Exterior Shape Factors From Interior Shape Factors
Shape factors for steady heat conduction enable quick and highly simplified calculations of heat transfer rates within bodies having a combination of isothermal and adiabatic boundary conditions. Many shape factors have been tabulated, and most undergraduate heat transfer books cover their derivatio...
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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ASME International
2019
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| Online Access: | https://hdl.handle.net/1721.1/121994 |
| _version_ | 1811083422782717952 |
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| author | Lienhard, John H |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Lienhard, John H |
| author_sort | Lienhard, John H |
| collection | MIT |
| description | Shape factors for steady heat conduction enable quick and highly simplified calculations of heat transfer rates within bodies having a combination of isothermal and adiabatic boundary conditions. Many shape factors have been tabulated, and most undergraduate heat transfer books cover their derivation and use. However, the analytical determination of shape factors for any but the simplest configurations can quickly come to involve complicated mathematics, and, for that reason, it is desirable to extend the available results as far as possible. In this paper, we show that known shape factors for the interior of two-dimensional objects are identical to the corresponding shape factors for the exterior of those objects. The canonical case of the interior and exterior of a disk is examined first. Then, conformal mapping is used to relate known configurations for squares and rectangles to the solutions for the disk. Both a geometrical and a mathematical argument are introduced to show that shape factors are invariant under conformal mapping. Finally, the general case is demonstrated using Green's functions. In addition, the “Yin-Yang” phenomenon for conduction shape factors is explained as a rotation of the unit disk prior to conformal mapping. |
| first_indexed | 2024-09-23T12:32:49Z |
| format | Article |
| id | mit-1721.1/121994 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:32:49Z |
| publishDate | 2019 |
| publisher | ASME International |
| record_format | dspace |
| spelling | mit-1721.1/1219942022-09-28T08:29:46Z Exterior Shape Factors From Interior Shape Factors Lienhard, John H Massachusetts Institute of Technology. Department of Mechanical Engineering Massachusetts Institute of Technology. Robust Design Group Shape factors for steady heat conduction enable quick and highly simplified calculations of heat transfer rates within bodies having a combination of isothermal and adiabatic boundary conditions. Many shape factors have been tabulated, and most undergraduate heat transfer books cover their derivation and use. However, the analytical determination of shape factors for any but the simplest configurations can quickly come to involve complicated mathematics, and, for that reason, it is desirable to extend the available results as far as possible. In this paper, we show that known shape factors for the interior of two-dimensional objects are identical to the corresponding shape factors for the exterior of those objects. The canonical case of the interior and exterior of a disk is examined first. Then, conformal mapping is used to relate known configurations for squares and rectangles to the solutions for the disk. Both a geometrical and a mathematical argument are introduced to show that shape factors are invariant under conformal mapping. Finally, the general case is demonstrated using Green's functions. In addition, the “Yin-Yang” phenomenon for conduction shape factors is explained as a rotation of the unit disk prior to conformal mapping. 2019-08-16T14:02:10Z 2019-08-16T14:02:10Z 2019-04 2019-01 Article http://purl.org/eprint/type/JournalArticle 0022-1481 https://hdl.handle.net/1721.1/121994 Lienhard, John H., V. et al. "Exterior Shape Factors From Interior Shape Factors." Journal of Heat Transfer 141, 6 (April 2019): 061301 © 2019 American Society of Mechanical Engineers en_US http://dx.doi.org/10.1115/1.4042912 Exterior Shape Factors From Interior Shape Factors Creative Commons Attribution 4.0 International license https://creativecommons.org/licenses/by/4.0/ application/pdf ASME International ASME |
| spellingShingle | Lienhard, John H Exterior Shape Factors From Interior Shape Factors |
| title | Exterior Shape Factors From Interior Shape Factors |
| title_full | Exterior Shape Factors From Interior Shape Factors |
| title_fullStr | Exterior Shape Factors From Interior Shape Factors |
| title_full_unstemmed | Exterior Shape Factors From Interior Shape Factors |
| title_short | Exterior Shape Factors From Interior Shape Factors |
| title_sort | exterior shape factors from interior shape factors |
| url | https://hdl.handle.net/1721.1/121994 |
| work_keys_str_mv | AT lienhardjohnh exteriorshapefactorsfrominteriorshapefactors |