity f the fin θ dimensinless temperature σ Stefan-Bltzmann cnstant δ b fin thickness at its base. References [] R.D. Cckfield, Structural ptimizatin f a space radiatr, J. Spacecraft Rckets 5 () (968) 4 4. [] J.E. Jr Wilkins Minimizing the Mass f Thin Radiating Fins, J. Aerspace Science, Vl. 7, 96, [3] N. M. Schnurr, Radiatin frm an array f lngitudinal fins f triangular prfile, AIAA Jurnal, Vl. 3, 975, [4] S. P. Venkateshan; Gpinath, Ashk, Asympttic analysis f a unifrm area radiating fin, Canadian Sciety fr Mechanical Engineering, Transactins, Vl., N., 978,3-8. [5] T. J. Lve and J. E. Francis, A linearized analysis fr lngitudinal fins with radiactive and cnvective exchange Therm physics and Heat Transfer Cnference, Pal Alt, Calif., AIAA, 978. [6] R. D. Karam and R. J. Eby, Linearized slutin f cnducting radiating fins, AIAA Jurnal (Fluid Mechanics and Heat Transfer), Vl. 6, 978, [7] N. M. Schnurr.; M. A. Twnsend.; A. B. Shapir, "Optimizatin f radiating fin arrays with respect t weight", ASME, Transactins, Series C - Jurnal f Heat Transfer, Vl. 98, 976, [8] R. G. Eslinger and B. T. F. Chung. Peridic heat transfer in radiating and cnvecting fins r fin arrays, American Institute f Aernautics and Astrnautics and American Sciety f Mechanical Engineers, Therm physics and Heat Transfer Cnference, Pal Alt, Calif., AIAA,978. [9] H. V. Trung and R. J. Mancus, Perfrmance predictins f radiating annular fins f varius prfile shapes, American Sciety f Mechanical Engineers and American Institute f Chemical Engineers, Jint Natinal Heat Transfer Cnference, Orland, Fla., ASME 5, 98. [] H. V. Chang, Optimizatin f a heat pipe radiatr design, Therm physics Cnference, Snwmass, CO, 984. [] F. G. Zaulichnyi, Pssibilities fr imprving the characteristics f a radiatr cler thrugh the use f finned heat pipes as radiating elements, Sibirskii Fizik- Tekhnicheskii Zhurnal (ISSN ), Categry: Fluid Mechanics and heat transfer, 99,-6. [] S. Sunil Kumar and Venkateshan S. P. "Optimized Tubular Radiatr with Annular Fins n a Nn-Isthermal Base, Int. J. Heat and Fluid Flw", Vl.5, , 994. [3] R. N. Krikkis and R. Panagitis Optimum Design f Spacecraft Radiatrs with Lngitudinal Rectangular and Triangular Fins, J. Heat Transfer, Vl.4, 85-8,. [4] R. J. Naumann, Optimizing the design f space radiatrs, Int. J. Thermphys. 5 (4)99 94 [5] A. Cihat, Optimum design f space radiatrs with temperature-dependent thermal cnductivity, Applied Thermal Engineering 6 (6) [6] M. J. Hsseini, M. Grji and M. Ghanbarpur, Slutin f Temperature Distributin in a Radiating Fin Using Hmtpy Perturbatin Methd, Mathematical Prblems in Engineering Vlume 9, Article ID [7] F. Bazdidi-Tehrani, and M. H. Kamrava, Cmbined Heat Transfer Calculatins in a Fin-Tube Radiatrs with Temperature-dependent Thermal Cnductivity, Internatinal Cnference n Mechanical and Industrial Engineering, WASET, (), Singapre, pp76-84.