Homotopi pertürbasyon yöntemi kullanılarak kesikli dairesel kanatların lineer olmayan ısıl analizi

Öz

Bu çalışmada, dairesel ve düz kısımlardan oluşan kesikli dairesel kanatçıkların ısıl performansları incelenmiştir. Kanatın ısı iletim katsayısının lineer olarak sıcaklığa bağlı olduğu kabul edilmiştir. Doğrusal olmayan diferansiyel denklemler, kesikli dairesel kanadın her iki bölümü için enerji dengesi denklemi uygulanarak elde edilmiş ve bu denklemler homotopi pertürbasyon yöntemi uygulanarak çözülmüştür. Yalıtılmış kanat ucu, sabit kanat taban sıcaklığı ve iki bölümün ara yüzü arasındaki ortak sınır koşulları göz önünde bulundurulmuştur. Isıl performansı ve sıcaklık dağılımını etkileyen kesik kanat yarıçap oranı (𝜀), kesik kanat yükseklik oranı (𝛿), termo-geometrik kanat parametresi (𝜓) ve ısıl iletkenlik parametresi (𝛽) incelenmiştir. Sonuçlar, homotopi pertürbasyon yönteminin, bu tür doğrusal olmayan diferansiyel denklemlerin çözümleri için güvenilir bir yöntem olduğunu göstermiştir. Homotopi pertürbasyon yönteminin sonuçları ile sayısal sonlu farklar yönteminin sonuçları arasında çok iyi bir uyum elde edilmiştir. Kesikli dairesel kanat veriminin dairesel ve dikdörtgen kanatçıklar arasında yer aldığı ve kesik kanat yükseklik oranının artmasıyla arttığı görülmektedir. Örneğin 𝜀 = 2, 𝜓1 = 1.0 ve 𝛽 = 0 durumunda 𝛿 = 0, 0.5 ve 1 için kanat verimi değerleri sırasıyla 0.692, 0.718 ve 0.762'dir.

Anahtar Kelimeler

• Homotopi pertürbasyon yöntemi
• Kesikli dairesel kanat
• Değişken ısıl iletkenlik

Nonlinear thermal analysis of serrated fins by using homotopy perturbation method

Abstract

Thermal analysis of serrated fins which are consist of annular and plain sections are investigated. Serrated fin’s thermal conductivity is assumed to change linearly with temperature. Nonlinear differential equations are obtained by applying the energy balance equation for both sections of the serrated fin and these equations are solved by applying homotopy perturbation method. Insulated fin tip, constant fin base temperature and common boundary conditions between the interface of two sections are considered. Serrated fin radii ratio (𝜀), segment height ratio (𝛿), thermo-geometric fin parameter (𝜓) and thermal conductivity parameter (𝛽) effecting the thermal performance and temperature distribution are investigated. The results showed that the homotopy perturbation is a reliable method for the solutions of such nonlinear differential equations. A very good agreement with the homotopy perturbation method and the numerical finite difference method are obtained. It is seen that, serrated fin efficiency lays between annular and rectangular fins and increases with the increase of segment height ratio and thermal conductivity parameter. Such as, fin efficiency values under the condition of 𝜀 = 2, 𝜓1 = 1.0 and 𝛽 = 0 for 𝛿 = 0, 0.5, and 1 are 0.692, 0.718, and 0.762, respectively.

Keywords

• Homotopy perturbation method
• Serrated fin
• Variable thermal conductivity

Kaynakça

1. [1] Kraus AD, Aziz A, Welty J. Extended Surface Heat Transfer. New York, USA, Wiley, 2001.
2. [2] Reid DR, Taborek J. “Selection criteria for plain and segmented finned tubes for heat recovery systems”. ASME Journal of Engineering for Gas Turbines and Power, 116(2), 406-410, 1994.
3. [3] Hashizume K, Morikawa R, Koyama T, Matsue T. “Fin Efficiency of Serrated Fins”. Heat Transfer Engineering, 23(2), 6-14, 2002.
4. [4] He JH. “Homotopy perturbation technique”. Computer Methods in Applied Mechanics and Engineering, 178(3-4), 257-262, 1999).
5. [5] He JH. “A coupling method of a homotopy technique and a perturbation technique for non-linear problems”. International Journal of Non-Linear Mechanics, 35(1), 37-43, 2000.
6. [6] He JH. “Homotopy perturbation method: A new nonlinear analytical technique”. Applied Mathematics and Computation, 135(1), 73-79, 2003.
7. [7] He JH. “Homotopy Perturbation Method for Solving Boundary Value Problems”. Physics Letters A, 350(1-2), 87-88, 2006).
8. [8] Ganji DD. “The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer”. Physics Letters A, 355(4-5), 337-341, 2006.

9. [9] Rajabi A, Ganji DD, Taherian H. “Application of homotopy perturbation method in nonlinear heat conduction and convection equations”. Physics Letters A, 360(3-4), 570-573, 2007.
10. [10] Hosseini MJ, Gorji M, Ghanbarpour M. “Solution of temperature distribution in a radiating fin using homotopy perturbation method”. Mathematical Problems in Engineering, 2009, 1-8, 2009.
11. [11] Cowdhury MSH, Hashim I. “Analytical solutions to heat transfer equations by homotopy perturbation method revisited”. Physics Letters A, 372(8), 1240-1243, 2008.
12. [12] Domairry G, Nadim N. “Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation”. International Communications in Heat and Mass Transfer, 35(1), 93-102, 2008.
13. [13] Arslantürk C. “Optimization of straight fins with a step change in thickness and variable thermal conductivity by homotopy perturbation method”. Journal of Thermal Science and Technology, 30(2), 09-19, 2010.
14. [14] Ganji DD, Ganji ZZ, Ganji HD. “Determination of temperature distribution for annual fins with temperature-dependent thermal conductivity by HPM”. Thermal Science, 15(1), 111-115, 2011.
15. [15] Saedodin S, Shahbabaei M. “Thermal Analysis of Natural Convection in Porous Fins with Homotopy Perturbation Method (HPM)”. Arabian Journal for Science and Engineering, 38(8), 2227-2231, 2013.
16. [16] Roy PK, Das A, Mondal H, Mallick, A. “Application of homotopy perturbation method for a conductiveradiative fin with temperature dependent thermal conductivity and surface emissivity”. Ain Shams Engineering Journal, 6(3), 1001-1008, 2015.
17. [17] Cuce E, Cuce PM. “A successful application of homotopy perturbation method for efficiency and effectiveness assessment of longitudinal porous fins”. Energy Conversion and Management, 93, 92-99, 2015.
18. [18] Arslantürk C. “Variation of parameters method for optimizing annular fins with variable thermal properties”. Pamukkale University Journal of Engineering Sciences, 24(1), 1-7, 2018.
19. [19] Venkitesh V, Mallick A. “Thermal analysis of a convectiveconductive-radiative annular porous fin with variable thermal parameters and internal heat generation”. Journal of Thermal Analysis and Calorimetry, 147, 1519-1533, 2022.