Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient

Abstract

Transient heat transfer through a longitudinal fin of various profiles is studied. The thermal conductivity and heat transfer coefficients are assumed to be temperature dependent. The resulting partial differential equation is highly nonlinear. Classical Lie point symmetry methods are employed and some reductions are performed. Since the governing boundary value problem is not invariant under any Lie point symmetry, we solve the original partial differential equation numerically. The effects of realistic fin parameters such as the thermogeometric fin parameter and the exponent of the heat transfer coefficient on the temperature distribution are studied.

Unsteady heat transfer through a longitudinal fin: An investigation into the effects of key parameters

“…Similar to Yeh and Liaw [8] we believed this occurrence to be related to thermal instability. In Moitsheki and Harley [9] similar unstable behaviour was found when considering a nonlinear partial differential equation modelling the unsteady heat transfer through a longitudinal fin with triangular profile. A dynamical systems analysis is conducted in this work as a means of establishing the credibility of the conclusions drawn in these previous works.…”

Heat Transfer in a Porous Radial Fin: Analysis of Numerically Obtained Solutions

“…The purpose of the asymptotic solution is to reveal the dominant physical mechanisms of the model. It can be seen in Moitsheki and Harley [16,35] that the impact of the thermogeometric parameter (M) in terms of its proportionality to the length of the fin ( ) was observed. They found that the heat transfer in the fin seemed to be unstable for small values of (M) due to the fact that M ∝ .…”

“…Hence, they conducted a conjugate conduction-convection analysis by solving the heat conduction equation. Moitsheki and Harley [16] studied the transient heat transfer through a longitudinal fin of various profiles by employing classical Lie point symmetry methods. They observed that for long periods of time the temperature profile becomes unusual for the heat transfer in longitudinal triangular and concave parabolic fins.…”

“…By investigating the asymptotic solution to the steady state heat transfer in a rectangular longitudinal fin, they were able to validate the above relationship and establish the importance of the fin length. Furthermore, the same authors [16,35] conducted a small scale dynamical analysis. In order to expand the analysis in [35], Harley [36] employed an in-depth dynamical analysis to monitor the behaviour especially at the fin tip.…”

Nonlinear numerical analysis of convective-radiative fin using MLPG method

“…This nonlinearity arises due to the reliance of thermal as well as material properties on the temperature or cross-sectional area of the fin. Optimal linearization method has been used by Jordan et al [12]; Frobenius expanding series has been employed by Kundu and Das [13]; homotopy analysis method has been presented by Khani et [26] and Hoshyar et al [27]; Aziz and Bouaziz [28] have employed method of least squares; Kirchoff's transformation method has been used by Moitsheki and Rowjee [29]; lie point symmetry method has been highlighted by Moitsheki and Harley [30], Mhlongo and Moitsheki [31], Ali et al [32] and Kader et al [33]; Hajabdollahi et al [34] has presented genetic algorithm; symbolic programming has been used by Fatoorehchi and Abolghasemi [35]; and Latif et al [36] have used symmetry reduction method successfully, to address nonlinear heat transfer problems of fins. Mahmoudi and Mejri [37] have investigated the effect of variable thermal conductivity and variable refractive index on transient conduction and radiation heat transfer by Lattice Boltzmann method.…”