Mathematical Problems in Engineering

Volume 2018, Article ID 4707860, 11 pages

https://doi.org/10.1155/2018/4707860

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

Department of Mechanical Systems Engineering, National Institute of Technology, Asahikawa College, 2-2-1-6 Shunkodai, Asahikawa 071-8142, Japan

Correspondence should be addressed to Ryoichi Chiba; pj.ca.tcn-awakihasa@abihc

Received 17 December 2017; Accepted 12 March 2018; Published 19 April 2018

Academic Editor: Filippo de Monte

Copyright © 2018 Ryoichi Chiba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. The composite slab, which has thermal contact resistance at 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.

#### 1. Introduction

Generally, heat transfer coefficients (HTCs) vary with time, and their variations can be random or, in certain cases, periodic. For example, because of irregular fluid motion around turbine blades, the HTC of the blade surfaces fluctuates [1]. The surface HTCs of the fuel elements in a boiling water reactor and a pebble bed reactor also vary over time [2]. The same holds for the 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]. In addition, during the thermal processing of foods, the pattern of air circulation may be altered around them, resulting in changes of the surface HTC [6]. The time variation in the surface aspects of objects is also one of the possible reasons (e.g., oxidation, dust contamination, and fissuring) [7]. Such a temporal change in the HTC leads to alterations in the object’s temperature.

Transient heat conduction with boundary conditions including a time-dependent HTC has been investigated since the late 1960s. Studies using (semi)numerical methods include the combined use of the finite difference and Laplace transform methods [8] and the parameter-group transformation followed by the Runge-Kutta shooting method [9]. Early studies using approximate methods include those of Ivanov et al. [10–12], which transformed the governing equation into a nonlinear equation using the change of variables and omitted the nonlinear term from the derived equation by restricting the research object to thin bodies. Kozlov [13] reduced the original problem to finding a solution to an infinite number of simultaneous ordinary differential equations, approximating it by a finite number. In addition, the application of the Laplace-Carson integral transform [7], finite integral transform methods with time-dependent eigenvalues [14, 15], and Lie point symmetry analysis [16] were reported. Further, an analytical method based on the Laplace transform and bifrequency transfer function was presented, but the inverse transform is extremely difficult [17]. An analytical solution to the heat conduction problem with arbitrary time-dependent HTC at the boundary had not been obtained for a long period because of the difficulty posed by eigenfunctions and eigenvalues that depend on time. However, Lee and colleagues [18] resolved this situation in 2010; they successfully derived explicit-form solutions by means of the shifting function method [19, 20].

On the other hand, there have been few studies with composite media. Il’chenko [21] obtained an approximate solution to the heat conduction problem of a two-layered plate with time-dependent HTC by replacing the continuous time function of the HTC with a piecewise function. Prikhod’ko [22] derived a transient temperature solution of a two-layer plate using a special series expansion of Green’s function, while assuming that the temperature is uniform across the thickness in one layer. Yener and Ozisik [23] extended an analytical method based on a finite integral transform [14] to solve the problem for finite composite media with variable boundary condition parameters. All of them are, however, approximate methods, leading to less rigorous analysis.

Composite (or multiregion) media have a wide application in many industrial, environmental, and biological fields (e.g., in turbines, heat exchangers, fuel cells, and brake systems). Thus, it is industrially important to study this kind of heat conduction problems for composite media. To be specific, 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. 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. They can be used to benchmark numerical solutions as well. The foregoing facts motivate the present study.

In this paper, we address the one-dimensional transient heat conduction problem of an infinite composite slab with general time-dependent HTC at one external boundary. 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. Main differences from the approach for single layered slab [20] are the initial shifting of solution in each layer, the addition of interface conditions, and the eigenfunction expansion of functions based on the Vodicka type of orthogonality relationship, which would be the novelty of the present analytical method. Numerical calculations are performed to verify the solution for several test cases. Subsequently, two numerical examples are given to quantify the effects of the temporal variations in the HTC on the temperature profiles in the slabs.

#### 2. Problem Formulation

Let us consider a composite slab consisting of layers, as shown in Figure 1, which has finite thickness but extends to infinity in the other two dimensions. The composite slab has an arbitrary initial temperature distribution and is subjected to two different, arbitrary time-varying ambient temperatures, and , at its external boundaries. The boundaries have different HTCs, and , one of which depends on time. At the interfacial boundaries, heat is transferred through different constant conductance coefficients , under the condition of heat flux continuity. A time- and space-dependent internal heat generation is also considered in each layer. The material of each layer is isotropic and homogeneous with constant thermophysical properties.