### Abstract

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 welltest 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.

Original language | English (US) |
---|---|

Article number | 121301 |

Journal | Journal of Heat Transfer |

Volume | 139 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2017 |

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### Keywords

- Green's function
- heat conduction
- improper integral
- numerical methods
- semi-infinite geometry

### ASJC Scopus subject areas

- Materials Science(all)
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

**Efficient Numerical Evaluation of Exact Solutions for One-Dimensional and Two-Dimensional Infinite Cylindrical Heat Conduction Problems.** / Pi, Te; Cole, Kevin; Beck, James.

Research output: Contribution to journal › Article

*Journal of Heat Transfer*, vol. 139, no. 12, 121301. https://doi.org/10.1115/1.4037081

}

TY - JOUR

T1 - Efficient Numerical Evaluation of Exact Solutions for One-Dimensional and Two-Dimensional Infinite Cylindrical Heat Conduction Problems

AU - Pi, Te

AU - Cole, Kevin

AU - Beck, James

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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 welltest 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.

AB - 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 welltest 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.

KW - Green's function

KW - heat conduction

KW - improper integral

KW - numerical methods

KW - semi-infinite geometry

UR - http://www.scopus.com/inward/record.url?scp=85026551145&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85026551145&partnerID=8YFLogxK

U2 - 10.1115/1.4037081

DO - 10.1115/1.4037081

M3 - Article

AN - SCOPUS:85026551145

VL - 139

JO - Journal of Heat Transfer

JF - Journal of Heat Transfer

SN - 0022-1481

IS - 12

M1 - 121301

ER -