ANALYTICAL INVESTIGATION OF NONLINEAR MODEL ARISING IN HEAT TRANSFER THROUGH THE POROUS FIN

Transcription

1 ANALYTICAL INVESTIGATION OF NONLINEAR MODEL ARISING IN HEAT TRANSFER THROUGH THE POROUS FIN Yasser ROSTAMIYAN a, Davood Domiri GANJI a*, Iman RAHIMI PETROUDI b, and Medi KHAZAYI NEJAD a a Department of Mecanical Engineering, Islamic Azad University, Sari Branc, Sari, Iran b Young researcers club, sari branc, Islamic Azad University, sari, Iran Introduction In tis letter simple analytical metods called omotopy perturbation metod(hpm), variation iteration metod(vim) and perturbation metod(pm) are employed to approac temperature distribution of porous fins. also energy balance and Darcy's model used to formulate te eat transfer equation. To study te termal performance, a type case considered is finite-lengt fin wit insulated tip. Te obtained results from variation iteration metod (VIM) are compared wit oter analytical tecniques proposed before. Tese metods are omotopy perturbation metod and perturbation metod (PM). Also BVP is applied as a numerical metod for validation. Te obtained results sows tat te VIM is more accurate, stable and more appropriate tan oter tecniques. Also it is found tat tis metod is powerful matematical tools and can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering specially some eat transfer equations. Keywords: Heat transfer, porous fin, Homotopy perturbation metod (HPM), Variational iteration metod (VIM), Perturbation metod (PM) Te eat transfer rate enancement in fins wit reducing size and cost is te aim of many researcers in engineering applications[1-3]. To acieve tis goals, convective eat transfer coefficient, surface area available and temperature difference between surface and surrounding fluid are suc as ways can be used. Nonlinear problems and penomena play an important role in applied matematics, pysics, engineering and oter brances of science specially some eat transfer equations. Except for a limited number of tese problems, most of tem do not ave precise analytical solutions; terefore, tese nonlinear equations sould be solved using approximation metods. Perturbation metod is one of te well-known metods to solve nonlinear problems, it is based on te existence of small/large parameters, te so-called perturbation quantity [4, 5]. Many nonlinear problems do not contain suc kind of perturbation quantity, and we can use non-perturbation metods, suc as te artificial small parameter metod [6], te δ-expansion metod [7], te Adomian s decomposition metod [8], te omotopy perturbation metod (HPM) [9 13], te variational iteration metod (VIM) [14 30], te optimal omotopy asymptopic metod (OHAM)[31-3] and te optimal omotopy perturbation metod (OHPM)[33]. Te present work, te basic idea of te omotopy perturbation metod, variational iteration metod, and perturbation metod are introduced and ten we ave applied to find te approximate solutions of nonlinear differential equations governing on porous fin. Result demonstrates te variational iteration metod is simple and offers superior accuracy compared wit te perturbation metod and omotopy perturbation metod. Also It is found tat tese metod are powerful matematical tools and tat tey * Corresponding autor; ddg_davood@yaoo.com

2 can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering. Nomenclature C P Specific eat Da Darcy number, kt / Nu Nusselt number, L / k f Gr K Grasof number Termal conductivity Kr Termal conductivity ratio, k eff / k f Pr Prandtl number, / M convection parameter Ra Rayleig number, Gr Pr Porous parameter S T( x ) T b Temperature at any point Temperature at fin base t Tickness of te fin Bi Biot Number, L c / k b V Velocity of fluid passing troug te fin any point wx w Widt of te fin x Axial coordinate X Dimensionless axial coordinate, x / L Greek symbols Coefficient of volumetric termal expansion Temperature difference Porosity ratio Stepen Boltzmann constant Dimensionless temperature, Base temperature difference, T T b Subscripts Kinematic viscosity Density b s f eff Solid properties Fluid properties Porous properties Governing equations

3 Figure 1. Scematic diagram of fin profile under consideration As sown in fig. 1, a rectangular fin profile is considered. Te dimensions of te fin are Lengt L, widt w and tickness t. Te cross section area of te fin is constant. Tis fin is porous to allow te flow of infiltrate troug it. Te following assumptions are made to solve tis problem. Te porous medium is isotropic and omogenous, Te porous medium is saturated wit single- pase fluid, Te surface radiant excange is neglected, Pysical properties of bot fluid and solid matrix are constant, Te temperature inside fin is only function of X, Tere is no temperature variation across te fin tickness, Te solid matrix and fluid are assumed to be at local termal equilibrium wit eac oter, Te interactions between te porous medium and te clear fluid can be simulated by te Darcy formulation. Apply an energy balance to te slice segment of te fin of tickness x sown in fig. 1, requires tat q( x) q( x x) mcp T( x) T ( px) T( x) T (1) Te mass flow rate of te fluid passing troug te porous material can be written as From te Darcy's model we ave Substitutions of eq. () and (3) into eq. (1) yields m w xw () gk w (3) T T q( x) q( x x) cp g k w x T ( x) T pt ( x) T (4) As, x 0 eq. (4) becomes From Fourier s Law of conduction, we ave dq cp g k w T ( x) T pt ( x) T (5) dx

4 Were A is te cross-sectional area of te fin A w. t conductivity of te porous fin given by 1 dt q k A eff dx (6) and k eff is te effective termal k k k eff f s Substitution eq. (6) into eq. (5) gives d T cp g k p T ( x) T T ( x) T 0 (7) dx t k k A eff Hence, wit applying energy balance equation at steady state condition as sown in fig., and T T X Introducing non-dimensional temperature function Were, T T ave eff b x and X into eq. (7) we L d dx L S M 0 (8) Da.. x Ra L Porous parameter, S and Convection parameter, p M were S is a porous kr t ks A parameter tat indicates te effect of te permeability of te porous medium as well as buoyancy effect so Higer value of S indicate iger permeability of te porous medium or iger buoyancy forces. M is a convection parameter tat indicates te effect of surface convecting of te fin. Case: finite-lengt fin wit insulated tip Variational iteration metod 1 1, 0 0 (9) To illustrate te basic idea of variational iteration metod, we consider te following general nonlinear system Lu Nu g x (10) were L is a linear operator, N nonlinear operator, g t a omogeneous term. According to te variational iteration metod, we can construct te following iteration formulation t u n1 t un t Lun N un g d 0 (11) were is a general Lagrangian multiplier [34-35], wic can be identified optimally via te variational teory [36-37],. Te subscript n indicates te n t approximation and u n is considered as a restricted variation [38,39], i.e., u n 0. Te application of Variational iteration metod (VIM)

5 First we construct a correction functional wic reads x d n n1 x n x L S n M n d 0 d (1) were is general Lagrange multiplier. Making te above correction functional stationary, we can obtain following stationary conditions t M t t t 0, 1 0, 0 t x (13) tx Te Lagrange multiplier, terefore, can be identified as 1 : e e M x M x M (14) As a result, we obtain te following iteration formula n1 x n x M x M x e e d n (15) L S n M n d M d x 1 0 Now we start wit an arbitrary initial approximation tat satisfies te initial condition x M Mx 0 sec cos (16) Using te above variational formula (15) for n 0, substituting eq. (16) into eq. (15) and after some simplifications, we ave x C sec M. cosmx Mx Mx 4. Mx. Mx. Mx L S. e. e e 1 6e e M cos M (17) Were 0 1 A C, tat A sec M cos M 3. M M 4. M. M. M L S. e. e e 1 6e e and M cos M so on. In te same way, te rest of te components of te iteration formula can be obtained. Analysis of He s Homotopy perturbation metod To illustrate te basic ideas of tis metod, we consider te following equation wit te boundary condition of Au f r 0, r (18)

6 u Bu, 0, r (19) n were A is a general differential operator, B a boundary operator, f() r a known analytical function and is te boundary of te domain. A can be divided into two parts, wic are L and N, were L is linear and N is nonlinear. eq. (8) can terefore be rewritten as follows Homotopy perturbation structure is sown as follows were Lu N u f r 0, r (0) H, p 1 p L L u p A f r 0 0 (1) r, p : 0,1 R () In eq. (1), p 0,1 is an embedding parameter and u 0 is te first approximation tat satisfies te boundary condition. We can assume tat te solution of eq. (8) can be written as a power series in p, as following and te best approximation for solution is 0 1 n i0 i i p p p (3) u lim p (4) Te application of HPM In tis section, we will apply te HPM to nonlinear ordinary differential eq. (8) wit a boundary condition (9). According to te HPM, we can construct omotopy of eq. (8) as follows 1 P ( x) ( x) p ( x) R ( x) M ( x) 0, (5) Tat R L S is constant. We consider θ as follows n i x x P x P x P x, (6) 0 1 From eq. (3), if te two terms approximations are sufficient, we will obtain wit substituting from eq. (6) into eq. (5) and some simplification and rearranging based on powers of p terms, wit assumption M 1 we ave i0 i p : x + x 0 1 1, 0 0 (7)

7 1 1 1( x) p : x R x 0 1 0, 0 0 (8) Solving eqs. (7), (8) wit boundary conditions, we ave x x x 0 ( ) e e x e e e e e R 1 6e e 1 x e e e e e e e e e e x 1 e R 1 6e e R1 6e e e e e e e e e e e e e 3 6 e 3 e 4 x 4x x (9) (30) Te solution of tis equation, wen p 1, will be as follows x x x (31) 0 1 Te application of Perturbation metod Wit cose of low value of S, we can use instead of it. now For very small, let us assume a regular perturbation expansion and calculate te first tree terms [5], tus we assume 0 1 (3) After substituting eq. (3) into eq. (8), collecting terms wit te powers of as 0, 1, and equating coefficients of eac power of on bot sides we ave: 0 d 0 x : 0 x 0 dx , d 1 x : 1 x [ 0x] 0 dx , 0 0 d x : x 0 x1 x 0 dx 1 0, 0 0 (33) (34) (35)

8 Te solution of eqs. (3) (34) are x e e x e e e e x 4 1 e e 6e 1 x e 1 e e e e e e e e e 1 e e 6e 1 e 1 6 e e e e e e e e ee e e e 3 6 e 3 e 1 1 x 8e 3e 34e 36 0 e 15 e 6 e 15 e 6 e e 1 x 4 x x 4x x 10 x 8 x x7 x3 x9 x3 x9 x7 180e x 60e x 60e x 60e x 60e x 180e x x5 x5 x5 x6 x8 3x3 x3 180e x 180e x 145e 80e 3e 3e 154e 4x 3e 154e x3 x4 x3 x1 10 1x 93x x5 x11 x7 x9 3x5 8 x 3x9 x 3e 154e 48e 5e 48e 5e 3e x 6 145e 5e 56e 34e 480e 9e 19e 8e 3e (36) (37) (38) 3x7 11 x 3x7 x10 x7 x4 x6 x 9e 5e 9e 8e 56e 3e 80e 8e 3x 5 3x3 9e 3 e ) And approximation solution obtained by perturbation metod will be as follow Results and discussion x x x x (39) 0 1

9 Figure. Te distribution of axial nondimensional temperature along te finite fin for different Values of S wit VIM. Figure 3. Te comparison of te results of te metods at M 1, S 0.1 Figure 4. Te comparison of te results of te metods at M 1, S 0.3 Figure 5. Te comparison of te results of te metods at M 1, S 0.6 An analytical solution for te temperature distributions in te fin by different metod approximant was obtained. Te results are compared wit omotopy perturbation metod (HPM), Variational iteration metod (VIM), Perturbation Metod(PM), and accurate numerical solution using (BVP). Fig. 3-5, depict te temperature distribution wit te axial distance along te fin for te tree metods. It is observed tat te variational iteration metod approximant solution is more accurate tan oter metods. Comparing fig. 3-5, gives closer results to numerical solution. It is interesting to note tat variational iteration metod is very close to te numerical results. Fig., sows te variation of dimensionless temperature distribution wit te axial distance along te fin wen te value of S is varying and M is constant. From fig., we can see tat te value of

10 dimensionless temperature decreases along te fin Lengt. It sould be noted tat as te value of increases, te temperature decreases rapidly and te fin quickly reaces te surrounding temperature. As te value of S increases, te fins cool down rapidly. S Conclusion Te present work, te basic idea of te variational iteration metod, omotopy perturbation metod and perturbation metod are introduced and ten we ave applied to metods to solve te temperature distribution of a porous fin because a second order non-linear ordinary differential equation as been derived as te governing equation for tis problem. Result demonstrates te variational iteration metod is simple and offers superior accuracy compared wit te perturbation metod and omotopy perturbation metod. Also It is found tat tese metod are powerful matematical tools and tat tey can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering specially some eat transfer equations. References [1] Kiwan, S., Al-Nmir, M.A., Using porous fins for eat transfer enancement, ASME J. Heat Transfer, 13 (001), pp s [] Nguyen, A., Aziz, A., Te eat transfer rates from convecting radiating fins for different profile sapes, Heat and Mass Transfer, 7 (199),, pp [3] Tadulkar, M., Misra, A., Te effect of combined radiation and convection eat transfer in porous cannel bounded by isotermal parallel plates, International Journal of Heat and Mass Transfer, 47 (004), 5, pp [4] Cole, J.D., Perturbation Metods in Applied Matematics, Blaisdell Waltam, MA, 1968 [5] Nayfe, A.H., Perturbation Metods, Wiley, New York, USA, 000 [6] Lyapunov, A.M., General Problem on Stability of Motion, Taylor & Francis, London, UK, 199 [7] Karmisin,, A.V., Zukov, A.I., Kolosov, V.G., Metods of Dynamics Calculation and Testing for Tin- Walled Structures, Masinostroyenie, Moscow, 1990 [8] Adomian, G., Solving Frontier Problems on Pysics: Te Decomposition Metod, Kluwer Academic,Dordrect, 1994 [9] He, J.H., A coupling metod of omotopy tecnique and perturbation tecnique for nonlinear problems, Internat. J. Non-Linear Mec. 35 (000), 1, pp [10] Davood Domiri GANJI, Zaman Ziabks GANJI, and Hosain Domiri GANJI, determination of temperature distribution for annular fins wit temperature dependent termal conductivity by HPM, termal scince, 15(011), pp.s111-s115 [11] He, J.H., Application of omotopy perturbation metod to nonlinear wave equations, Caos Solitons Fractals, 6 (005), pp [1] Esmaeilpour, M., Ganji, D.D., Application of He's omotopy perturbation metod to boundary layer flow and convection eat transfer over a flat plate, Pys. Lett. A, 37(007), 1, pp [13] Esmaeilpour, M., Ganji, D.D., Moseni, E., Application of omotopy perturbation metod to micropolar flow in a porous cannel, J. Porous Media, 1 (009), 5, pp

11 [14] SB, Coskun., MT, Atay., Fin efficiency analysis of convective straigt fin wit temperature dependent termal conductivity using variational iteration metod. Appl Term Eng, 8 (008), pp [15] Miansari MO, Ganji DD, Miansar ME. Application of He s variational iteration metod to nonlinear eat transfer equations. Pys Lett A, 37 (008), pp [16] Ganji,D.D., Sadigi, A., Application of omotopy-perturbation and variational iteration metods to nonlinear eat transfer and porous media equations, J. Comput. Appl. Mat, 07(007), pp [17] Ganji, D.D., Jannatabadi, M., Moseni, E., Application of He s variational iteration metod to nonlinear Jaulent Miodek equations and comparing it wit ADM, J. Comput. Appl. Mat, 07(007), pp [18] He, J.H., Variational iteration metod some recent results and new interpretations, Journal of Computational and Applied Matematics, 07 (007), 1, pp [19] He, J.H., Wu, X.H., Construction of solitary solution and compaction-like solution by variational iteration metod, Caos Solitons & Fractals, 9 (006), 1, pp [0] Odibat, Z.M., Momani, S., Application of variational iteration metod to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7 (006), 1, pp [1] Xu, Lan., Variational principles for coupled nonlinear Scr dinger equations, Pysics Letters A, 359 (006), pp [] Momani, S., Abuasad, S., Application of He s variational iteration metod to Helmoltz equation, Caos Solitons & Fractals, 7 (006), 5, pp [3] Ganji, D.D., Jamsidi, N., Ganji, Z.Z., HPM and VIM metods for finding te exact solutions of te nonlinear dispersive equations and sevent-order Sawada Kotera equation, International Journal of Modern Pysics B, 3 (009), 1, pp [4] Ganji, D.D., Afrouzi, G.A., Talarposti,R.A., Application of He s variational iteration metod for solving te reaction diffusion equation wit ecological parameters, Computers and Matematics wit Applications, 54 (007), pp [5] Ganji, D.D., Tari, Hafez., Baksi Jooybari, M., Variational iteration metod and omotopy perturbation metod for nonlinear evolution equations, Computers and Matematics wit Applications, 54 (007), pp [6] Rafei, M., Ganji, D.D., Daniali, H., Pasaei, H., Te variational iteration metod for nonlinear oscillators wit discontinuities, Journal of Sound and Vibration, 305 (007), pp [7] Ganji, D.D., Afrouzi, G.A., Talarposti, R.A., Application of variational iteration metod and omotopy-perturbation metod for nonlinear eat diffusion and eat transfer equations, Pysics Letters A, 368 (007), pp [8] He, J.H., Variational iteration metod a kind of nonlinear analytical tecnique: Some examples, International Journal of Non-linear Mecanics, 34 (1999), 4, pp [9] He, J.H., Approximate analytical solution for seepage wit fractional derivatives in porous media, Computational Metods in Applied Mecanics and Engineering, 167 (1998), pp

12 [30] GANJI, D. D., SAJJADI, H., new analytical solution for natural convection of darcian fluid in porous media prescribed surface eat flux, termal scince, 15( 011), pp. S1-S7 [31] Esmaeilpour M, Ganji D.D.,solution of te Jeffery-Hamel flow problem by optimal omotopy asymptotic metod, computers and matematics wit applications, 59 (010), pp [3] Herisanu N, Marinca V, Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia, Meccanica, 45 (010), pp [33] Marinca, V., Herisanu, N, Nonlinear dynamic analysis of an electrical macine rotor-bearing system by te optimal omotopy perturbation metod, computers and matematics wit applications, 61 (011), pp [34] Bildik,N., Konuralp, A., Te use of variational iteration metod, differential transform metod and adomian decomposition metod for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci.Numer. Simul, 7 (006), pp [35] Tari, Hafez., Ganji, D.D., Babazade, H., Te application of He's variational iteration metod to nonlinear equations arising in eat transfer, Pys. Lett, A, 363 (007), 3, pp [36] Inokuti, M., General use of te Lagrange multiplier in non-linear matematical pysics, in: S. Nemat- Nasser (Ed.), Variational Metod in te Mecanics of Solids, Pergamon, Oxford, (1978), pp [37] He, J.H., Semi-inverse metod of establising generalized variational principles for fluid mecanics wit empasis on turbo macinery aerodynamics, Int J Turbo Jet Engines, 14(1997), (1),pp. 3 8 [38] Finlayson, B.A., Te Metod of Weigted Residuals and Variational Principles, Academic Press, New York, USA, 197 [39] He, J.H., Variational iteration metod a kind of non-linear analytical tecnique: some examples, Int. J. Non-linear Mec, 34 (1999), pp