Steady Thermal Analysis of Two‐Dimensional Cylindrical Pin Fin with a Nonconstant Base Temperature

Abstract

Steady heat transfer through a pin fin is studied. Thermal conductivity, heat transfer coefficient, and the source or sink term are assumed to be temperature dependent. In the model considered, the source or sink term is given as an arbitrary function. We employ symmetry techniques to determine forms of the source or sink term for which the extra Lie point symmetries are admitted. Method of separation of variables is used to construct exact solutions when the governing equation is linear. Symmetry reductions result in reduced ordinary differential equations when the problem is nonlinear and some invariant solution for the linear case. Furthermore, we analyze the heat flux, fin efficiency, and the entropy generation.

Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended Surfaces

“…The difficulty for group-invariant solutions is the satisfaction of the imposed or prescribed boundary conditions. This has been seen in two dimensional steady state problems [7,16], and 1 + 1 D transient problems [25]. Perhaps the most successful attempt in in [26].…”

“…Perhaps the most successful attempt in in [26]. For nonlinear steady state problems, some transformation such as Kirchoff [7,16], may linearise the two dimensional problems which then becomes easier to solve using standard methods. Linearisation of nonlinear steady state one dimensional problems is possible when thermal conductivity is a differential consequence of heat transfer coefficient [5].…”

“…Few exact solutions are recorded in the literature, for example for one dimensional problems [2][3][4][5][9][10][11][12][13][14][15], two dimensions [6,7,16,17]. An attempt to construct exact solutions for the three dimensional problems is found in [8], however these were general solutions.…”

A Lie Symmetry Classification of a Nonlinear Fin Equation in Cylindrical Coordinates

“…To determine consistency, we now use the last of the determining equations, that is, (17). This requirement gives…”

“…In the same series of studies Moitsheki and Harley [17] have considered a two-dimensional pin fin equation with length and radius , having the form…”