Application of the two-dimensional differential transform method to heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins. (English) Zbl 1308.35046

Ndlovu, Partner L.; Moitsheki, Raseelo J.

Commun. Nonlinear Sci. Numer. Simul. 18, No. 10, 2689-2698 (2013).

Summary: In this article, approximate analytical (series) solutions for the temperature distribution in a longitudinal rectangular and convex parabolic fins with temperature dependent thermal conductivity and heat transfer coefficient are derived. The transient heat conduction problem is solved for the first time using the two-dimensional differential transform method (2D DTM). The effects of some physical parameters such as the thermo-geometric parameter, exponent and thermal conductivity gradient on temperature distribution are studied. Furthermore, we study the temperature profile at the fin tip.

MSC:

35C05	Solutions to PDEs in closed form
35K59	Quasilinear parabolic equations
35Q79	PDEs in connection with classical thermodynamics and heat transfer
80A20	Heat and mass transfer, heat flow (MSC2010)

Keywords:

two-dimensional DTM; analytical solutions; fins; rectangular and convex profiles; temperature dependent thermal properties

References:

[1]	Kraus, A. D.; Aziz, A.; Welty, J., Extended surface heat transfer, (2001), Wiley New York
[2]	Moradi, A., Analytical solutions for fin with temperature dependent heat transfer coefficient, Int J Eng Appl Sci, 3, 2, 1-12, (2011)
[3]	Moitsheki, R. J.; Hayat, T.; Malik, M. Y., Some exact solutions of the fin problem with the power law temperature dependent thermal conductivity, Nonlinear Anal Real World Appl, 11, 5, 3287-3294, (2010) · Zbl 1261.35140
[4]	Mhlongo, M. D.; Moitsheki, R. J.; Makinde, O. D., Transient response of longitudinal rectangular fins to step change in base temperature and in base heat flow conditions, Int J Heat Mass Transfer, 57, 117-125, (2013)
[5]	Zhou, J. K., Differential transform method and its applications for electric circuits, (1986), Huarjung University Press Wuuhahn, China
[6]	Moradi, A.; Ahmadikia, H., Investigation of effect thermal conductivity on straight fin performance with DTM, Int J Eng Appl Sci, 3, 1, 42-54, (2011)
[7]	Chiba, R., Application of differential transform method to thermoelastic problem for annular disks of variable thickness with temperature-dependent parameters, Int J Thermophys, 33, 363-380, (2012)
[8]	Sadri, S.; Raveshi, M. R.; Amiri, S., Efficiency analysis of straight fin with variable heat transfer coefficient and thermal conductivity, J Mech Sci Technol, 26, 4, 1283-1290, (2012)
[9]	Mosayebidorcheh, S.; Mosayebidorcheh, T., Series solution of convective radiative conduction equation of the nonlinear fin with temperature dependent thermal conductivity, Int J Heat Mass Transfer, 55, 6589-6594, (2012)
[10]	Chen, C. K.; Ho, S. H., Solving partial differential equations by two-dimensional differential transform method, Appl Math Comput, 106, 171-179, (1999) · Zbl 1028.35008
[11]	Chen, C. K.; Ho, S. H., Solving partial differential equations by two-dimensional differential transform method, Appl Math Comput, 2, 2076-7366, (2010)
[12]	Yaghoobi, H.; Torabi, M., The application of differential transform method to nonlinear equations arising in heat transfer, Int Commun Heat Mass Transfer, 38, 815-820, (2011)
[13]	Abdel-Halim Hassan, I. H., Comparision differential transformation technique with Adomian decomposition method for linear and nonlinear initial valuer problem, Chaos Solitons Fractals, 36, 53-65, (2008) · Zbl 1152.65474
[14]	Peker, H. A.; Karaoğlu, O.; Oturanç, G., The differential transformation method and pade approximation for a form of Blasius equation, Math Comput Appl, 16, 2, 507-513, (2011) · Zbl 1290.65069
[15]	Bagheri, M.; Bagheri, M.; Miralikatouli, E., Comparison differential transform method with homotopy perturbation method for nonlinear integral equations, J Math Comput Sci, 5, 4, 288-296, (2012)
[16]	Ghafoori, S.; Motevalli, M.; Nejad, M. G.; Shakeri, F.; Ganhji, D. D.; Jalaal, M., Efficiency of differential transformation method for nonlinear oscillation comparison with HPM and VIM, Curr Appl Phys, 11, 965-971, (2011)
[17]	Torabi, M.; Yaghoobi, H., Two dominant analytical methods for thermal analysis of convective step fin with variable thermal conductivity, Therm Sci Int Sci J, 1, 46, (2012)
[18]	Ndlovu PL, Moitsheki RJ. Analytical solutions for steady heat transfer in longitudinal fins with temperature dependent properties, submitted for publication. · Zbl 1296.80008
[19]	Jafari, H.; Alipour, M.; Tajadodi, H., Two-dimensional differential transform method for solving nonlinear partial differential equations, Int J Res Rev Appl Sci, 106, 171-179, (1999)
[20]	Bhadauria, R.; Singh, A. K.; Singh, D. P., A mathematical model to solve reaction diffusion equation using differential transformation method, Int J Math Trends Technol, 2, 2, 26-31, (2011)
[21]	Ünal, H. C., An analytical study of boiling heat transfer from a fin, Int J Heat Mass Transfer, 31, 7, 1483-1496, (1988) · Zbl 0669.76056
[22]	Moitsheki, R. J.; Harley, C., Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient, Pramana J Phys, 77, 3, 519-532, (2011)
[23]	Ayaz, F., On the two-dimensional differential transform method, Appl Math Comput, 143, 361-374, (2003) · Zbl 1023.35005
[24]	Kangalgil, F.; Ayaz, F., Solution of linear and nonlinear heat equations by differential transform method, Selcuk J Appl Math, 8, 1, 75-85, (2007) · Zbl 1229.65185
[25]	Mills, A. F., Basic heat and mass transfer, (1995), Irwin Inc. Chicago · Zbl 0861.35081
[26]	Aziz, A.; Torabi, M., Convective-radiative fins with simultaneous variation of thermal conductivity, heat transfer coefficient, and surface emissivity with temperature, Heat Transfer Asian Res, 41, 2, 99-113, (2012)