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Research Papers: Conduction

A Two-Dimensional Cylindrical Transient Conduction Solution Using Green's Functions

[+] Author and Article Information
Robert L. McMasters

Department of Mechanical Engineering,
Virginia Military Institute,
Lexington, VA 24450
e-mail: mcmastersrl@vmi.edu

James V. Beck

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: jvb@beckeng.com

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 24, 2013; final manuscript received June 14, 2014; published online July 15, 2014. Assoc. Editor: William P. Klinzing.

J. Heat Transfer 136(10), 101301 (Jul 15, 2014) (8 pages) Paper No: HT-13-1438; doi: 10.1115/1.4027882 History: Received August 24, 2013; Revised June 14, 2014

There are many applications for problems involving thermal conduction in two-dimensional (2D) cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green's functions where the principle of superposition is employed in combining a one-dimensional (1D) cylindrical case with a 1D Cartesian case to provide a temperature solution for a 2D cylindrical. Green's functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

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Figures

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Fig. 1

Geometry for nonhomogeneous problem

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Fig. 7

Transient temperature at x = 0 as a function of dimensionless time for several values of dimensionless time. In this plot, b/L = 5, a/b = 0.5, Bix = 0.5, and Bir = 2.5.

Grahic Jump Location
Fig. 8

Transient temperature at x = 1 as a function of dimensionless time for several values of the radial position, r/b. In this plot, b/L = 5, a/b = 0.5, Bix = 0.5, and Bir = 2.5.

Grahic Jump Location
Fig. 6

Transient temperature at x = 1 as a function of dimensionless time for several values of the radial position, r/b. In this plot, b/L = 5, a/b = 0.5, Bix = 0.5, and Bir = 2.5.

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Fig. 2

Dimensionless steady-state temperature as a function of dimensionless radial position, where b/L = 20, a/b = 0.5, Bix = 0.3, Bir = 6.0, x/L = 0

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Fig. 3

Steady-state temperature distribution at x = 0 as a function of the radius ratio r/b for several values of the x-direction Biot number, Bix. a/b = 0.5 and b/L = 20

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Fig. 4

Steady-state temperature distribution at x = 0 as a function of the radius ratio r/b for several values of the x-direction Biot number, Bix. a/b = 0.5 and b/L = 1

Grahic Jump Location
Fig. 5

Transient temperature at x = 0 as a function of dimensionless time for several values of the radial position, r/b. In this plot, b/L = 5, a/b = 0.5, Bix = 0.5, Bir = 2.5, and x/L = 0.

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