Journal of Applied Mathematics

Volume 2015, Article ID 203404, 9 pages

http://dx.doi.org/10.1155/2015/203404

## Analytical Solution of Heat Conduction for Hollow Cylinders with Time-Dependent Boundary Condition and Time-Dependent Heat Transfer Coefficient

^{1}Department of Mechanical Engineering, Air Force Institute of Technology, No. 198 Jieshou W. Road, Gangshan Township, Kaohsiung 820, Taiwan^{2}Department of Mechanical Engineering, National Cheng Kung University, No. 1 University Road, Tainan 701, Taiwan

Received 5 May 2015; Accepted 21 July 2015

Academic Editor: Assunta Andreozzi

Copyright © 2015 Te-Wen Tu and Sen-Yung Lee. 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 for the heat transfer in hollow cylinders with time-dependent boundary condition and time-dependent heat transfer coefficient at different surfaces is developed for the first time. The methodology is an extension of the shifting function method. By dividing the Biot function into a constant plus a function and introducing two specially chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions only. The transformed system is thus solved by series expansion theorem. Limiting cases of the solution are studied and numerical results are compared with those in the literature. The convergence rate of the present solution is fast and the analytical solution is simple and accurate. Also, the influence of physical parameters on the temperature distribution of a hollow cylinder along the radial direction is investigated.

#### 1. Introduction

The problems of transient heat flow in hollow cylinders are important in many engineering applications. Heat exchanger tubes, solidification of metal tube casting, cannon barrels, time variation heating on walls of circular structure, and heat treatment on hollow cylinders are typical examples. It is well known that if the temperature and/or the heat flux are prescribed at the boundary surface, then the heat transfer system includes heat conduction coefficient only; on the other hand, if the boundary surface dissipates heat by convection on the basis of Newton’s law of cooling, the heat transfer coefficient will be included in the boundary term.

For the problem of heat conduction in hollow cylinders with time-dependent boundary conditions of any kinds at inner and outer surfaces, the associated governing differential equation is a second-order Bessel differential equation with constant coefficients. After conducting a Hankel transformation, the analytical solutions can be obtained and found in the textbook by Özisik [1].

For the heat transfer in hollow cylinders with mixed type boundary condition and time-dependent heat transfer coefficient simultaneously, the problem cannot be solved by any analytical methods, such as the method of separation of variable, Laplace transform, and Hankel transform. Few studies in Cartesian coordinate system can be found and various approximated and numerical methods were proposed. By introducing a new variable, Ivanov and Salomatov [2, 3] together with Postol’nik [4] transformed the linear governing equation into a nonlinear form. After ignoring the nonlinear term, they developed an approximated solution, which was claimed to be valid for the system with Biot number being less than 0.25. Moreover, Kozlov [5] used Laplace transformation to study the problems with Biot function in a rational combination of sines, cosines, polynomials, and exponentials. Even though it is possible to obtain the exact series solution of a specified transformed system, the problem is the computation of the inverse Laplace transformation, which generally requires integration in the complex plane. Becker et al. [6] took finite difference method and Laplace transformation method to study the heating of the rock adjacent to water flowing through a crevice. Recently, Chen and his colleagues [7] proposed an analytical solution by using the shifting function method for the heat conduction in a slab with time-dependent heat transfer coefficient at one end. Yatskiv et al. [8] studied the thermostressed state of cylinder with thin near-surface layer having time-dependent thermophysical properties. They reduced the problem to an integrodifferential equation with variable coefficients and solved it by the spline approximation.

In addition, different approximation methods such as the iterative perturbation method [9], the time-varying eigenfunction expansion method with finite integral transforms [10, 11], generalized integral transforms [12], and the Lie point symmetry analysis [13] were used to study this kind of problems. Various inverse schemes for determining the time-dependent heat transfer coefficient were developed by some researchers [14–20].

According to the literature, because of the complexity and difficulty of the methodology, none of any analytical solutions for the heat conduction in a hollow cylinder with time-dependent boundary condition and time-dependent heat transfer coefficient existed. This work extends the methodology of shifting function method [7, 21, 22] to develop an analytical solution with closed form for the heat transfer in hollow cylinders with time-dependent boundary condition and time-dependent heat transfer coefficient simultaneously. By setting the Biot function in a particular form and introducing two specially chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions and can be solved by series expansion theorem. Examples are given to demonstrate the methodology and numerical results are compared with those in the literature. And last but not least, the influence of physical parameters on the temperature profile is studied.

#### 2. Mathematical Modeling

Consider the transient heat conduction in heat exchanger tubes as shown in Figure 1. A fluid with time-varying temperature is running inside the hollow cylinder and the heat is dissipated by the time-dependent convection at the outer surface into an environment of zero temperature. The governing differential equation of the system iswhere is the temperature, is the space variable, is the thermal diffusivity, is the time variable, and and denote inner and outer radii, respectively. The boundary and initial conditions of the boundary value problem areHere, is a time-dependent temperature function at the inner surface, is the thermal conductivity, is a time-dependent heat transfer coefficient function, and is an initial temperature function. For consistence in initial temperature field, must be equal to . The above problem can be normalized by definingwhere is a constant reference temperature, and the dimensionless boundary value problem will then become