Optimal fin shape in finned double pipe with fully developed laminar flow

Abstract

Optimal shape of longitudinal fins augmented to the outer surface of the inner pipe enclosed within a concentric outer pipe, is investigated for maximizing the Nusselt number in a steady laminar and fully-developed flow. The wall-fin assembly is made up of highly conductive material and is subjected to the uniform heat flux boundary condition. Taking triangular cross-section as the initial profile, the fin shape is represented by Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) at each step of the optimization process with control points as the design variables. Genetic algorithm and trust region method have been used as the optimizers with FEM as the solver of the governing equations. The results show that the optimal profile is strongly dependent on the number of fins, the ratio of radii, the number of control points and the characteristic length. It gives improvement in the Nusselt number upto 138%, 312% and 263% over the conventional fins of trapezoidal, triangular and parabolic shapes for the case of equivalent diameter and for the case of hydraulic diameter it gives such improvements upto 212%, 59% and 90% respectively. For large number of fins the optimal fin shape based on the equivalent diameter is usually triangular on smaller inner pipes and wavy on larger inner pipes. More and more articulates appear as the larger number of higher fins is employed on bigger inner pipes. For the case of hydraulic diameter, it may either be regarded as parabolic or wavy. Comparison of the present results with the available literature results proves their validity and accuracy.