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Research Papers

# Numerical Solution of Unsteady Conduction Heat Transfer in Anisotropic Cylinders

[+] Author and Article Information
Aslib Imane, Hamza Hamid, Lahjomri Jawad, Zniber Khalid, Oubarra Abdelaziz

Laboratory of Mechanics,
Faculty of Science Aïn Chock,
University Hassan II,
Casablanca 20100, Morocco

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received January 29, 2016; final manuscript received April 19, 2016; published online May 17, 2016. Assoc. Editor: John C. Chai.

J. Thermal Sci. Eng. Appl 8(3), 031013 (May 17, 2016) (9 pages) Paper No: TSEA-16-1026; doi: 10.1115/1.4033467 History: Received January 29, 2016; Revised April 19, 2016

## Abstract

This paper investigates a numerical solution of 2D transient heat conduction in an anisotropic cylinder, subjected to a prescribed temperature over the two end sections and a convective boundary condition over the whole lateral surface. The analysis of this anisotropic heat conduction problem is tedious because the corresponding partial differential equation contains a mixed-derivative. In order to overcome this difficulty, a linear coordinate transformation is used to reduce the anisotropic cylinder heat conduction problem to an equivalent isotropic one, without complicating the boundary conditions but with a more complicated geometry. The alternating-direction implicit finite-difference method (ADI) is used to integrate the isotropic equation combined with boundary conditions. Inverse transformation provides profile temperature in the anisotropic cylinder for full-field configuration. The numerical code is validated by the analytical heat conduction solutions available in the literature such as transient isotropic solution and steady-state orthotropic solution. The aim of this paper is to study the effect of cross-conductivity on the temperature profile inside an axisymmetrical anisotropic cylinder versus time, radial Biot number ($Bir$), and principal conductivities. The results show that cross-conductivity promotes the effect of $Bir$ according to the principal conductivities. Furthermore, the anisotropy increases the time required to achieve the steady-state heat conduction.

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## Figures

Fig. 1

Geometry and boundary conditions of the solid cylinder

Fig. 2

Geometry studied of the solid cylinder in the plane (r,z)

Fig. 3

Isotropic geometry studied of the solid cylinder after a linear coordinate transformation

Fig. 4

Integration domain of Eq. (7)

Fig. 5

History of temperature profile in an isotropic cylinder for θR=θL=θF=0, and θi=1

Fig. 6

Steady-state temperature profiles in orthotropic cylinder for θR=θF=0 and θL=1

Fig. 7

Full-fields temperature distribution for symmetric configuration (θL=θR=1 and θF=θi=0), for kz/kr=1

Fig. 8

Full-fields temperature distribution for symmetric configuration (θL=θR=1 and θF=θi=0), for kz/kr=2

Fig. 9

Full-fields temperature distribution for symmetric configuration (θL=θR=1 and θF=θi=0), for kz/kr=0.5

Fig. 10

History for center dimensionless temperature for Bir = 0.1 and 10

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