Homotopy perturbation method for temperature distribution, fin efficiency and fin effectiveness of convective straight fins

Transcription

1 *Corresponding author: Homotopy perturbation method for temperature distribution, fin efficiency and fin effectiveness of convective straight fins... Pinar Mert Cuce 1 *, Erdem Cuce 2 and Cemalettin Aygun 3 1 Department of Mathematics Education, Karadeniz Technical University, 618, Trabzon, Turkey; 2 Institute of Sustainable Energy Technology, The University of Nottingham, University Park Campus, Nottingham NG7 2RD, UK; 3 Department of Mechanical Engineering, Karadeniz Technical University, 618, Trabzon, Turkey... Abstract Homotopy perturbation method is a novel approach that provides an approximate analytical solution to differential equations in the form of an infinite power series. In this work, homotopy perturbation method has been used to evaluate temperature distribution, efficiency and effectiveness of straight fins exposed to convection. The fin efficiency and the fin effectiveness have been obtained as a function of thermogeometric fin parameter. The results have revealed that the homotopy perturbation method is a very effective and practical approach for a rapid assessment of physical systems. The resulting correlation equations can assist thermal design engineers for designing of straight fins with both constant and temperature-dependent thermal conductivity. Keywords: homotopy perturbation method; fins; efficiency; effectiveness Received 24 June 212; revised 9 July 212; accepted 25 July INTRODUCTION Heat transfer is a discipline of thermal engineering that deals with the generation, consumption, conversion and exchange of thermal energy between physical systems. It is a very broad scientific field and hence numerous studies are carried out every year on various specific areas of heat transfer science. Among the popular topics that are extensively studied all over the world, heat transfer from extended surfaces (fins) stands out with its wide concept and rapidly developing applications. Fins are extensively used to enhance the rate of heat transfer from a hot surface especially in thermal engineering applications where cooling is required. Besides the traditional applications, such as internal combustion engines, compressors and heat exchangers, fins also prove effective in heat rejection systems in space vehicles and in cooling of electronic components [1]. An extensive review on the industrial applications of extended surfaces was presented by Kern and Kraus [2]. In recent years, several attempts have been made to use fins in renewable energy technologies such as photovoltaics to minimize irreversibilities during the energy conversion [3]. Depending on the growing significance of environmental issues [4, 5], fins have been recommended for cost-effective cooling applications. HPM is a novel analytical approach to solve both linear and nonlinear differential equations encountered as modeling physical systems. In recent years, the application of the HPM in nonlinear problems has been devoted by scientists and engineers because the approach provides a fast assessment with high accuracy [6]. Arslanturk [7] investigated the convective fins with step change in thickness via HPM. Temperature distribution within the fin was evaluated. The optimum geometry that maximizes the heat transfer rate for a given fin volume was analyzed. The results demonstrated that the maximum heat dissipation is always higher for a step fin than that of a constant thickness fin for the identical design condition. Aziz and Hug [8] utilized the regular perturbation method to compute a closed-form solution for a straight convecting fin with temperature-dependent thermal conductivity. Domairry and Fazeli [9] investigated the efficiency of convective fins by using HPM. The results have been compared with those of the exact solution and Adomian s decomposition method (ADM) performed by Arslanturk. Chowdhury [1] compared the performances of modified HPM and ADM for solving nonlinear heat transfer equations. The local heat transfer coefficient was considered to vary with a power-law function of temperature. Comparisons were made between 13-term ADM International Journal of Low-Carbon Technologies 214, 9, 8 84 # The Author 212. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( by-nc/3./), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com doi:1.193/ijlct/cts62 Advance Access Publication 22 August 212 8

2 Homotopy perturbation method for temperature distribution and 6-term modified HPM. It was observed that modified HPM provides higher accuracy than ADM when only 6-term is used in the series expansion. Ganji and Rajabi [11] utilizedhpmto solve an unsteady nonlinear convective radiative equation and a nonlinear convective radiative conduction equation containing two small parameters. Ganji [12] evaluated HPM which does not need small parameters in the equations, and compared the prescribed approach with the perturbation and numerical methods in the heat transfer field. Bouaziz et al. [13] presented the efficiency of longitudinal fins with temperature-dependent thermophysical properties. Recently, homotopy theory has become an effective mathematical tool, when it was successfully coupled with perturbation theory [14 18]. HPM has also been used to determine the roots of nonlinear equations [19 21]. Some researchers have preferred different techniques to solve energy equations of fins as Chang [22]. Ghorbani and Saberi-Nadjafi [23] andgorbani [24] wereabletoovercomethedifficultyin ADM through the so-called He polynomials. Through the literature review, it is clear that several attempts have been made to solve energy balance equations of extended surfaces via HPM. However, any dimensionless expression for fin effectiveness has not been developed so far. The goal of this study is to evaluate the effectiveness of HPM for a well-known heat transfer problem. Temperature distribution within the fin, fin efficiency and fin effectiveness are determined via HPM, and the results are compared with those of the exact solution. The thermogeometric fin parameter considered as a variable in the analysis is a term that emerges in nondimensionalization and includes heat transfer coefficient as well as cross-sectional area, thermal conductivity, length and perimeter of the fin. 2 PROBLEM DESCRIPTION The general form of the energy equation for an extended surface under the influence of natural convection is given as follows [25]: d 2 T f dx 2 þ 1 A c da c dx dtf dx 1 A c h k da s dx ðt f T 1 Þ¼ where x is the direction of heat transfer, T f is the fin temperature at any x, A c is the cross-sectional area of the fin, h is the heat convection coefficient of the ambient air, k is the thermal conductivity of the fin, A s is the surface area measured from base to x, T 1 is the ambient temperature. For the prescribed problem, A c is constant and A s ¼ Px where P is the fin perimeter. From this point of view, with da c =dx ¼ andda s =dx ¼ P,Equation(1)reducesto: d 2 T f Ph ðt dx2 f T 1 Þ¼ ð2þ ka c The fin profile considered in the study is illustrated in Figure 1. The aforementioned fin is attached to a hot surface with fin base temperature T b, extends into a fluid media with temperature T 1 and the fin tip is insulated. The thermal conductivity of the fin is considered constant in the study. ð1þ Figure 1. Schematic of a convective straight fin. By employing the following dimensionless parameters: f ¼ T T 1 ; c ¼ x 1=2 T b T 1 L ; d ¼ hpl2 ka c The formation of the energy equation reduces to: d 2 f dc 2 d2 f ¼ ð3þ where f is the dimensionless temperature; c is the dimensionless coordinate; d is the thermogeometric fin parameter. which is subject to f ðþ ¼! at the fin tip fð1þ ¼1! at the fin base: 3 HOMOTOPY PERTURBATION METHOD The homotopy perturbation method was first proposed by the Chinese mathematician J. Huan He [26 28]. To illustrate the basic ideas of this method, we consider the following general nonlinear differential equation: AðuÞ f ðrþ ¼; r [ V with boundary conditions B ¼ r [ z ð5þ where A is a general differential operator, B is a boundary operator, f(r) is a known analytic function, and z is the boundary of the domain V. The operator A can be divided into linear and nonlinear parts called L and N. Hence, Equation (4) can be rearranged as follows: LðuÞþNðuÞ f ðrþ ¼ Then a homotopy nðr; pþ : V ½; 1Š! R is constructed which satisfies Hðn; pþ ¼LðnÞ Lðu ÞþpLðu Þþp½NðnÞ f ðrþš ¼ ð7þ where p [ ½; 1Š is an embedding parameter and u is the first ð4þ ð6þ International Journal of Low-Carbon Technologies 214, 9,

3 P. M. Cuce et al. approximation that satisfies the boundary condition. The process of changes in p from zero to unity is that of n(r, p) changing from u to u(r). n is considered as follows: n ¼ n þ pn 1 þ p 2 n 2 þ...þ p j n j ð8þ The method assumes that n ; n 1 ; n 2 ;...; n j can be determined analytically through the functions occurred in the analysis. However, it may not be possible all the time. Finally, the best approximation for solution is obtained as follows: u ¼ lim p!1 n þ n 1 þ n 2 þ...þ n j ð9þ 4 RESULTS AND DISCUSSION 4.1 Fin temperature distribution HðÌ; pþ ¼LðÌÞ Lðf ÞþpLðf Þþp½NðÌÞ f ðrþš ¼ Ì ¼ Ì þ pì 1 þ p 2 Ì 2 þ p 3 Ì 3 : ð1þ ð11þ By rearranging the homotopy equation with the following terms, LðÌÞ ¼ d2 Ì dc 2 ; NðÌÞ ¼ d2 Ì ; Lðf Þ¼ d2 f dc 2 ðì þ pì 1 þ p 2 Ì 2 þ p 3 Ì 3 Þ pd 2 ðì þ pì 1 þ p 2 Ì 2 þ p 3 Ì 3 Þ ¼ Through the power of p terms p : Ì ¼ dì Ì ¼ 1 for c ¼ 1 p 1 : Ì 1 d2 Ì ¼ ð12þ ð13þ ð14þ ð15þ ð16þ The accuracy of the solution can be enhanced by increasing the number of terms. Solving Equations (14, 17, 2 and 23) via a simple MATLAB code, Ì, Ì 1, Ì 2 and Ì 3 are determined as follows: Ì ¼ 1 Ì 1 ¼ d2 2 c2 d2 2 Ì 2 ¼ d4 24 c4 d4 4 c2 þ 5d4 24 Ì 3 ¼ d6 72 c6 d6 48 c4 þ 5d6 48 c2 61d6 72 Finally, the solution is obtained as follows: Ì ¼ 1 þ d2 2 c2 d2 2 þ d4 24 c4 d4 4 c2 þ 5d4 24 þ d6 72 c6 d6 48 c4 þ 5d6 48 c2 61d6 72 ð25þ ð26þ ð27þ ð28þ ð29þ Figure 2 illustrates the dimensionless temperature distribution within the fin with respect to the different values of thermogeometric fin parameter. An excellent agreement between HPM results and exact solution is observed. The results indicate that if the thermogeometric fin parameter increases the mean temperature of the fin decreases. 4.2 Fin efficiency As reported by Bergman et al. [25], efficiency of an extended surface is determined by the following equation: h ¼ Q f Q m ð3þ The amount of heat dissipation from the entire fin is found by using Newton s law of cooling as follows: dì 1 Ì 1 ¼ for c ¼ 1 p 2 : Ì 2 d2 Ì 1 ¼ dì 2 Ì 2 ¼ for c ¼ 1 p 3 : Ì 3 d2 Ì 2 ¼ dì 3 Ì 3 ¼ for c ¼ 1 ð17þ ð18þ ð19þ ð2þ ð21þ ð22þ ð23þ ð24þ Figure 2. Temperature distribution within the fin for different thermogeometric fin parameter. 82 International Journal of Low-Carbon Technologies 214, 9, 8 84

4 Homotopy perturbation method for temperature distribution Q f ¼ ð L PðT T 1 Þdx ð31þ Maximum heat dissipation is obtained if the fin base temperature is kept throughout the fin. Q m ¼ PLðT b T 1 Þ ð32þ From this point of view, Equation (3) can be rearranged as follows: Ð L h ¼ PðT T 1Þdx ¼ PLðT b T 1 Þ ð 1 c¼ fðcþdc ð33þ Figure 3 depicts the variation of the fin efficiency with thermogeometric fin parameter. It can be clearly seen that h remarkably decreases with the increasing values of d. The results obtained from HPM are in excellent agreement with those of the exact solution. 4.3 Fin effectiveness The effectiveness of an extended surface is determined by the following equation [25]: 1 ¼ Q f Q fb ð34þ where Q fb refers to the amount of heat dissipation from the area of fin base. If the width of the fin (w) is considered quite bigger than the fin thickness (t), Q fb can be expressed by the following equation: Q fb ¼ Pt 2 ðt b T 1 Þ ð35þ Equation (34) can be rearranged by using Equations (31 and 35) as follows: Figure 3. Variation of the fin efficiency with thermogeometric fin parameter. Figure 4. Variation of the fin effectiveness with thermogeometric fin parameter and length/thickness ratio. Ð L 1 ¼ 2PðT T 1Þdx ¼ PtðT b T 1 Þ ð 1 c¼ 2tfðcÞdc ð36þ where t ¼ L/t; t is the length/thickness ratio of the fin. Figure 4 illustrates the results obtained by HPM and exact solution for the fin effectiveness. As it is firmly seen that HPM results are in good agreement with those of the exact solution for each value of t. The dimensionless fin effectiveness increases with increasing t. On the other hand, it slightly decreases with increasing thermogeometric fin parameter. 5 CONCLUSIONS In this study, homotopy perturbation method has been used to evaluate temperature distribution, efficiency and effectiveness of straight fins exposed to convection. A dimensionless expression for the fin effectiveness has been defined. The fin efficiency and the fin effectiveness have been obtained as a function of thermogeometric fin parameter. The results indicated that HPM is a very effective method which provides a simply approximate exact solution without any assumption of linearization. The expressions presented may be very useful for engineers in terms of providing a fundamental key before dealing with heat conduction problems with strong nonlinearities. In further studies, it is hoped to apply HPM to different fin profiles with variable cross-sectional area to make the problem more elaborative. Moreover, real-time conditions such as considering thermal conductivity of the fin material is a function of fin temperature will be considered to assess the HPM under strong nonlinearity. Furthermore, efficiency of the method will be compared with the alternative approaches such as ADM and Galerkin methods. Thus, the HPM can be applied to various physical systems in heat transfer science. International Journal of Low-Carbon Technologies 214, 9,

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