Heat Distribution in Rectangular Fins Using Efficient Finite Element and Differential Quadrature Methods

ShahNor BASRI, M. M. FAKIR, F. MUSTAPHA, D. L. A. MAJID, A. A. JAAFAR
DOI: 10.4236/eng.2009.13018                

Abstract

Finite element method (FEM) and differential quadrature method (DQM) are among important numerical techniques used in engineering analyses. Usually elements are sub-divided uniformly in FEM (conventional FEM, CFEM) to obtain temperature distribution behavior in a fin or plate. Hence, extra computational complexity is needed to obtain a fair solution with required accuracy. In this paper, non-uniform sub-elements are considered for FEM (efficient FEM, EFEM) solution to reduce the computational complex-ity. Then this EFEM is applied for the solution of one-dimensional heat transfer problem in a rectangular thin fin. The obtained results are compared with CFEM and efficient DQM (EDQM), with non-uniform mesh generation). It is found that the EFEM exhibit more accurate results than CFEM and EDQM showing its potentiality.

Keywords

Efficient Finite Element Method, Efficient Differential Quadrature Method, Heat Transfer Problem

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	G. Strang and G. J. Fix, “An analysis of the finite element method,” Prentice-Hall, Inc., 1997.
[2]	R. Tirupathi and P. E. Chandrupatla, Introduction to finite elements in engineering, Prentice-Hall International, 1997.
[3]	“Mathematics of finite element method,” 2004, Available at: http://math.nist.gov/mcsd/savg/tutorial/ansys/FEM/ (Accessed on October 19, 2006).
[4]	B. Li, “Numerical method for a parabolic stochastic partial differential equation,” Chalmers Finite Element Center, 2004.
[5]	F. Fairag, “Numerical computations of viscous, incompressible flow problems using a two-level finite element method,” arXiv: Math.NA/0109109, Vol. 1, No. 17, September 2001.
[6]	S. Park, “Development and applications of finite elements in time domain,” PhD Thesis, Faculty of the Virginia Polytechnic Institute and State University, Virginia, December, 1996, Available in the Library, Virginia Polytechnic Institute and State University.
[7]	R. Bellman and J. Casti, “Differential quadrature and long-term integration,” J Math and Appl., Vol. 34, pp. 235–238, 1971.
[8]	R. Bellman and J. Casti, “Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations,” J. Comput Phys, Vol. 10, pp. 40–52, 1972.
[9]	C. W. Bert, S. K. Jang, and A. G. Striz, “Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature,” J. Comput Mech, Vol. 5, pp. 217–226, 1989.
[10]	C. W. Bert and M. Malik, “Fast computing technique for the transient response of gas-lubricated journal bearings,” Proceedings of the U.S. National Congress of Applied Mechanics, Seattle, WA, pp. 298, June 26-July 1, 1994.
[11]	C. Shu, W. Chen, H. Xue, and H. Du, “Numerical analysis of grid distribution effect on the accuracy of differential quadrature analysis of beams and plates by error estimation of derivative approximation,” Int. Journal of Numer, Methods Engineering, Vol. 51, No. 2, pp. 159–179, 2001.
[12]	M. M. Fakir, M. K. Mawlood, W. Asrar, S. Basri, and A. A. Omar, “Triangular fin temperature distribution by the method of differential quadrature,” Journal Mekanikal, Malaysia, Vol. 15, pp. 20–31, 2003.
[13]	M. M. Fakir, M. K. Mawlood, W. Asrar, S. Basri, and A. A. Omar, “Rectangular fin temperature distribution by the method of differential quadrature,” The Journal of the Institute of Engineers, Malaysia (IEM), Vol. 63, No. 4, pp. 41–47, 2002.
[14]	E. Hinton and D. R. J. Owen, “An introduction to finite element computations,” Pineridge Press Limited, Swansea, U.K., 1985.
[15]	S. Tiwari, G. Biswas, P. L. N. Prasad, and S. Basu, “Numerical prediction of flow and heat transfer in a rectangular channel with a built-in circular tube,” ASME Journal of Heat Transfer, Vol. 125, No. 3, pp. 413–421, 2003.
[16]	B. L. Wang, and Z. H. Tian, “Application of finite element-finite difference method to the determination of transient temperature field in functionally graded materials,” Journal of Finite Elements in Analysis and Design, Vol. 41, pp. 335–349, 2005.
[17]	S. H. Lo and W. X. Wang, “Finite element mesh generation over intersecting curved surfaces by tracing of neighbours,” Journal of Finite Elements in Analysis and Design, Vol. 41, pp. 351–370, 2005.
[18]	M. N. Ozisik, “Heat transfer: A basic approach,” McGraw-Hill, 1985.