كوهساركنده

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كوهساركنده

  1. 1. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2009, Article ID 831362, 8 pages doi:10.1155/2009/831362 Research Article Solution of Temperature Distribution in a Radiating Fin Using Homotopy Perturbation Method M. J. Hosseini, M. Gorji, and M. Ghanbarpour Department of Mechanical Engineering, Noshirvani University of Technology, P.O. Box 484, Babol, Iran Correspondence should be addressed to M. Gorji, gorji@nit.ac.ir Received 27 November 2008; Accepted 22 January 2009 Recommended by Saad A. Ragab Radiating extended surfaces are widely used to enhance heat transfer between primary surface and the environment. The present paper applies the homotopy perturbation to obtain analytic approximation of distribution of temperature in heat fin radiating, which is compared with the results obtained by Adomian decomposition method ADM . Comparison of the results obtained by the method reveals that homotopy perturbation method HPM is more effective and easy to use. Copyright q 2009 M. J. Hosseini et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Most scientific problems and phenomena such as heat transfer occur nonlinearly. Except a limited number of these problems, it is difficult to find the exact analytical solutions for them. Therefore, approximate analytical solutions are searched and were introduced 1–5 , among which homotopy perturbation method HPM 6–12 and Adomain decomposition method ADM 13, 14 are the most effective and convenient ones for both weakly and strongly nonlinear equations. The analysis of space radiators, frequently provided in published literature, for example, 15–22 , is based upon the assumption that the thermal conductivity of the fin material is constant. However, since the temperature difference of the fin base and its tip is high in the actual situation, the variation of the conductivity of the fin material should be taken into consideration and includes the effects of the variation of the thermal conductivity of the fin material. The present analysis considers the radiator configuration shown in Figure 1. In the design, parallel pipes are joined by webs, which act as radiator fins. Heat flows by conduction from the pipes down the fin and radiates from both surfaces.
  2. 2. 2 Mathematical Problems in Engineering 2w x W b Figure 1: Schematic of a heat fin radiating element. Here, the fin problem is solved to obtain the distribution of temperature of the fin by homotopy perturbation method and compared with the result obtained by the Adomian decomposition method, which is used for solving various nonlinear fin problems 23–25 . 2. The Fin Problem A typical heat pipe space radiator is shown in Figure 1. Both surfaces of the fin are radiating to the vacuum of outer space at a very low temperature, which is assumed equal to zero absolute. The fin is diffuse-grey with emissivity ε, and has temperature-dependent thermal conductivity k, which depends on temperature linearly. The base temperature Tb of the fin and tube surfaces temperature is constant; the radiative exchange between the fin and heat pipe is neglected. Since the fin is assumed to be thin, the temperature distribution within the fin is assumed to be one-dimensional. The energy balance equation for a differential element of the fin is given as 26 : d dT 2w k T − 2εσT 4 0, 2.1 dx dx where k T and σ are the thermal conductivity and the Stefan-Boltzmann constant, respectively. The thermal conductivity of the fin material is assumed to be a linear function of temperature according to K T Kb 1 λ T − Tb , 2.2 where kb is the thermal conductivity at the base temperature of the fin and λ is the slope of the thermal conductivity temperature curve. Employing the following dimensionless parameters: 3 T εσb2 Tb x θ , ψ , ξ , β λTb , 2.3 Tb kw b the formulation of the fin problem reduces to 2 d2 θ dθ d2 θ β βθ − ψθ4 0, 2.4a dξ2 dξ dξ2
  3. 3. Mathematical Problems in Engineering 3 with boundary conditions dθ 0 at ξ 0, 2.4b dξ θ 1 at ξ 1. 2.4c 3. Basic Idea of Homotopy Perturbation Method In this study, we apply the homotopy perturbation method to the discussed problems. To illustrate the basic ideas of the method, we consider the following nonlinear differential equation, A θ −f r 0, 3.1 where A θ is defined as follows: A θ L θ N θ , 3.2 where L stands for the linear and N for the nonlinear part. Homotopy perturbation structure is shown as the following equation: H θ, P 1 − p L θ − L θ0 p A θ −f r 0 3.3 Obviously, using 3.3 we have H θ, 0 L θ − L θ0 0, 3.4 H θ, 1 A θ −f r 0, where p ∈ 0, 1 is an embedding parameter and θ0 is the first approximation that satisfies the boundary condition. We Consider θ and as M θ pi θi θ0 pθ1 p2 θ2 p3 θ3 p4 θ4 ··· . 3.5 i 0 4. The Fin Temperature Distribution Following homotopy-perturbation method to 2.4a , 2.4b , and 2.4c , linear and non-linear parts are defined as d2 θ L θ , dξ2 2 4.1 dθ d2 θ N θ β βθ 2 − ψθ4 , dξ dξ with the boundary condition given in 2.4b , θ 0 is any arbitrary constant, C.
  4. 4. 4 Mathematical Problems in Engineering Then we have dθ 0 at ξ 0, θ C at ξ 0. 4.2 dξ Substituting 3.5 in to 4.1 and then into 3.3 and rearranging based on power of p-terms, we have the following p0 ∂2 θ0 ξ 0, 4.3a ∂ξ2 dθ0 0 at ξ 0, θ0 C at ξ 0; 4.3b dξ p1 2 ∂2 ∂2 ∂ θ1 ξ βθ0 ξ θ0 ξ β θ0 ξ − ψθ0 ξ 4 0, 4.4a ∂ξ2 ∂ξ2 ∂ξ dθ1 0 at ξ 0, θ1 0 at ξ 0; 4.4b dξ p2 ∂2 ∂2 ∂2 θ2 ξ βθ0 ξ θ1 ξ βθ1 ξ θ0 ξ ∂ξ2 ∂ξ2 ∂ξ2 4.5a ∂ ∂ 2β θ0 ξ θ1 ξ − 3 4ψθ0 ξ θ1 ξ 0, ∂ξ ∂ξ dθ2 0 at ξ 0, θ2 0 at ξ 0; 4.5b dξ p3 2 ∂2 ∂2 ∂2 ∂ θ3 ξ βθ2 ξ θ0 ξ βθ0 ξ θ2 ξ β θ1 ξ ∂ξ2 ∂ξ2 ∂ξ2 ∂ξ ∂2 ∂ ∂ 4.6a βθ1 ξ θ1 ξ 2β θ0 ξ θ2 ξ − 4ψθ0 ξ θ2 ξ 3 ∂ξ2 ∂ξ ∂ξ − 6ψθ0 ξ θ1 ξ 2 2 0, dθ3 0 at ξ 0, θ3 0 at ξ 0; 4.6b dξ
  5. 5. Mathematical Problems in Engineering 5 1 ψ 1 0.8 Dimensionless temperatrure, θ 0.6 ψ 10 0.4 ψ 100 0.2 0 0 0.2 0.4 0.6 0.8 1 Dimensionless coordinate, ξ HPM ADM Figure 2: Comparison of the HPM and ADM for β 1.0 and M 14. p4 ∂2 ∂2 ∂2 θ4 ξ βθ3 ξ θ0 ξ βθ0 ξ θ3 ξ − 4ψθ0 ξ θ3 ξ 3 ∂ξ2 ∂ξ2 ∂ξ2 ∂2 ∂ ∂ βθ2 ξ θ1 ξ 2β θ0 ξ θ3 ξ − 12ψθ0 ξ θ2 ξ θ1 ξ 2 4.7a ∂ξ2 ∂ξ ∂ξ ∂2 ∂ ∂2 β θ1 ξ θ2 ξ βθ1 ξ θ2 ξ − 4ψθ0 ξ θ1 ξ 3 0, ∂ξ2 ∂ξ ∂ξ2 dθ4 0 at ξ 0, θ4 0 at ξ 0; 4.7b dξ and so forth. By increasing the number of the terms in the solution, higher accuracy will be obtained. Since the remaining terms are too long to be mentioned in here, the results are shown in tables. Solving 4.3a , 4.4a , 4.5a , 4.6a , and 4.7a results in θ ξ . When p → 1, we have 1 1 2 7 4 1 5 2 13 3 10 6 θ ξ C ψC4 ξ2 ψ C ξ − βC ψξ ψ C ξ 2 6 2 180 11 2 8 4 1 2 6 2 23 4 13 8 19 3 11 6 − ψ C βξ β C ψξ ψ C ξ − ψ C βξ 4.8 24 2 720 60 7 2 9 2 4 1 3 7 2 β C ψ ξ − β C ψξ · · · . 8 2
  6. 6. 6 Mathematical Problems in Engineering Table 1: The dimensionless tip temperature for ψ 1. Number of the terms M 5 M 7 M 10 M 14 in the solution M Method HPM ADM HPM ADM HPM ADM HPM ADM β 0.6 0.827821 0.819185 0.826552 0.825079 0.8267319 0.825063 0.82675 0.825052 β 0.4 0.813389 0.814489 0.813351 0.813279 0.8133683 0.810866 0.813369 0.813236 β 0.2 0.797708 0.80021 0.797712 0.799025 0.7977122 0.797812 0.797712 0.797809 β 0 0.779177 0.777778 0.779147 0.777765 0.7791452 0.776554 0.779145 0.775333 β −0.2 0.757702 0.752987 0.75698 0.752967 0.7568182 0.754144 0.756802 0.754132 β −0.4 0.819743 0.814465 0.816013 0.813252 0.8132926 0.810831 0.812155 0.813201 β −0.6 0.710294 0.715063 0.703219 0.700812 0.6988481 0.696134 0.696764 0.694918 Table 2: The dimensionless tip temperature for ψ 1000. Number of the terms M 5 M 7 M 10 M 14 in the solution M Method HPM ADM HPM ADM HPM ADM HPM ADM β 0.6 0.139382 0.144312 0.135502 0.137711 0.1330265 0.134125 0.132616 0.132718 β 0.4 0.138351 0.143412 0.134243 0.136526 0.1315594 0.132756 0.130091 0.131026 β 0.2 0.137341 0.142026 0.133009 0.134937 0.1301168 0.131213 0.129486 0.129613 β 0 0.136349 0.141055 0.131797 0.133785 0.1286995 0.129663 0.127504 0.127327 β −0.2 0.135376 0.140165 0.13061 0.132654 0.127308 0.128143 0.125348 0.125431 β −0.4 0.134422 0.139565 0.129445 0.131856 0.1259426 0.127342 0.124118 0.124432 β −0.6 0.133485 0.138374 0.128303 0.130525 0.1246034 0.125336 0.122715 0.122936 5. Results The coefficient C representing the temperature at the fin tip can be evaluated from the boundary condition given in 2.4c using the numerical method. Tables 1 and 2 show the dimensionless tip temperature, that is, coefficient C, for different thermal conductivity parameters, β. The tables state that the convergence of the solution for the higher thermo-geometric fin parameter, ψ, is faster than the solution with lower fin parameter. It is clear from the tables that the solution is convergent. In order to investigate the accuracy of the homotopy solution, the problem is compared with decomposition solution 26 , also the corresponding results are presented in Figure 2. It should be mentioned that the homotopy results in the tables are arranged for first 5, 7, 10, 14 terms of the solution M . It is seen that the results by homotopy perturbation method, and adomian decomposition method are in good agreement. The results of the comparison show that the difference is 3.1% in the case of the strongest nonlinearity, that is, β 1.0 and ψ 100. 6. Conclusions In this work, homotopy perturbation method has been successfully applied to a typical heat pipe space radiator. The solution shows that the results of the present method are in excellent agreement with those of ADM and the obtained solutions are shown in the figure
  7. 7. Mathematical Problems in Engineering 7 and tables. Some of the advantage of HPM are that reduces the volume of calculations with the fewest number of iterations, it can converge to correct results. The proposed method is very simple and straightforward. In our work, we use the Maple Package to calculate the functions obtained from the homotopy perturbation method. References 1 E. M. Abulwafa, M. A. Abdou, and A. A. Mahmoud, “The solution of nonlinear coagulation problem with mass loss,” Chaos, Solitons & Fractals, vol. 29, no. 2, pp. 313–330, 2006. 2 A. Aziz and T. Y. Na, Perturbation Methods in Heat Transfer, Series in Computational Methods in Mechanics and Thermal Sciences, Hemisphere, Washington, DC, USA, 1984. 3 J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006. 4 M. M. Rahman and T. Siikonen, “An improved simple method on a collocated grid,” Numerical Heat Transfer Part B, vol. 38, no. 2, pp. 177–201, 2000. 5 A. V. Kuznetsov and K. Vafai, “Comparison between the two- and three-phase models for analysis of porosity formation in aluminum-rich castings,” Numerical Heat Transfer Part A, vol. 29, no. 8, pp. 859–867, 1996. 6 J.-H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 695–700, 2005. 7 J.-H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 207–208, 2005. 8 J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999. 9 J.-H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000. 10 J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000. 11 J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. 12 J.-H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004. 13 G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988. 14 G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. 15 J. G. Bartas and W. H. Sellers, “Radiation fin effectiveness,” Journal of Heat Transfer, vol. 82C, pp. 73–75, 1960. 16 J. E. Wilkins Jr., “Minimizing the mass of thin radiating fins,” Journal of Aerospace Science, vol. 27, pp. 145–146, 1960. 17 B. V. Karlekar and B. T. Chao, “Mass minimization of radiating trapezoidal fins with negligible base cylinder interaction,” International Journal of Heat and Mass Transfer, vol. 6, no. 1, pp. 33–48, 1963. 18 R. D. Cockfield, “Structural optimization of a space radiator,” Journal of Spacecraft Rockets, vol. 5, no. 10, pp. 1240–1241, 1968. 19 H. Keil, “Optimization of a central-fin space radiator,” Journal of Spacecraft Rockets, vol. 5, no. 4, pp. 463–465, 1968. 20 B. T. F. Chung and B. X. Zhang, “Optimization of radiating fin array including mutual irradiations between radiator elements,” Journal of Heat Transfer, vol. 113, no. 4, pp. 814–822, 1991. 21 C. K. Krishnaprakas and K. B. Narayana, “Heat transfer analysis of mutually irradiating fins,” International Journal of Heat and Mass Transfer, vol. 46, no. 5, pp. 761–769, 2003. 22 R. J. Naumann, “Optimizing the design of space radiators,” International Journal of Thermophysics, vol. 25, no. 6, pp. 1929–1941, 2004. 23 C. H. Chiu and C. K. Chen, “A decomposition method for solving the convective longitudinal fins with variable thermal conductivity,” International Journal of Heat and Mass Transfer, vol. 45, no. 10, pp. 2067–2075, 2002. 24 C. H. Chiu and C. K. Chen, “Application of Adomian’s decomposition procedure to the analysis of convective-radiative fins,” Journal of Heat Transfer, vol. 125, no. 2, pp. 312–316, 2003.
  8. 8. 8 Mathematical Problems in Engineering 25 D. Lesnic and P. J. Heggs, “A decomposition method for power-law fin-type problems,” International Communications in Heat and Mass Transfer, vol. 31, no. 5, pp. 673–682, 2004. 26 C. Arslanturk, “Optimum design of space radiators with temperature-dependent thermal conductiv- ity,” Applied Thermal Engineering, vol. 26, no. 11-12, pp. 1149–1157, 2006.
  9. 9. Journal of Applied Mathematics and Decision Sciences Special Issue on Decision Support for Intermodal Transport Call for Papers Intermodal transport refers to the movement of goods in Before submission authors should carefully read over the a single loading unit which uses successive various modes journal’s Author Guidelines, which are located at http://www of transport (road, rail, water) without handling the goods .hindawi.com/journals/jamds/guidelines.html. Prospective during mode transfers. Intermodal transport has become authors should submit an electronic copy of their complete an important policy issue, mainly because it is considered manuscript through the journal Manuscript Tracking Sys- to be one of the means to lower the congestion caused by tem at http://mts.hindawi.com/, according to the following single-mode road transport and to be more environmentally timetable: friendly than the single-mode road transport. Both consider- ations have been followed by an increase in attention toward Manuscript Due June 1, 2009 intermodal freight transportation research. Various intermodal freight transport decision problems First Round of Reviews September 1, 2009 are in demand of mathematical models of supporting them. Publication Date December 1, 2009 As the intermodal transport system is more complex than a single-mode system, this fact offers interesting and challeng- ing opportunities to modelers in applied mathematics. This Lead Guest Editor special issue aims to fill in some gaps in the research agenda Gerrit K. Janssens, Transportation Research Institute of decision-making in intermodal transport. (IMOB), Hasselt University, Agoralaan, Building D, 3590 The mathematical models may be of the optimization type Diepenbeek (Hasselt), Belgium; Gerrit.Janssens@uhasselt.be or of the evaluation type to gain an insight in intermodal operations. The mathematical models aim to support deci- sions on the strategic, tactical, and operational levels. The Guest Editor decision-makers belong to the various players in the inter- Cathy Macharis, Department of Mathematics, Operational modal transport world, namely, drayage operators, terminal Research, Statistics and Information for Systems (MOSI), operators, network operators, or intermodal operators. Transport and Logistics Research Group, Management Topics of relevance to this type of decision-making both in School, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, time horizon as in terms of operators are: Belgium; Cathy.Macharis@vub.ac.be • Intermodal terminal design • Infrastructure network configuration • Location of terminals • Cooperation between drayage companies • Allocation of shippers/receivers to a terminal • Pricing strategies • Capacity levels of equipment and labour • Operational routines and lay-out structure • Redistribution of load units, railcars, barges, and so forth • Scheduling of trips or jobs • Allocation of capacity to jobs • Loading orders • Selection of routing and service Hindawi Publishing Corporation http://www.hindawi.com
  10. 10. Fixed Point Theory and Applications Special Issue on Impact of Kirk’s Results in the Development of Fixed Point Theory Call for Papers Kirk’s fixed point theorem published in 1965 had a profound Guest Editor impact in the development of the fixed point theory over the Tomas Domínguez-Benavides, Departamento de Análisis last 40 years. Through the concept of generalized distance, Matemático, Universidad de Sevilla, 41080 Sevilla, Spain; Tarski’s classical fixed point theorem may be seen as a variant tomasd@us.es of Kirk’s fixed point theorem in discrete sets. This shows among other things the power of this theorem. This special issue will focus on any type of applications of Kirk’s fixed point theorem. It includes: • Nonexpansive mappings in Banach and metric spaces • Multivalued mappings in Banach and metric spaces • Monotone mappings in ordered sets • Multivalued mappings in ordered sets • Applications to nonmetric spaces like modular func- tion spaces • Applications to logic programming and directed gra- phs Before submission authors should carefully read over the journal’s Author Guidelines, which are located at http://www .hindawi.com/journals/fpta/guidelines.html. Articles publis- hed in this Special Issue shall be subject to a reduced Article Processing Charge of C200 per article. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sys- tem at http://mts.hindawi.com/ according to the following timetable: Manuscript Due October 1, 2009 First Round of Reviews January 1, 2010 Publication Date April 1, 2010 Lead Guest Editor M. A. Khamsi, Department of Mathematical Science, University of Texas at El Paso, El Paso, TX 79968, USA; mohamed@utep.edu Hindawi Publishing Corporation http://www.hindawi.com
  11. 11. International Journal of Mathematics and Mathematical Sciences Special Issue on Selected Papers from ICMS’10 Call for Papers The First International Conference on Mathematics and Ayman Badawi, American University of Sharjah, P.O. Box Statistics (ICMS’10), which will be held in Sharjah, UAE 26666, Sharjah, UAE; abadawi@aus.edu (March 18–21, 2010) aims at bringing together researchers Carl Cowen, Mathematical Sciences Building, Purdue and scientists working in the fields of pure and applied University, 150 N. University Street West, Lafayette, IN mathematics, mathematics education, and statistics. The 47907, USA; ccowen@iupui.edu proposed technical program of the conference will include paper presentations and keynote lectures in algebra, analysis, Hana Sulieman, American University of Sharjah, P.O. Box applied mathematics, applied statistics, differential equa- 26666, Sharjah, UAE; hsulieman@aus.edu tions, discrete mathematics, financial mathematics, mathe- matics education, number theory, numerical analysis, prob- ability theory, statistics, stochastic differential equations, topology, and geometry. Organizing this conference will help build bridges between institutions and encourage interaction among researchers from different disciplines from the region and worldwide. Authors will be required to rewrite their conference papers as full-length manuscripts that extend substantially from their conference submission. All papers will be rigorously reviewed, and three review reports for each paper will be solicited. Before submission authors should carefully read over the journal’s Author Guidelines, which are located at http://www .hindawi.com/journals/ijmms/guidelines.html. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sys- tem at http://mts.hindawi.com/ according to the following timetable: Manuscript Due May 1, 2010 First Round of Reviews August 1, 2010 Publication Date November 1, 2010 Lead Guest Editor Mahmoud Anabtawi, American University of Sharjah, P.O. Box 26666, Sharjah, UAE; manabtawi@aus.edu Guest Editors Zayid Abdulhadi, American University of Sharjah, P.O. Box 26666, Sharjah, UAE; zahadi@aus.edu Hindawi Publishing Corporation http://www.hindawi.com

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