Solution of Two-dimensional Transient Heat Conduction in a Hollow Sphere under Harmonic boundary condition

M.A Abdous,N Moallemi

doi:10.19026/rjaset.7.815

Abstract

In this study, an analytical modeling of two dimensional heat conduction in a hollow sphere, subjected to time dependent periodic boundary condition at the inner and the outer surfaces, is performed. The thermo physical properties of the material are assumed to be isotropic and homogenous. Also, the effects of the temperature oscillations frequency on the boundaries, the thickness variation of the hollow sphere and thermo physical properties of the ambient and the sphere involved in some dimensionless numbers are studied. The results show that the obtained temperature distribution contains two characteristics, the dimensionless amplitude and the dimensionless phase difference. Comparison between the present results and the findings of the previous study as related to a two-dimensional solution of the hollow sphere subjected to the simple harmonic condition shows a good agreement.

Highlights

The heat conduction analysis in spherical solids is important, because these geometries have some special features such as symmetry and minimum surface energy
Heat transfer problems in transient form are very common in engineering applications, e.g., hydro cooling of spherical food products (Dincer, 1995b) or fast transient heat conduction in sphere subjected to the sudden and violent thermal effects on its surface used in many engineering fields such as aeronautics, electronics, metallurgy (Baïri and Laraqi, 2003; Dincer, 1995c)
The heat conduction problems with periodic boundary conditions have some applications in engineering, like periodic heat conduction through composite spheres consisting of shells (Lit, 1987), periodic radial heat conduction through a sphere (Sengupta et al, 1993) and in a solid homogeneous finite cylinder (Cossali, 2009)

Summary

INTRODUCTION

The heat conduction analysis in spherical solids is important, because these geometries have some special features such as symmetry and minimum surface energy. Transient heat conduction is studied in sphere by using Laplace transforms (Youming et al, 2003; Ostrogorsky, 2008) or in polar coordinates with multiple layers in radial direction (Suneet et al, 2008). Analytical method to solve transient heat conduction in spherical coordinates with time-dependent boundary conditions (Prashant et al, 2010), the problem of evaluating the dynamic heat storage capacity of a solid sphere (Cossali, 2007) and the analytic solution of the periodic heat conduction in a homogeneous cylinder are some of the solutions in term of Fourier transform which are solved by researchers (Atefi et al, 2009). The main purpose of this study is to derive a general analytical solution for two-dimensional heat conduction in a hollow sphere subjected to a periodic boundary condition at the inner and the outer surfaces. The effects of the inner boundary conditions on temperature distribution are discussed

METHODOLOGY

CONCLUSION