Analytical Solution for Fin with Temperature Dependent Heat Transfer Coefficient

Abstract

In this study, the differential transformation method (DTM) has been used for thermal characteristics of straight rectangular fin for all type of heat transfer and numerical comparison between DTM and adomain decomposition method (ADM) and exact analytical solution method as well as problem is solved numerically with fourth order Rang- Kutta method using shooting method because one of the boundary conditions is boundary value. Local heat transfer coefficient is assumed to vary power-law function of temperature. The obtained differential transform approximate analytic solution is in the form of an infinite power series, so that with obtained explicit form of temperature profile, the fin tip temperature, fin base heat transfer rate, and fin efficiency can be calculated directly from temperature profile easily, but in the exact analytical method obtained implicit form of temperature profile. Results showed that present results have excellent agreement with ADM results and exact analytical method results

Keywords

• DTM, ADM, analytical solution, fin, efficiency

References

1. [1] Q.D. Kern, D.A. Kraus, Extended Surface Heat Transfer, McGraw-Hill, New York, 1972.
2. [2] Chang M. H., A decomposition solution for fins with temperature dependent surface heat flux, Int. J. Heat Mass Transfer, 48, 1819-1824, 2005.
3. [3] Joneidi A. A., Ganji D. D. and Babaelahi M., Differential transformation method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity, Int. Commun. Heat Mass Transfer, 36, 757-762, 2009.
4. [4] Unal H. C., Determination of the temperature distribution in an extended surface with a non-uniform heat transfer coefficient, Int. J. Heat Mass Transfer, 28, 2279-2284, 1986.
5. [5] Unal H. C., A simple method of dimensioning straight fins for nucleate pool boiling, Int. J. Heat Mass Transfer, 29, 640-644, 1987.
6. [6] Unal H. C., An analytic study of boiling heat transfer from a fin, Int. J. Heat Mass Transfer, 30, 341-349, 1987.
7. [7] Unal H. C., The effect of the boundary at a fin tip on the performance of the fin with and without internal heat generation, Int. J. Heat Mass Transfer, 31, 1483-1496, 1988.
8. [8] Liaw S. P. and Yeh R. H., Fins with temperature dependent surface heat flux –I: Single heat transfer mode, Int. J. Heat Mass Transfer, 37, 1509-1515, 1994.

9. [9] Liaw S. P. and Yeh R. H., Fins with temperature dependent surface heat flux –II: Multiboiling heat transfer, Int. J. Heat Mass Transfer, 37, 1517-1524, 1994.
10. [10] Abbasbandy S. and Shivanian E., Exact analytical solution of a nonlinear equation arising in heat transfer, Phys. Lett, A 374, 567-574, 2010.
11. [11] Kou H. S., Lee J. J. and Lai C. Y., Thermal analysis of a longitudinal fin with variable thermal properties by recursive formulation, Int. J. Heat Mass Transfer, 48, 2266-2277, 2005.
12. [12] Khani F. and Aziz A., Thermal analysis of a longitudinal trapezoidal fin with temperaturedependent thermal conductivity and heat transfer coefficient, Communication in Nonlinear Science Numerical Simulation, 15, 590-601, 2010.
13. [13] Zhou J. K., Differential Transformation method and its Application for Electrical Circuits, Hauzhang Univ. press, Wuhan, China, 1986.
14. [14] Aziz A. A. and Hug E., Perturbation solution for convecting fin with variable thermal conductivity, J. Heat Transfer, 91, 300-310, 1995.
15. [15] Domairry G. and Fazeli M., Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity, Communication in Nonlinear Science and Numerical, Simulation 14, 489-499, 2009.
16. [16] Khani F., Ahmadzade Raji M. and Hamidi Nejad H., Analytical solution and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat Transfer coefficient, Communication in Nonlinear Science Numerical Simulation, 14, 3327-3338, 2009.
17. [17] Mostafa Inc., Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Mathematics and Computers in Simulation, 79, 189-200, 2008.
18. [18] Arslanturk C., A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Int. Commun. Heat Mass Transfer, 32, 831-841, 2005.
19. [19] Rajabi A., Homotopy perturbation method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Phys. Lett. A 364, 33-37, 2007.
20. [20] Lesnic D. and Heggs P. J., Adomain decomposition method for power-law fin-type problems, Int. Commun. Heat Mass Transfer, 31, 673-682, 2004.
21. [21] Chiu C. H. and Chen C. K., A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, Int. J. Heat Mass Transfer, 45, 2067-2075, 2002.
22. [22] Kundu B. and Das P. K., Performance analysis and optimization of straight taper fins with variable heat transfer coefficient, Int. J. Heat Mass Transfer. 45, 4739-4751, 2002.
23. [23] Mokhimer M. A., Performance of annular fins with different profiles subject to variable heat transfer coefficient, Int. J. Heat Mass Transfer, 45, 3631-3642, 2002.
24. [24] Rashidi M. M. and Erfani E., New analytical method for solving Burgers, and nonlinear heat transfer equation and comparison with HAM, Comput. Phys. Commun, 180, 1539-1544, 2009.
25. [25] Chiam T. C., Heat transfer in a fluid with variable thermal conductivity over a linearly stretching sheet, Acta Mechanica, 129, 63-72, 1998.