Simplified Integral Calculations for Radial Fin with Temperature-Dependent Thermal Conductivity

Okey Oseloka Onyejekwe

doi:10.4236/jamp.2019.73037

Abstract

Numerical solution of a radiative radial fin with temperature-dependent thermal conductivity is presented. Calculations are implemented along the lines of a boundary integral technique coupled with domain discretization. Localized solutions of the nonlinear governing differential equation are sought on each element of the problem domain after enforcing inter-nodal connectivity as well as the boundary conditions for the dependent variables. A finite element-type assembly of the element equations and matrix solution yield the scalar profile. Comparison of the numerical results with those found in literature validates the formulation. The effects of such problem parameters as radiation-sink temperature, thermal conductivity, radiation-conduction fin parameter, volumetric heat generation, on the scalar profile were found to be in conformity with the physics of the problem. We also observed from this study that the volumetric heat generation plays a significant role in the overall heat transfer activity for a fin. For relatively high values of internal heat generation, a situation arises where a greater percentage of this energy can not escape to the environment and the fin ends up gaining energy instead of losing it. And the overall fin performance deteriorates. The same can also be said for the radiation-conduction parameter , whose increases can only give physically realistic results below a certain threshold value.

Highlights

IntroductionFins are vastly used in different heat transfer applications such as air-conditioning
Comparison of the numerical results with those found in literature validates the formulation. The effects of such problem parameters as radiation-sink temperature, thermal conductivity, radiation-conduction fin parameter, volumetric heat generation, on the scalar profile were found to be in conformity with the physics of the problem
The validity of the formulation developed was tested by comparing the numerical results with those of Torabi et al [25]. They applied a seminumerical-analytic technique known as the differential transformation method (DTM) to calculate the temperature profile of a radiative radial fin

Summary

Introduction

Fins are vastly used in different heat transfer applications such as air-conditioning. They considered the thermal conductivity of the annular fin to obey a power-law function; and replaced the variable coefficients related to the second and third terms of the governing differential equation by their mean-values Their studies led them to the conclusion that the temperature gradient for the functionally graded annular fin is lower for the non-graded or homogeneous case. Oguntala and Sobamowo [24] used the Galerkin’s method of weighted residual to study the temperature distribution of a rectangular fin with temperature-dependent thermal properties and internal heat generation Their results displayed a monotonic drop in scalar profiles for various thermo-geometric, thermal conductivity and convective heat transfer parameters. Greek Symbols β : Thermal conductivity parameter ε : Emissivity η : Fin efficiency λ : Slope of the thermal conductivity temperature curve, K−1 σ : Stefan-Boltzman constant, Wm−2K−4 θ : Dimensionless temperature θs : Dimensionless radiation sink temperature ψ : Radiation-conduction fin parameter

Problem Formulation

Elemental Discretization and Integration of the Fin Equation

Results and Discussion

Conclusion