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Research Papers: Conduction

Efficient Numerical Evaluation of Exact Solutions for One-Dimensional and Two-Dimensional Infinite Cylindrical Heat Conduction Problems

[+] Author and Article Information
Te Pi

Mechanical & Materials Engineering Department,
University of Nebraska Lincoln,
W342C NH,
Lincoln, NE 68588-0526
e-mail: alitem0829@gmail.com

Kevin Cole

Professor
Mechanical & Materials Engineering Department,
University of Nebraska Lincoln,
W342C NH,
Lincoln, NE 68588-0526
e-mail: kcole1@unl.edu

James Beck

Professor Emeritus
College of Engineering,
Michigan State University,
Engineering Building, 428 S. Shaw Lane,
East Lansing, MI 48824-1226
e-mail: jameserebeck@gmail.com

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received September 14, 2016; final manuscript received May 9, 2017; published online July 25, 2017. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 139(12), 121301 (Jul 25, 2017) (10 pages) Paper No: HT-16-1582; doi: 10.1115/1.4037081 History: Received September 14, 2016; Revised May 09, 2017

Estimation of thermal properties or diffusion properties from transient data requires that a model is available that is physically meaningful and suitably precise. The model must also produce numerical values rapidly enough to accommodate iterative regression, inverse methods, or other estimation procedures during which the model is evaluated again and again. Bodies of infinite extent are a particular challenge from this perspective. Even for exact analytical solutions, because the solution often has the form of an improper integral that must be evaluated numerically, lengthy computer-evaluation time is a challenge. The subject of this paper is improving the computer evaluation time for exact solutions for infinite and semi-infinite bodies in the cylindrical coordinate system. The motivating applications for the present work include the line-source method for obtaining thermal properties, the estimation of thermal properties by the laser-flash method, and the estimation of aquifer properties or petroleum-field properties from well-test measurements. In this paper, the computer evaluation time is improved by replacing the integral-containing solution by a suitable finite-body series solution. The precision of the series solution may be controlled to a high level and the required computer time may be minimized, by a suitable choice of the extent of the finite body. The key finding of this paper is that the resulting series may be accurately evaluated with a fixed number of terms at any value of time, which removes a long-standing difficulty with series solution in general. The method is demonstrated for the one-dimensional case of a large body with a cylindrical hole and is extended to two-dimensional geometries of practical interest. The computer-evaluation time for the finite-body solutions are shown to be hundreds or thousands of time faster than the infinite-body solutions, depending on the geometry.

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Figures

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Fig. 1

(a) Geometry for R20 with R1<r<∞ and (b) geometry for R22 with R1<r<R2

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Fig. 2

Plot integrand of R20’s solution F(β,t̃) with t̃=0.5,1,5,10

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Fig. 3

Plot of integrand in improper integral from R20’s solution F¯(z,t̃) for 1/b = 10 with t̃=0.5,1,5,10

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Fig. 4

(a) R20Z22 geometry model with radial domain (R1<r<∞) and (b) R21Z22 geometry model with radial domain (R1<r<R2)

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Fig. 5

(a) Dimensionless temperature of R20Z22 case versus z̃ at t̃=0.0001,0.0003,0.0005,0.001 with accuracy A = 3; (b) dimensionless temperature of R20Z22 case versus z̃ at t̃=0.0001,0.0003,0.0005,0.001 with accuracy A = 4; and (c) dimensionless temperature of R20Z22 case versus z̃ at t̃=0.0001,0.0003,0.0005,0.001 with accuracy A = 5

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Fig. 6

(a) R00Z22 geometry model with infinite radial domain (0<r<∞) and (b) R02Z22 geometry model with finite radial domain (0<r<R2)

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