ANALYTICAL SOLUTION TO CONVECTION-RADIATION OF A CONTINUOUSLY MOVING FIN WITH TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY

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1 ANALYTICAL SOLUTION TO CONVECTION-RADIATION OF A CONTINUOUSLY MOVING FIN WITH TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY by Ami MORADI *, Rez RAFIEE b Islmic Azd Univesity, Ak Bnch, Young Reseches Club, Ak, In b Resech& Development Cente, Imm Khomeini Oil Refiney Compny, Shznd, In In this ticle, the simultneous convection-dition het tnsfe of moving fin of vible theml conductivity is studied. The diffeentil tnsfomtion method (DTM is pplied fo n nlytic solution fo het tnsfe in fin with two diffeent pofiles. Fin pofiles e ectngul nd exponentil. The ccucy of nlytic solution is vlidted by comping it with the numeicl solution tht is obtined by fouth ode Runge-Kutt method. The nlyticl nd numeicl esults e shown fo diffeent vlues of the embedding pmetes. DTM esults show tht seies convege pidly with high ccucy. The esults indicte tht the fin tip tempetue inceses when mbient tempetue inceses. Convesely, the fin tip tempetue deceses with n incese in the Peclet numbe, convection-conduction nd dition-conduction pmetes. It is shown tht the fin tip tempetue of the exponentil pofile is highe thn the ectngul one. The esults indicte tht the numeicl dt nd nlyticl method e in good geement with ech othe. Key wods: DTM, fin, fin tip tempetue, exponentil, dition 1. Intoduction In ecent yes, the het tnspot fom moving continuous sufce hs ttcted the ttention of some eseches. This phenomenon is n impotnt poblems tht occuing in numbe of industil pplictions. Extusion, hot olling, glss fibe dwing nd csting e exmple of continuous moving sufce. In industil pocesses, contol of cooling te of the sheets (o the fibes is vey impotnt to obtin desied mteil stuctue. These types of poblems hve solution substntilly diffeent fom tht of boundy lye flow ove semi-infinite plte. The velocity of moving mteil is elted to its ppliction. Fo exmple the velocity of the mteil cn be extemely low (few centimetes pe hou such in cystl gowth o vey fst (few metes pe second s in opticl fibe dwing. Flow nd het tnsfe in boundy lye on continuous moving sufce hve been studied by some eseches. The new clss of boundy lye flow ove moving sufce ws fist intoduced by Skidis [1]. He studied *Coesponding utho; Tel.: +98( , Fx: +98( , e-mil ddess:mimodi_hs@yhoo.com 1

2 the momentum tnsfe occuing when flt sufce continuously moves though quiescent fluid t the constnt sufce velocity. Eickson et l. [] developed this poblem to the cse with suction nd blowing t the moving sufce. The next eseches in litetue coveing sufce mss tnsfe, Newtonin nd non-newtonin fluid, mgnetic nd electic effects, diffeent theml boundy conditions, combined fee nd foce convection, combined convection nd dition het tnsfe, etc. Cotell [3] studied het tnsfe in moving fluid ove moving sufce numeiclly by mens of fouth-ode Rung-Kutt method. Ephim nd Abhm [4] investigted the stemwise vition of the tempetue of moving sheet in the pesence of moving fluid. They pplied n itetive method fo solving boundy lye equtions. Thei solution does not depend on the popeties of sheet nd fluid. Ming et l. [5] studied conjugte het tnsfe fom continuous, moving flt plte numeiclly by employing the cubic spline colloction. They investigted effects of Pndtl numbe, the convectionconduction pmetes nd the Peclet numbe on the het tnsfe fom continuous, moving plte. The investigtion of mixed convection het tnsfe long continuously moving heted veticl plte with suction nd blowing ws cied out by Al-Sne [6]. He pplied the finite volume method to solve boundy lye equtions. He used the published esults vilble unde specil condition to vlidte numeicl dt, nd the compison indicted n excellent geement. The buoyncy foce nd theml dition effects in MHD boundy lye visco-elstic fluid flow ove continuously moving sufce wee pefomed by Abel et l. [7]. Lee nd Tsi [8] studied cooling of continuous moving sheet of finite thickness. The effect of the buoyncy foce is lso tken into ccount. They obtined the tempetue distibution long the solid-fluid intefce by solving numeiclly conjugte het tnsfe poblem tht consists of het conduction inside the sheet nd induced mixed convection djcent to the sheet sufce. Othe conjugte convection-conduction eseches hve been pesented by Choudhy nd Jlui [9], nd Mendez nd Tevino [1], mong othes. The het tnsfe of moving mteil in non-newtonin fluid ws fist studied by Fox et l. [11]. They pplied n exct solution fo boundy lye equtions. Howell et l. [1] studied het tnsfe on continuous moving plte in non-newtonin powe lw fluid. They pplied Mek-Cho seies expnsion to genete odiny diffeentil eqution fom the ptil diffeentil momentum nd het tnsfe equtions in ode to obtin univesl velocity nd tempetue functions. Tobi et l. [13] investigted convective-ditive non-fouie het conduction with vible coefficients by employing HPM (homotopy petubtion method Some of the othe studies tht investigted the het tnsfe of continuous moving mteil in powe lw fluid hve been epoted by Shu et l. [14] nd, Zheng nd Zhng [15]. Tempetue distibution fo nnul fins with tempetue-dependent theml conductivity ws studied by Gnji et l. [16]. They employed HPM fo solving govening eqution. The effects of tempetue-dependent theml conductivity of moving fin nd dded ditive component to the sufce het loss hve been studied by Aziz nd Khni [17]. As hs been mentioned in [17], these impovements hve not been pusued in the litetue. They pplied the homotopy nlysis method (HAM to solve govening equtions. They comped the nlyticl nd numeicl esults to ech othe nd obseved n excellent geement. The im of this study is obtining n nlyticl solution fo tempetue distibution of moving fin with tempetue-dependent theml conductivity. The effect of the theml dition is lso

3 consideed hee. With the inclusion of dition nd vible theml conductivity, thee new pmetes, in ddition to the Peclet numbe nd Biot numbe, emege, nmely theml conductivity pmete, dition-conduction pmete nd n envionment tempetue pmete. The effect of the embedding pmetes on the tempetue distibution is shown in the moving mteil. The discepncy of pesent study with Aziz s esech [17] is tht in this study two diffeent pofiles (ectngul nd exponentil pofiles e consideed fo moving fin. As well s the diffeentil tnsfomtion method (DTM is pplied to solve nonline poblem nlyticlly. To vlidte nlyticl esults, the obtined DTM esults e comped with numeicl dt tht e obtined by the fouth-ode Rung-Kutt method. The concept of diffeentil tnsfomtion method ws fist intoduced by Zho [18] in 1986 nd it ws used to solve both line nd nonline initil vlue poblems in electic cicuit nlysis. The min benefit of this method is tht it cn be used diectly fo line nd nonline diffeentil eqution without equiing lineiztion, discetiztion, o petubtion. The DTM hs been egded by mny eseches, Rshidi nd Efni [19] used of DTM fo solving Buge s eqution nd het conduction poblem in fin with tempetue dependent theml conductivity. Gnji t el. [] pplied diffeentil tnsfomtion method to detemine fin efficiency of convective stight fins with tempetue dependent theml conductivity. Modi nd Ahmdiki [1] pplied the diffeentil tnsfomtion method to solve the enegy eqution fo fin with thee diffeent pofiles nd tempetue-dependent theml conductivity. The new lgoithm to clculte one nd two-dimensionl diffeentil tnsfom of nonline functions ws pesented by Hsing nd Ling [, 3]. Jng [4] solved the line nd nonline initil vlue poblems by the pojected diffeentil tnsfom method, tht this method cn be esily pplied to the initil vlue poblem by less computtionl wok. The novel nlyticl method, nmely DTM-Pde to solve MHD stgntion-point flow in poous medi with het tnsfe ws pesented by Rshidi nd Efni [5]. When thee is n infinite boundy in the poblem, the DTM could not to obtin the ccute solution. Becuse of the Pde ppoximtion is employed to solve poblem. Complementy infomtion bout this method hs been pesented in Ref [6].. Fundmentls of diffeentil tnsfomtion method Conside the nlytic function y( t in domin D wheet = ti epesent ny point in it. The function y( t is epesented by powe seies t centet i. The Tylo seies expnsion function of y( t is in the following fom []: ( j j ( t ti d y t y ( t = t D j j= j! dt t= ti (1 The pticul cse of Eq. (1 is when t i = nd is efeed to s the Mcluin seies of y(t expessed s: ( j j t d y t y ( t = t D j j= j! dt t= ( 3

4 As explined by Fnco [7], diffeentil tnsfomtion of the function y(t is defined s: Y ( j ( j j H d y t = j j= j! dt t=, (3 whee y( t is the oiginl function nd Y ( j is the tnsfomed function. The diffeentil spectum of Y( j is confined within the intevl t [, H] whee H is constnt. The diffeentil invese tnsfom of Y ( j is defined s: t y t = Y j j= H ( ( j Some of the oiginl functions nd tnsfomed functions e shown in tb 1. It is cle tht the concept of diffeentil tnsfomtion is the Tylo seies expnsion. Fo the solution with highe ccucy, moe tems in the seies in eq. (4 should be etined. Tble 1 The fundmentl opetions of diffeentil tnsfom method Oiginl function Tnsfomed function f ( x = α g( x ± βh( x Fk ( = αgk ( ± β Hk ( f ( x = g( xhx ( k Fk ( = GiHk ( ( i i= ( = g x Fk ( = ( k+ 1( k+ ( k+ ngk ( + n f ( x ( n n f ( x = x 1 k = n Fk ( = δ ( k n = k n f ( x = exp( α x k α Fk ( = k! f ( x = (1 + x n kk ( 1 ( k m 1 Fk ( = k! (4 3. Mthemticl fomultion Conside moving fin of the length L, with coss-section e A(x while it moves hoizontlly with constnt velocityu s depicted in fig. 1. Fin sufce is exposed to convective nd ditive envionment t tempetue T nd the bse tempetue of fin is T b > T. The locl het tnsfe coefficient h long the fin sufce is constnt nd the sufce of the moving fin is ssumed to be gy nd diffuse with constnt emissivityε. The ole of dition component could be moe sensible if the foce convection is wel o bsent o when only ntul convection occus. Since the mteil undegoing the tetment expeience lge chnge in its tempetue duing theml pocess, the theml conductivity of the mteil could not be constnt. Fo most mteils, the theml conductivity vies with the 4

5 tempetue linely. The one dimensionl stedy stte enegy eqution fo the fin moving with constnt speed nd losing het by simultneous convection nd dition cn be expessed s: d 4 4 kt ( Ax ( dt pht ( T pt ( T caxu ( dt dx εσ ρ = dx (5 dx whee p is the peiphey of the fin, T is the mbient tempetue, ε is the emissivity,σ is the Boltzmn constnt, ρ is the density of mteil c is the specific het nd k(t is defined s: [ λ ] kt ( = k 1 + ( T T (6 b whee, k b is the fin theml conductivity t mbient tempetue, nd λ is constnt. The fin pofile is defined ccoding to vition of the fin thickness long its extended length. Fo exmple, the coss section e of the fin my vy s: A( x = bt( x (7 wheeb is the width of the fin, t(x is the fin thickness long the length. The t(x fo diffeent pofiles cn be defined s follows: ( fo ectngul pofile tx ( = tb x (b fo exponentil pofile tx ( = tb exp γ ( L (8 by employing the following dimensionless pmetes: T T x hpl εσ plt k UL θ =, θ =, X =, N =, N =, α =, Pe= T T L K A k A ρ c α 3 b b c b b b b b b (9 whee A b is the bse e, Nc is the convection-conduction pmete (moe popully known s the Biot numbe, N is the dition-conduction pmete,α is the theml diffusivity of fin nd Pe is the Peclet numbe which epesent the dimensionless speed of the moving fin nd Pe = epesents sttiony fin. Thus, the enegy eqution fo two pofiles e educes to: θ θ θ ( N N Pe = d d d dx dx dx 4 4 ( ( θ θ c( θ θ ( θ θ ectngul pofile (1 5

6 dθ d θ dθ dx dx dx 4 4 dθ N( θ θ exp( γx Pe = dx ( 1+ ( θ θ γ + + exp( γx ( Nc( θ θ exponentil pofile (11 whee = λtb in which T b is the bse tempetue nd fin tip is insulted. Theefoe, boundy conditions fo this poblem e defined s follows: X = dθ = dx (1 X = 1 θ = 1 (13 Rectngul pofile b-1 Exponentil pofile γ < b-1 Exponentil pofile γ > Figue 1 Schemtic of diffeent moving fin pofiles 4. Solution by diffeentil tnsfomtion method (DTM The one dimensionl tnsfom of eqs. (1, 11 consideed by using the elted definition in Tble 1, we hve the following: ( Rectngul pofile ( j 1( j Θ( j ( 1 θ Θ( i( j i 1( j i Θ( j i j i= i= ( 1 Θ( 1( 1 Θ( 1 c( Θ ( θη ( i+ i+ j i+ j i+ N j j j j j i j i s N Θ i Θ s Θ z Θ j i s z j Pe j+ Θ j+ = i= s= z= (b Exponentil pofile 4 ( ( ( ( θη( ( 1 ( 1 (14 6

7 j i γ 1 ( j i+ 1( j i+ Θ( j i+ + i! ( θ s= i= i= j j s s γ ( j i s+ 1( j i s+ Θ( i Θ( j i s+ + s! j i j j s s γ γ 1 θ γ j i+ 1 Θ j i+ 1 + γ ( j i s+ 1 Θ( i Θ( j i s+ 1 i! s! ( ( ( s= i= i= s= i= j j s s γ + ( i+ 1( j i s+ 1 Θ ( i+ 1 Θ( j i s+ 1 Nc( Θ ( j θη ( j s! j j i j i s 4 N Θ( i Θ( s Θ( z Θ( j i s z θη( j i= s= z= j i γ Pe ( j i+ 1 Θ( j i+ 1 = i! i= (15 In the bove equtions Θ ( j is tnsfomed function of θ ( X. The tnsfomed boundy condition tkes the fom: Θ( 1 = (16 i= ( Θ i = 1 (17 Θ = nd using the Eq. (16 nd eq. (17, nothe vlue of ( Supposing ( β Θ i fo two pofiles cn be clculted. The vlue of β cn be clculted using the eq. (17. Thus end up hving the following: ( Rectngul pofile N Θ ( = 4 4 ( β θ + N ( β θ ( + β θ 1 Pe N Θ (3 = 3 1 c 4 4 ( c( β θ + N ( β θ ( + β θ ( + β θ ( c( β θ ( β θ 1 Pe 6 N + N ( Nc( β θ + N( β θ ( Nc + β N + 4 ( 1+ β θ Θ (4 = (18 (b Exponentil pofile 7

8 N Θ ( = 4 4 ( β θ + N ( β θ c ( + β θ ( Pe 1 N N ( ( ( ( β θ γ + β + γθ c + β + β θ + βθ + θ Θ (3 = 61 ( + β θ The continuing bove pocess, substituting eq. (18 fo eq. (4 fo H=1, we cn obtin the closed fom of the solution: ( Rectngul pofile Pe N ( ( ( c( ( c β θ + N N N β θ β θ + β θ θ( X = β + X + 3 X 1+ β θ 1+ β θ ( 3 ( ( + β θ ( c( β θ ( β θ 1 Pe 6 N + N ( Nc( β θ + N ( β θ ( Nc + β N + 4 ( 1+ β θ X + In ode to obtin the vlue β we used eq. (17. Then we will hve ( 4 4 ( c( β θ + N ( β θ Pe 4 4 Nc( β θ + N ( β θ N θ(1 = β β θ 1+ β θ ( ( + β θ ( c( β θ ( β θ (19 ( Pe 6 N + N ( Nc( β θ + N( β θ ( Nc + β N + 4 ( 1+ β θ + + = 1 (1 1 Solving eq. (1 by MATHEMATICA softwe, gives the vlue of β. Fo the exponentil pofile the sme pocess is used to obtin the vlue of β nd tempetue distibution. 5. Results nd discussions The nlyticl esults e shown fo 5 tems of the finl powe seies hee. The effect of theml conductivity pmete on the tempetue distibution fo both ectngul nd exponentil ( γ = 1 pofiles is shown in fig. fo Nc = 4, N = 4, Pe= 3 ndθ =.. The bottom line ( = epesent the constnt theml conductivity fo the ectngul pofile. As shown in fig., with n incese in the theml conductivity pmete, the fin tip tempetue inceses s well. The fin tip tempetue fo the exponentil pofile is highe thn the ectngul one. Fo vlidting DTM esults, the nlyticl solution is comped with the numeicl solution which is obtined by the fouth-ode Rung-Kutt scheme. Figues 3 nd 4 show how the tempetue distibutions of fin e ffected by the chnges in the convection-conduction pmete fo ectngul nd exponentil ( γ = 1 pofiles, espectively, 8

9 when N =.5, = 1, Pe= ndθ =.5. As depicted in figs. 3, 4, when the convection-conduction pmete inceses, the losing het of the fin by convection gets stonge, the cooling becomes moe effective, thus the tempetue of fin decese. Figue Tempetue pofile fo the ectngul nd exponentil ( γ = 1 pofiles fo diffeent vlues of when: N = 4, N = 4, Pe= 3 ndθ =. c Figue 3 Tempetue distibution of the ectngul pofile fo diffeent vlues of Nc when: N =.5, = 1, Pe= ndθ =.5 9

10 Figue 4 Tempetue distibution of the exponentil pofile ( γ = 1 fo diffeent vlues of Nc when: N =.5, = 1, Pe= ndθ =.5 The effect of dition-conduction pmete on the tempetue pofiles fo ectngul nd exponentil ( γ =.5 pofiles e shown in figs. 5, 6, espectively, with the est of the pmetes fixed t Nc =.5, =.4, Pe= 1.5ndθ =.3. With n incese in the dition-conduction pmete s the convection-conduction pmete, the ditive tnsfe becomes stonge, thus s shown in figs. 3.4, the fin tempetue deceses. Figue 7 depicts the tempetue pofiles fo both ectngul nd exponentil ( γ =.5 pofiles when the mbient tempetueθ is ssigned vlues of.,.4,.6 nd.8. The othe pmetes e fixed t Nc = 4, N = 4, =. nd Pe = 3. As shown in fig. 7, when the mbient tempetue inceses, the tempetue diffeence between the fin nd mbient tempetue becomes shote. Consequently, the fin tempetue inceses. As illustted, the obtined fin tempetue fo the exponentil pofile is highe thn in the ectngul one. While the discepncy between ectngul nd exponentil pofiles becomes shote fo the lge vlues of mbient tempetue (see fig. 7. The effect of the exponentil pmeteγ on the tempetue pofile is illustted in fig. 8 when Nc =, N =, =.4, Pe= ndθ =.5. As the exponentil pmete becomes lge, the fin tempetue inceses. The esults indicte tht fo positive vlues of the exponentil pmete, the fin tempetue of exponentil pofile is lge the ectngul one. While fo the negtive vlues of the exponentil pmete, this esult is evese. The effect of the Peclet numbe Pe (dimensionless speed on the tempetue distibution fo both ectngul nd exponentil ( γ =.3 pofiles is shown in fig. 9 with the emining pmetes fixed t N =.5, N = 1, =.6 ndθ =.6. c 1

11 Figue 5 Tempetue distibution of the ectngul pofile fo diffeent vlues of N when: N =.5, =.4, Pe= 1.5ndθ =.3 c Figue 6 Tempetue distibution of the exponentil pofile ( γ =.5 fo diffeent vlues of N when: N =.5, =.4, Pe= 1.5ndθ =.3 c 11

12 Figue 7 Tempetue distibution fo the ectngul nd exponentil ( γ =.5 pofiles fo diffeent vlues of θ when: Nc = 4, N = 4 =. nd Pe = 3 With n incese in the Peclet numbe, the losing het fom fin sufce becomes stonge, thus the fin tempetue deceses. The compison between DTM nd numeicl esults fo both pofiles when Nc = 1.5, N =.5, =.5 nd Pe = 1.5 is shown in tb. In ll figues nd tb, the geement between nlyticl nd numeicl esults is obseved tht it confims the ccucy of the diffeentil tnsfomtion method to solve nonline boundy vlue poblems. Tble Compison between nlyticl nd numeicl esults fo both the ectngul nd exponentil pofile when: Nc = 1.5, N =.5, Pe= 1.5, =.5 ndθ =.3 Rectngul pofile Exponentil pofile γ = 1 γ = 1 X θ ( X DTM θ ( X NS θ ( X DTM θ ( X NS θ ( X DTM θ ( X NS

13 Figue 8 Tempetue distibution fo diffeent vlues of exponentil pmete when: N =, N = =.4, Pe= ndθ =.5 c Figue 9 Tempetue distibution fo the ectngul nd exponentil ( γ =.3 pofiles fo diffeent vlues of Peclet numbe when: N =.5, N = 1=.6 ndθ =.6 c 13

14 Conclusions In this study, the diffeentil tnsfomtion method ws pplied to solve simultneous convection nd dition het tnsfe poblem in continuously moving fin with tempetue theml conductivity. The ectngul nd exponentil pofiles wee consideed fo moving fin. This method hs been pplied fo the line nd nonline diffeentil equtions. This method is n infinite powe-seies fom nd hs high ccucy nd fst convegence. To vlidte the nlyticl esults, DTM esults e comped with numeicl dt obtined using the fouth ode Runge-Kutt method. The esults illustte how the tempetue distibutions in the moving fin e ffected by the chnges in the embedding pmetes. The esults indicte tht the fin tempetue inceses with n incese in the exponentil pmete s well s the fin tempetue of exponentil pofile fo the positive vlues of exponentil pmete is lge thn the ectngul one. In genel, DTM hs good ppoximte nlyticl solution fo the line nd nonline engineeing poblems without ny ssumption nd lineiztion. Nomencltue A(x fin coss-section [m ] theml expnsion coefficient [K -1 ] b width of the fin c specific het [J Kg -1 K -1 ] H constnt h het tnsfe coefficient [W m - K -1 ] k theml conductivity [W m -1 K -1 ] L fin length [m] N c hpl convection-conduction fin pmete ( =, [-] KbAb 3 N εσ pltb dition-conduction fin pmete ( =, [-] ka b b p peiphey of the fin coss section [m] Pe UL Peclet numbe ( =, [-] α T tempetue [K] t fin thickness [m] U velocity of fin [m s -1 ] X non-dimensionl spce coodinte x dimensionl spce coodinte [m] Y tnsfomed function y(t oiginl nlytic function Geek lettes α k theml diffusivity ( = b, [m s -1 ] ρc γ exponentil pmete λ dimensionl constnt [K -1 ] ε emissivity Θ tnsfomed tempetue 14

15 θ dimensionless tempetue ρ density [kg m -3 ] σ Stefn Boltzmnn constnt [W m K -4 ] Subscipts mbient popety b fin bse Refeences [1] Skidis, B. C., Boundy-lye Behviou on Continuous Solid Sufce: I. Boundy Lye Equtions fo Two Dimensionl nd Axisymmetic Flow, AICHE. J., 7 (1961, 6. [] Eickson, L. E., fn, L. T., Fox, V. G., Het nd Mss Tnsfe on Continuous Moving Flt Plte with Suction o Injection. Ind. Eng. Chem. Fund., 5 (1966, 19. [3] Cotell, R., Flow nd Het Tnsfe in Moving Fluid ove Moving Flt Sufce, Theoeticl nd Computtionl Fluid Dynmics, 1 (7, pp [4] Spow, E. M., Abhm, J. P., Univesl Solutions fo the Stemwise Vition of the Tempetue of Moving Sheet in the Pesence of Moving Fluid, Int. J. Het Mss Tnsfe, 48 ( 5, pp [5] Ch, M. I., Chen, C. K., Cleve, J. W., Conjugte Foce Convection Het Tnsfe fom Continuous, Moving Flt Sheet, Int. J. Het nd Fluid Flow, 11(3 (199, pp [6] Al-Sne, S. A., Mixed Convection Het Tnsfe long Continuously Moving Heted Veticl Plte with Suction o Injection, Int. J. Het Mss Tnsfe, 47 (4, pp [7] Abel, S., Psd K. V., Mhboob A., Buoyncy Foce nd Theml Rdition Effects in MHD Boundy Lye Visco-elstic Fluid Flow ove Continuously Moving Stetching Sufce, Int. J. Theml Science, 44 (5, pp [8] Lee S. L., Tsi, J. S., Cooling of Continuous Moving Sheet of Finite Thickness in the Pesence of Ntul Convection, Int. J. Het Mss Tnsfe, 33 (199, 3, pp [9] Choudhuy, S. R., Jlui, Y., Foced Convective Het Tnsfe fom Continuously Moving Heted Cylindicl Rod in Mteil Pocessing, ASME Jounl of Het Tnsfe, 116 (1994 pp [1] Mendez, F., Tevino, C., Het Tnsfe Anlysis on Moving Flt Sheet Emeging into Quiescent Fluid, J. Themophysics nd Het Tnsfe, 16 (, pp [11] Fox, V. G., Eickson, L. E., fn, L. T., The Lmin Boundy Lye on Moving Continuous Flt Sheet Immesed in Non-Newtonin Fluid, AICHE. J., 15 (1969, 3, pp [1] Howell, T. G., Jeng, D. R., De Witt, K. J., Momentum nd Het Tnsfe on Continuous Moving Sufce in Powe Lw Fluid, Int. J. Het Mss Tnsfe, 4 (1997, 8, pp [13] Tobi, M., Yghoobi, H., Sedodin, S., Assessment of Homotopy Petubtion Method in Nonline Convective-Rditive Non-Fouie Conduction Het Tnsfe Eqution with Vible Coefficient, Theml Science, (In Pess [14] Shu, A. K., Mthu, M. N., Chtuni, P., Bhtiy, S. S., Momentum nd Het Tnsfe fom Continuous Sufce to Powe-Lw Fluid, Act Mechnic, 14 (, pp

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