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Research Papers: Conduction

# Transversal Method of Lines for Unsteady Heat Conduction With Uniform Surface Heat Flux

[+] Author and Article Information
Antonio Campo

Department of Mechanical Engineering,
College of Engineering,
The University of Texas at San Antonio,
San Antonio, TX 78249
e-mail: campanto@yahoo.com

José Garza

Department of Mechanical Engineering,
College of Engineering,
The University of Texas at San Antonio,
San Antonio, TX 78249

Recall that TMOL is unaffected by the dimensionless space coordinate interval ΔX.

For brevity, the dimensionless time interval $Δτ$ is replaced by $τ$ from here on.

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received November 13, 2013; final manuscript received July 21, 2014; published online August 26, 2014. Assoc. Editor: William P. Klinzing.

J. Heat Transfer 136(11), 111302 (Aug 26, 2014) (7 pages) Paper No: HT-13-1580; doi: 10.1115/1.4028082 History: Received November 13, 2013; Revised July 21, 2014

## Abstract

The transversal method of lines (TMOL) is a general hybrid technique for determining approximate, semi-analytic solutions of parabolic partial differential equations. When applied to a one-dimensional (1D) parabolic partial differential equation, TMOL engenders a sequence of adjoint second-order ordinary differential equations, where in the space coordinate is the independent variable and the time appears as an embedded parameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have constant or variable coefficients. In this work, TMOL is applied to the unsteady 1D heat equation in simple bodies (large plate, long cylinder, and sphere) with temperature-invariant thermophysical properties, constant initial temperature and uniform heat flux at the surface. In engineering applications, the surface heat flux is customarily provided by electrical heating or radiative heating. Using the first adjoint quasi-stationary heat equation for each simple body with one time jump, it is demonstrated that approximate, semi-analytic TMOL temperature solutions with good quality are easily obtainable, regardless of time. As a consequence, usage of the more involved second adjoint quasi-stationary heat equation accounting for two consecutive time jumps come to be unnecessary.

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## Figures

Fig. 1

Computational domain for TMOL

Fig. 2

Comparison of the dimensionless surface, midplane and mean temperatures in a large plate between the approximate, semi-analytic TMOL solution and the exact, analytic solution from Luikov [3]

Fig. 3

Comparison of the dimensionless surface, centerline and mean temperatures in a long cylinder between the approximate, semi-analytic TMOL solution and the exact, analytic solution from Luikov [3]

Fig. 4

Comparison of the dimensionless surface, center, and mean temperatures in a sphere between the approximate, semi-analytic TMOL solution and the exact, analytic solution from Luikov [3]

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