APPLICATION OF VIM, HPM AND CM TO THE SYSTEM OF STRONGLY NONLINEAR FIN PROBLEM. Islamic Azad University, Sari, Iran

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1 Journal of Engineering and Tehnology APPLICATION OF VIM, HPM AND CM TO THE SYSTEM OF STRONGLY NONLINEAR FIN PROBLEM M. R. Shirkhani,H.A. Hoshyar *, D.D. Ganji Departent of Mehanial Engineering, Sari Branh, Islai Azad University, Sari, Iran Reeived: 3 Otober: 8 Deeber, Aepted: Deeber 3 ABSTRACT The nonlinear fin proble with teperature-dependent theral ondutivity and heat transfer oeffiient is analytially studied. Colloation ethod (CM), Variation iteration ethod (VIM) and Hootopy Perturbation Method (HPM) are used to solve the present proble. Also, fourth order Runge Kutta nuerial ethod is applied as a nuerial ethod for validation. Analytial results are presented through the graphs and the tables in various values of paraeters. The results reveal that the CM is very effetive, siple and ore aurate than other tehniques. Furtherore, we analyze the effets of soe physial appliable paraeters in this proble suh as theral ondutivity paraeter ( ), thero-geoetri fin paraeter ( M ) and heat transfer ode ( ). KEYWORDS: Colloation ethod (CM); Variational iteration ethod (VIM); Hootopy Perturbation Method (HPM); Heat transfer; Fin. INTRODUCTION Fins or etended surfaes are frequently used to enhane the heat transfer between a solid surfae and its surrounding ediu. Etend surfaes are etensively used in various industrial appliations; for eaple, liquid-gas heat ehangers, air-ooled internal obustion engines, the eletrial apparatus, nuleate boiling, et. Sine Most of probles and sientifi phenoenon suh as heat transfer probles for the fins (Kiwan, 7; Gorla Reddy & Bakier, ; Doairry & Fazeli, ; Ganji, ; Khani et al, 9), are inherently of nonlinearity. Therefore, the differential equation for a onvetive fin does not adit an eat solution and these nonlinear equations should be solved using other ethods, Suh as nuerial analysis or analytial ethod. In the analytial perturbation ethod, we should eert a sall paraeter in the equation. Therefore, finding this paraeter and eerting it into the equation is diffiulties of this ethod. In reent years, sientists have presented soe new ethods for solving nonlinear differential equations; for instane, δ-epansion ethod (Ganji & Hashei, a), Adoian s deoposition ethod (Ganji & Hashei, b), Hootopy perturbation ethod (HPM) (He, 999; He 5a; He, 5b; Esaeilpour & Ganji, 7; He, ; Ganji & Rajabi, 6; Ganji, Ganji & Ganji, ) and Variational iteration ethod (VIM) (He, 7; He & Wu, 6; Ganji, Rostaiyan, Petroudi & Nejad, 4; Ganji, Tari & Jooybari, 4; He, 999). One of the other sei-eat ethods is the weighted residual ethods (WRMs). Colloation ethod (CM), Galerkin ethod (GM), and least square ethod (LSM) are eaples of WRMs. These ethods are the ost effetive and * Corresponding Eail: hoshyarali@yail.o

2 onvenient ones for both linear and nonlinear equations. Stern and Rasussen used olloation ethod for solving a third order linear differential equation (Stern and Rasussen, 996). Hu and Li and Herrera et al. applied olloation ethod for Poisson s equation and advetion diffusion equation respetively (Hu & Li, 6; Herrera, Diazviera, Yates, 4). Arnau et al., presented a new alternative ethod based in polynoial olloation and used it to odel the flow in intake and ehaust of internal obustion engine. Hendi and Albugai solved Fredhol Volterra integral equation using olloation ethod (Hendi & Albugai, ). Reently Hatai and Ganji applied CM on non-newtonian nanofluid passing through the porous edia between two oaial ylinders (Hatai and Ganji, 3). In this artile, the nonlinear fin proble with teperature-dependent theral ondutivity and heat transfer oeffiient is solved through the three ethods: Colloation ethod, Hootopy perturbation ethod and the Variation iteration ethod. Also Runge-Kutta ethod is used to evaluate the eellene and auray of the proposed ethods. The oputations show that CM allows us to obtain approiations with an error relative to the nuerial solution saller than the errors obtained using other ethods.. METHODOLOGY. Governing equations Consider a straight fin with an arbitrary onstant ross-setional area ; perieter and length. The fin is attahed to a base surfae of teperature, etends into a fluid of teperature, and its tip is insulated. The one-diensional energy balane equation is given: L T a T b A p d dt A k ph T Ta dx dx X L, () The theral ondutivity of the fin aterial is assued to be a linear funtion of teperature aording to: k k T ka T T a () where k a is the theral ondutivity at the abient fluid teperature of the fin and is the paraeter desribing the theral ondutivity variation. Also It is onsidered that vary with teperature funtion by the following equations: h T T a h ht hb T b T a (3) where h b is the heat transfer oeffiient at the base teperature. The eponent depends on the heat transfer ode. Typial values of n are / 4 for lainar fil boiling or

3 ondensation, onstant for heat transfer oeffiient, / 4 / 3 for lainar natural onvetion, for turbulent natural onvetion, for nuleate boiling, and 3 for radiation. In order to siplify the energy equation, the diensionless paraeters are defined as follows: T T X k h pl a b,, k, Tb Ta, M Tb Ta L ka ka A (4) Hene, the energy Equation () will take the for (Khani, Ahadzadeh Raji, Haedi Nejad, 9): d d k M dx dx ( ), (5) The boundary onditions are: () at the tip, at the bae (6) The governing Equation (5) an be rewritten into the following for: M (7) ( ) ( ). Basi idea of Colloation ethod Colloation ethod is one of the approiation tehniques for solving differential equations alled the Weighted Residual Methods (WRMs). For the oneption of the ain idea of this ethod, suppose a differential operator D is ated on a funtion u to produe a funtion p (Hatai and Ganji, 3): D( u( )) p( ) (8) We wish to approiate u by a funtion, whih is a linear obination of basi funtions hosen fro a linearly independent set. That is: u u u n ii (9) i Now, when substituted into the differential operator, D, the result of the operations is not p (). Hene an error or residual will eist: E( ) R( ) D( u( ) p( )) () The notion in the Colloation is to fore the residual to zero in soe average sense over the doain. That is: R ( ) W i( ), i,,..., n () where the nuber of weight funtions W i is eatly equal the nuber of unknown onstants i in u. The result is a set of n algebrai equations for the unknown onstants

4 . For Colloation ethod, the weighting funtions are taken fro the faily of Dira funtions in the doain. That is, w ( ) ( ). The Dira funtion has the property of: i i i if i ( i ) other whise () The residual funtion in Equation () ust be fored to be zero at speifi points..3 Appliation of olloation ethod We wish to obtain an approiate solution for this proble in the interval. To onstrut a trial solution, we hoose the basi funtion to polynoial in. The trial solution ontains four undeterined oeffiients and satisfies the onditions for all values of as follows: ( ) ( ) 3 ( ) 4 ( ) (3) Whereas the trial solution satisfies the boundary ondition of Equation (6). The auray of the solution an be iproved by inreasing the nuber of its ters. When is introdued into differential equation it yields residual as follows: R M R( ) M (4) Now the proble of finding approiate solution of the proble in the interval beoes one adjusting the values of,, 3 and. So that residual stays lose to zero throughout the interval. The basi assuption is that the residual does not deviate uh fro zero between olloation loations. For reahing to this ai, four speifi points should be hosen. These points are: R 3 4,,,, 5 R 5 R 5 R 5 (5) R( ) M M (6)

5 R( ) M M R( ) M M R( ) M M (7) (8) (9) Thus we an obtain oeffiient for different value of, M and. For eaple, Using Colloation ethod with., M =,, is as follows: = ().4 Variational iteration ethod To illustrate the basi idea of variational iteration ethod, we onsider the following general nonlinear syste: Lu Nu g t () where L is a linear operator, N nonlinear operator, a hoogeneous ter. Aording to the variational iteration ethod, we an onstrut the following iteration forulation: t u t u t Lu N u g d () n n n n where is a general Lagrangian ultiplier, whih an be identified optially via the variational theory. The subsript n indiates the nth approiation and is onsidered as a restrited variation, i.e., u..5 Appliation of Variational iteration ethod First we onstrut a orretion funtional whih reads: n g t u n M d. n n n n n n (3)

6 Now we start with an arbitrary initial approiation that satisfies the initial ondition. For eaple, when M,.4 and, is as follows: e e e e e e (4) Also, aking the above orretion funtional stationary, we an obtain following stationary onditions: t t, t, t (5) t t The Lagrangian ultiplier an therefore be identified as: e e (6) Substituting and into Equation (3) and after soe siplifiations, we have: () =.99e.99e e e e e e e e e e where A, that A.4685 (7) () = sinh( ).6766 e.6766 e e e e e.4638osh(3 3).4638osh(3 3) osh(3 4) (8) where B, that B In a siilar anner, we will obtain other solutions for different ases M results are presented graphially..6 Analysis of He s Hootopy perturbation ethod To eplain the basi ideas of this ethod, onsider the following equation;, and. The Au f r, r (9) with the boundary ondition of: u Bu,, r n (3) where A is a general differential operator, B a boundary operator, f ( r) a known analytial funtion and is the boundary of the doain. A an be divided into two parts, whih are L and N, where L is linear and N is nonlinear Equation () an therefore be rewritten as follows;

7 Lu N u f r, r (3) Hootopy perturbation struture is shown as follows; H, p p L L u pa f r (3) where r, p :, R (33) is an ebedding paraeter and is the first approiation that satisfies the boundary ondition. It an be assued the solution of Equation (4) an be written as a power series in, as follows; In Equation (4), p, p u n i p p i p (34) i and the best approiation for the solution is: u li p (35).7 Appliation of Hootopy perturbation ethod In this setion, we will apply the HPM to nonlinear ordinary differential syste (). aording to Equation (7),Using HPM, when M,.4 and leads to: H, p P ( ) ( ) p (. ).( ) (36) We onsider ( ) as following: p p (37) By substituting fro Equation (37) into Equation (36) and after soe siplifiations and rearrangeents based on powers of -ters, we have: p :,, p, (38).4 ( ).4, : p,, ( ).4 ( ).4 ( ), :.8 ( ) p ( ), Solving Equations (38) to (4) with boundary onditions, we have: (39) (4)

8 e e e e e e e e e e 5 e e e e e e e e e e e e e e e e e e e e e e e 3 e e 76e 5e 5e 33e 9e e 5 6e e 4e e e 4e e e e 4e e e ee 6ee 4ee e e 4e 6e 4e e e e e e (4) (4) (44) The solution of this equation, when p, will be as follows: (44) In a siilar anner, we will obtain other solutions for different ases M, results are presented in net setion. and. The 3. RESULTS AND DISCUSSION In this anusript the olloation ethod suh as analytial tehnique is eployed to find an analytial solution of the nonlinear fin proble. The results are opared with other analytial ethods suh as VIM and HPM. For validation all these results are opared with the nuerial solution. Figures -3 and Table show the teperature distribution with the aial distane along the fin for the three ethods. It is interesting to note that olloation ethod is very lose to the nuerial results and the results of HPM and VIM are signifiantly in error. Figure. Coparison between the CM, VIM, HPM and nuerial solution for, M,.4

9 Figure. Coparison between the CM, VIM, HPM and nuerial solution for M,,.4 Figure 3. Coparison between the CM, VIM, HPM and nuerial solution for M 3,,.4 Table The results of CM, VIM, HPM, and nuerial ethods for, M,.4 CM VIM HPM NUM ERROR VIM ERROR HPM ERROR CM

10 Figures 4, 5 and 6 show oparison between the nuerial solutions and CM solutions in prediting for different values of theral ondutivity ( ), thero-geoetri fin paraeter ( M ) and heat transfer ode ( ) respetively. We an see a very good agreeent between the olloation and the nuerial results. In addition, It an be seen in these figures that by inreasing the and, veloity profiles inreases, but the veloity profiles dereases with the inrease in M. Figure 4. Effet of thero-geoetri fin paraeter ( M ) on when / 4,. Figure 5. Effet of thero-geoetri fin paraeter ( ) on when / 4, M Figure 6. Effet of thero-geoetri fin paraeter ( ) on when M,.

11 4. CONCLUSION In this paper, nonlinear heat transfer equations for fin with teperature-dependent theral ondutivity and heat transfer oeffiient are presented and the CM has been suessfully applied to find the ost eat analytial solution. Furtherore, the obtained solutions by olloation ethod are opared with VIM, HPM and nuerially solutions. The results deonstrate that the CM is powerful atheatial tools and has eellent agreeent with nuerial outoes. Also auray of the solution an be inreased by inreasing the stateents of the trial funtions. The CM is very effetive, sipler and offers superior auray opared with the variation iteration ethod and Hootopy Perturbation Method. It does not need any perturbation, linearization or sall paraeter versus Hootopy Perturbation Method and Variation iteration ethod. ACKNOWLEDGEMENT The authors would like to thank to everyone and all organisation espeially Islai Azad University that supports this work. REFERENCES Kiwan, S. (6). Theral Analysis of Natural Convetion Porous Fins. Transport In Porous Media, 67(), 7-9. Gorla, R., & Bakier, A. (). Theral analysis of natural onvetion and radiation in porous fins. International Couniations In Heat And Mass Transfer, 38(5), Doairry, G., & Fazeli, M. (9). Hootopy analysis ethod to deterine the fin effiieny of onvetive straight fins with teperature-dependent theral ondutivity. Couniations In Nonlinear Siene And Nuerial Siulation, 4(), Doiri, G., Ziabkhsh, G., & Doiri, G. (). Deterination of teperature distribution for annular fins with teperature dependent theral ondutivity by HPM. Theral Siene, 5(suppl. ), -5. Khani, F., Raji, M., & Nejad, H. (9). Analytial solutions and effiieny of the nonlinear fin proble with teperature-dependent theral ondutivity and heat transfer oeffiient. Couniations In Nonlinear Siene And Nuerial Siulation, 4(8), Ganji, D.D., Hashei Kahapi, S.H. (). Analytial and Nuerial Methods in Engineering and Applied Sienes, Progress in Nonlinear Siene, 3, Ganji, D.D. Hashei Kahapi. S.H. (). Analysis of Nonlinear Equations in Fluids. Progress in Nonlinear Siene,; 3, -94.

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