Geometrical optimization of a straight longitudinal fin with multiple nonlinearities.

Abstract

Extended surfaces, also known as cooling fins are widely used in a variety of industrial fields for cooling down electronic and mechanical components while they are employed in the area of the thermal design for heating up houses using a number of solar devices such as evacuated solar tubes and flat plate collectors. Fins can have numerous shapes, such as convex, concave, triangular, and rectangular, however, the most popular fins are the straight rectangular profiles due to their simplicity of adaptation to the other equipments and their low manufacturing cost. We develop in the present thesis a powerful shooting method for solving the onedimensional convective-radiative straight rectangular longitudinal fin with multiple nonlinearities where the physical properties like the thermal conductivity, the convection heat transfer coefficient and the radiation surface emissivity are considered to be functions of the temperature. The fin base is maintained at a constant temperature while the fin tip is taken to be nonadiabatic and subjected to heat dissipation by a combination of convection and radiation. Such a highly nonlinear thermal fin boundary value problem is primarily transformed into a dimensionless form, then converted to an equivalent initial value problem, described by a system of two coupled first order ordinary differential equations of the fin temperature and its derivative. We employ a mixture of fundamental numerical techniques like the four order Runge-Kutta method, finite difference method and Secant method to successfully integrate this strongly nonlinear initial value problem. The present thermal fin problem is essentially governed by Biot number which is well known in the field of the heat transfer and fluid mechanics and also by two new dimensionless numbers, introduced for the first time in this kind of fin thermal investigation, which are the Stark number and the geometrical number. Results for the fin temperature distribution are computed via a double precision FORTRAN code and then validated by a comparative process using four different numerical and semi-numerical approaches, where the computed absolute error has been found to be very small. We generate several comparative solutions such as the fin temperature distribution, the mean fin temperature, the rate of the heat transfer, the entropy generation and the fin efficiency for two identical fins with different tip boundary condition, an adiabatic tip and a nonadiabatic tip, representing the most realistic situation in order to visualize the thermal response of the type of the tip boundary condition to the overall fin thermal performance. Numerical calculations have demonstrated that there exists a strong sensibility to the type of the tip boundary condition, mainly depicted in the fin efficiency where a very dissimilar thermal behavior corresponding to the nonadiabatic tip was observed. Such a remarkable characteristic could allow effective geometrical and thermal optimization of the fin problem for a better cooling in the industry.