Design of a longitudinal cooling fin with minimum volume by a modified Newton-Raphson method

In this paper, the minimal volume of nonlinear longitudinal cooling fin design problem by using a modified Newton-Raphson method is presented. The profile of the fin is built by B-spline curve in which the control points of the B-spline curve are regarded as optimization variables. Additionally, a mechanism called “volume updating” is added into the modified Newton-Raphson algorithm to obtain the minimum volume of the fin. Four cases with the different boundary conditions and thermal properties of the longitudinal fin are presented to demonstrate the proposed method. The results show that the optimal fin obtained by the proposed method is in good agreement with Schmidt's (1926) result and is better than that of Azarkish, et al. (2010). It is concluded that the B-spline with the second degree and three control points could be used enough to find the minimum volume of the longitudinal fin for the linear and nonlinear fin design problems. From the results obtained in four cases, it appears that the proposed method is an efficient and accurate method in finding the minimum volume of the nonlinear longitudinal cooling fin design problem.