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Research Papers: Conduction

# Analytical Solution to Transient Asymmetric Heat Conduction in a Multilayer Annulus

[+] Author and Article Information
Prashant K. Jain1

Department of Nuclear, Plasma and Radiological Engineering, University of Illinois at Urbana–Champaign, 216 Talbot Lab, 104 South Wright Street, Urbana, IL 61801

Suneet Singh, Rizwan-uddin

Department of Nuclear, Plasma and Radiological Engineering, University of Illinois at Urbana–Champaign, 216 Talbot Lab, 104 South Wright Street, Urbana, IL 61801

1

Corresponding author.

J. Heat Transfer 131(1), 011304 (Oct 20, 2008) (7 pages) doi:10.1115/1.2977553 History: Received February 12, 2008; Revised April 21, 2008; Published October 20, 2008

## Abstract

In this paper, we present an analytical double-series solution for the time-dependent asymmetric heat conduction in a multilayer annulus. In general, analytical solutions in multidimensional Cartesian or cylindrical $(r,z)$ coordinates suffer from existence of imaginary eigenvalues and thus may lead to numerical difficulties in computing analytical solution. In contrast, the proposed analytical solution in polar coordinates (2D cylindrical) is “free” from such imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of transverse (radial) eigenvalues on those in the other direction.

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## Figures

Figure 1

Schematic of an n-layer annulus. ith layer has an inner and outer radii equal to ri−1 and ri, respectively.

Figure 2

Asymmetric heat conduction in a three-layer annulus. Each layer has a different thermal conductivity (ki) and thermal diffusivity (αi). The lower-half of the annulus (π⩽θ⩽2π) is kept insulated, while the upper-half (0⩽θ⩽π) is subjected to a θ-dependent incoming heat-flux.

Figure 3

Transient isotherms in three-layer annulus: (a) t=5, (b) t=10, (c) t=15, and (d) steady state

Figure 4

Transient temperature variation in the radial direction at (a) θ=0, (b) θ=π∕2, and (c) θ=3π∕2

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