Optimization of convective–radiative fins by using differential quadrature element method P. Malekzadeh a , H. Rahideh b , G. Karami c, * a Department of Mechanical Engineering, School of Engineering, Persian Gulf University, Bushehr 75168, Iran b Department of Chemical Engineering, School of Engineering, Persian Gulf University, Bushehr 75168, Iran c Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, P.O. Box 5285, Fargo, ND 58105-5285, USA Received 9 February 2005; accepted 13 August 2005 Available online 21 September 2005 Abstract A first endeavor to exploit the differential quadrature element method (DQEM) as a simple, accurate and computationally efficient numerical tool for the shape optimization of convective–radiating extended surfaces or fins is made. The formula- tions are general so that the spatial and spatial-temperature dependent geometrical and thermal parameters can easily be implemented. The thermal conductivity of the fin is assumed to vary as a linear function of the temperature. The effects of a convective–radiative condition at the fin tip and effective convective condition at the fin base are considered. The optimi- zation of fins with uniform and step cross-sections is investigated. The accuracy of the method is demonstrated by comparing its results with those generated by AdomianÕs decomposition technique, Taylor transformation technique and finite difference method. It is shown that, using few grid points, highly accurate results are obtained. Less computational effort of the method with respect to the finite difference method is shown. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Fins; Energy conservation; Differential quadrature method 1. Introduction Fins are extensively used in various industrial applications [1]. For this reason, the optimization of fins is an important engineering topic. It is well known [1] that a straight fin with a concave parabolic profile provides the maximum heat dissipation for a given profile area. However, the concave parabolic shape is difficult and costly to manufacture. Thus, for most applications, a rectangular profile is preferred for the sake of simplicity in fabrication, even though it does not utilize the material most efficiently. Extensive literature exists on the above optimum problem for convective fins. Aziz [2] presented a discus- sion and a literature survey on this subject in a review article. Aziz [3,4] studied the optimum design of 0196-8904/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2005.08.003 * Corresponding author. Tel.: +1 701 231 5859; fax: +1 701 231 8913. E-mail address: G.Karami@ndsu.edu (G. Karami). Energy Conversion and Management 47 (2006) 1505–1514 www.elsevier.com/locate/enconman