An Analytical Solution for Transient Heat Conduction in a Composite Slab with Time-Dependent Heat Transfer Coefficient

Ryoichi Chiba

doi:10.1155/2018/4707860

Abstract

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of n layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. The composite slab, which has thermal contact resistance at n-1 interfaces, as well as an arbitrary initial temperature distribution and internal heat generation, convectively exchanges heat at the external boundaries with two different time-varying surroundings. To obtain the analytical solution, the shifting function method is first used, which yields new partial differential equations under conventional types of external boundary conditions. The solution for the derived differential equations is then obtained by means of an orthogonal expansion technique. Numerical calculations are performed for two composite slabs, whose heat transfer coefficient on the heated surface is either an exponential or a trigonometric function of time. The numerical results demonstrate the effects of temporal variations in the heat transfer coefficient on the transient temperature field of composite slabs.

Highlights

Heat transfer coefficients (HTCs) vary with time, and their variations can be random or, in certain cases, periodic
The same holds for the heat transfer coefficients (HTCs) between a casting and its metal moulds [3], on diesel fuel droplets subjected to transient heating [4], and on solids enveloped by pulsating flows of liquid or gas in internal combustion engines [5]
An analytical method based on the shifting function method has been presented for solving the one-dimensional transient heat conduction problem with time-dependent HTC for a composite slab consisting of an arbitrary number of layers

Summary

Introduction

Heat transfer coefficients (HTCs) vary with time, and their variations can be random or, in certain cases, periodic. The need to analyse multiregion problems including contact resistances has been underlined owing to the canning of fuel elements in a reactor [24] and design of thermal insulation systems [21, 22] In those systems, ambient temperature may vary with time as well as HTC. Ambient temperature may vary with time as well as HTC In studying such engineering problems, analytical solutions are highly valuable as they provide greater insight into the solution behaviour, which is typically missing in any numerical solution. The shifting function method developed by Chen et al [18] and an orthogonal expansion method [25, 26] are used to obtain an analytical solution for the multiregion heat conduction caused by time-varying ambient temperatures. Two numerical examples are given to quantify the effects of the temporal variations in the HTC on the temperature profiles in the slabs

Problem Formulation

Analysis

Verification of Analytical Solution

Illustrative Examples

Findings

Conclusions