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Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

[+] Author Affiliations
Robert L. McMasters

Virginia Military Institute, Lexington, VA

Filippo de Monte

University of L’Aquila, L’Aquila, Italy

James V. Beck

Michigan State University, East Lansing, MI

Donald E. Amos

Sandia National Laboratories, Albuquerque, NM

Paper No. HT2016-7103, pp. V002T15A013; 14 pages
doi:10.1115/HT2016-7103
From:
  • ASME 2016 Heat Transfer Summer Conference collocated with the ASME 2016 Fluids Engineering Division Summer Meeting and the ASME 2016 14th International Conference on Nanochannels, Microchannels, and Minichannels
  • Volume 2: Heat Transfer in Multiphase Systems; Gas Turbine Heat Transfer; Manufacturing and Materials Processing; Heat Transfer in Electronic Equipment; Heat and Mass Transfer in Biotechnology; Heat Transfer Under Extreme Conditions; Computational Heat Transfer; Heat Transfer Visualization Gallery; General Papers on Heat Transfer; Multiphase Flow and Heat Transfer; Transport Phenomena in Manufacturing and Materials Processing
  • Washington, DC, USA, July 10–14, 2016
  • Conference Sponsors: Heat Transfer Division
  • ISBN: 978-0-7918-5033-6
  • Copyright © 2016 by ASME

abstract

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance.

The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems.

This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Copyright © 2016 by ASME

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