Some Exact Solutions of Nonlinear Fin Problem for Steady Heat Transfer in Longitudinal Fin with Different Profiles

M D Mhlongo,R J Moitsheki

doi:10.1155/2014/947160

Abstract

One-dimensional steady-state heat transfer in fins of different profiles is studied. The problem considered satisfies the Dirichlet boundary conditions at one end and the Neumann boundary conditions at the other. The thermal conductivity and heat coefficients are assumed to be temperature dependent, which makes the resulting differential equation highly nonlinear. Classical Lie point symmetry methods are employed, and some reductions are performed. Some invariant solutions are constructed. The effects of thermogeometric fin parameter, the exponent on temperature, and the fin efficiency are studied.

Highlights

Heat transfer through extended surfaces has been studied quite extensively [1], perhaps because of its frequent applications in engineering
Analytical, and approximate schemes for solving differential equations have been devised through considerable effort, those arising in heat conduction through one-dimensional fin problems
Moitsheki et al [10] constructed the exact solutions of the one-dimensional fin problem given nonlinear thermal conductivity and heat transfer coefficient

Summary

Introduction

Heat transfer through extended surfaces has been studied quite extensively [1], perhaps because of its frequent applications in engineering. Moitsheki et al [10] constructed the exact solutions of the one-dimensional fin problem given nonlinear thermal conductivity and heat transfer coefficient. This work has been extended in [11] whereby the introduction of the Kirchhoff transformation linearized the one-dimensional fin problem when heat transfer is a differential consequence of thermal conductivity. Moitsheki and Harley [17] considered fins of various profiles with both heat transfer coefficient and thermal conductivity being given as temperature dependent. The transient problem is considered for a fin of arbitrary profile in [20] Both thermal conductivity and heat transfer are considered to be constants. We determine exact solutions of nonlinear fin problem for steady heat transfer in longitudinal fin of various profiles where the thermal conductivity is related to temperature by a power law.

Mathematical Models

Exact Solutions

Lie Point Symmetry Analysis

Symmetry Reductions and Invariant Solutions

Some Discussions

Concluding Remarks