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**Analytical solutions for steady heat transfer in longitudinal fins with temperature-dependent properties.**
*(English)*
Zbl 1296.80008

Summary: Explicit analytical expressions for the temperature profile, fin efficiency, and heat flux in a longitudinal fin are derived. Here, thermal conductivity and heat transfer coefficient depend on the temperature. The differential transform method (DTM) is employed to construct the analytical (series) solutions. Thermal conductivity is considered to be given by the power law in one case and by the linear function of temperature in the other, whereas heat transfer coefficient is only given by the power law. The analytical solutions constructed by the DTM agree very well with the exact solutions even when both the thermal conductivity and the heat transfer coefficient are given by the power law. The analytical solutions are obtained for the problems which cannot be solved exactly. The effects of some physical parameters such as the thermogeometric fin parameter and thermal conductivity gradient on temperature distribution are illustrated and explained.

### MSC:

80A20 | Heat and mass transfer, heat flow (MSC2010) |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

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\textit{P. L. Ndlovu} and \textit{R. J. Moitsheki}, Math. Probl. Eng. 2013, Article ID 273052, 14 p. (2013; Zbl 1296.80008)

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### References:

[1] | A. D. Kraus, A. Aziz, and J. Welty, Extended Surface Heat Transfer, Wiley, New York, NY, USA, 2001. |

[2] | R. J. Moitsheki, T. Hayat, and M. Y. Malik, “Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 3287-3294, 2010. · Zbl 1261.35140 |

[3] | R. J. Moitsheki, “Steady one-dimensional heat flow in a longitudinal triangular and parabolic fin,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 3971-3980, 2011. · Zbl 1229.34043 |

[4] | R. J. Moitsheki, “Steady heat transfer through a radial fin with rectangular and hyperbolic profiles,” Nonlinear Analysis. Real World Applications, vol. 12, no. 2, pp. 867-874, 2011. · Zbl 1205.80035 |

[5] | Mo. Miansari, D. D. Ganji, and Me. Miansari, “Application of He’s variational iteration method to nonlinear heat transfer equations,” Physics Letters A, vol. 372, no. 6, pp. 779-785, 2008. · Zbl 1217.80067 |

[6] | 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. · Zbl 1011.80011 |

[7] | A. Rajabi, “Homotopy perturbation method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity,” Physics Letters A, vol. 364, no. 1, pp. 33-37, 2007. · Zbl 1203.74148 |

[8] | G. Domairry and M. Fazeli, “Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 489-499, 2009. |

[9] | A. Campo and R. J. Spaulding, “Coupling of the methods of successive approximations and undetermined coefficients for the prediction of the thermal behaviour of uniform circumferential fins,” Heat and Mass Transfer, vol. 34, no. 6, pp. 461-468, 1999. |

[10] | A. Campo and F. Rodrfguez, “Approximate analytic temperature solution for uniform annular fins by adapting the power series method,” International Communications in Heat and Mass Transfer, vol. 25, no. 6, pp. 809-818, 1998. |

[11] | R. Chiba, “Application of differential transform method to thermoelastic problem for annular disks of variable thickness with temperature-dependent parameters,” International Journal of Thermophysics, vol. 33, pp. 363-380, 2012. |

[12] | S. Sadri, M. R. Raveshi, and S. Amiri, “Efficiency analysis of straight fin with variable heat transfer coefficient and thermal conductivity,” Journal of Mechanical Science and Technology, vol. 26, no. 4, pp. 1283-1290, 2012. |

[13] | M. Torabi, H. Yaghoori, and A. Aziz, “Analytical solution for convective-radiative continously moving fin with temperature dependent thermal conductivity,” International Journal of Thermophysics, vol. 33, pp. 924-941, 2012. |

[14] | M. Torabi and H. Yaghoobi, “Two dominant analytical methods for thermal analysis of convective step fin with variable thermal conductivity,” Thermal Science. In press. |

[15] | A. Moradi, “Analytical solutions for fin with temperature dependant heat transfer coefficient,” International Journal of Engineering and Applied Sciences, vol. 3, no. 2, pp. 1-12, 2011. |

[16] | A. Moradi and H. Ahmadikia, “Analytical solution for different profiles of fin with temperature-dependent thermal conductivity,” Mathematical Problems in Engineering, vol. 2010, Article ID 568263, 15 pages, 2010. · Zbl 1204.74039 |

[17] | H. Yaghoobi and M. Torabi, “The application of differential transformation method to nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 38, no. 6, pp. 815-820, 2011. |

[18] | S. Ghafoori, M. Motevalli, M. G. Nejad, F. Shakeri, D. D. Ganji, and M. Jalaal, “Efficiency of differential transformation method for nonlinear oscillation: Comparison with HPM and VIM,” Current Applied Physics, vol. 11, no. 4, pp. 965-971, 2011. |

[19] | A. A. Joneidi, D. D. Ganji, and M. Babaelahi, “Differential transformation method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity,” International Communications in Heat and Mass Transfer, vol. 36, no. 7, pp. 757-762, 2009. |

[20] | J. K. Zhou, Differential Transform Method and Its Applications for Electric circuIts, Huazhong University Press, Wuhan, China, 1986. |

[21] | S. Mukherjee, “Reply to comment on ‘solutions of the duffing-van der pol oscillator equation by the differential transform method’,” Physica Scripta, vol. 84, Article ID 037003, 2 pages, 2011. · Zbl 1320.34060 |

[22] | C. Bervillier, “Status of the differential transformation method,” Applied Mathematics and Computation, vol. 218, no. 20, pp. 10158-10170, 2012. · Zbl 1246.65107 |

[23] | A. Arikoglu and I. Ozkol, “Solution of fractional differential equations by using differential transform method,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1473-1481, 2007. · Zbl 1152.34306 |

[24] | Z. Odibat, S. Momani, and V. S. Erturk, “Generalized differential transform method: application to differential equations of fractional order,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 467-477, 2008. · Zbl 1221.34022 |

[25] | C. W. Bert, “Application of differential transform method to heat conduction in tapered fins,” Journal of Heat Transfer, vol. 124, no. 1, pp. 208-209, 2002. |

[26] | F. Khani and A. Aziz, “Thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 590-601, 2010. |

[27] | H. C. Ünal, “An anlytical study of boiling heat transfer from a fin,” International Journal of Heat and Mass Transfer, vol. 31, no. 7, pp. 1483-1496, 1988. · Zbl 0669.76056 |

[28] | M. H. Chang, “A decomposition solution for fins with temperature dependent surface heat flux,” International Journal of Heat and Mass Transfer, vol. 48, no. 9, pp. 1819-1824, 2005. · Zbl 1189.76519 |

[29] | A. Jezowski, B. A. Danilchenko, M. Boćkowski et al., “Thermal conductivity of GaN crystals in 4.2-300 K range,” Solid State Communications, vol. 128, no. 2-3, pp. 69-73, 2003. |

[30] | S. Vitanov, V. Palankovski, S. Maroldt, and R. Quay, “High-temperature modeling of AlGaN/GaN HEMTs,” Solid-State Electronics, vol. 54, no. 10, pp. 1105-1112, 2010. |

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