0

Full Content is available to subscribers

Subscribe/Learn More  >

Integral Methods in Solving Governing Nonlinear Partial Differential Equations in the Undergraduate Heat Transfer Course

[+] Author Affiliations
Salim M. Haidar

Grand Valley State University, Allendale, MI

AliReza Mohammadzadeh

Grand Valley State University, Grand Rapids, MI

Paper No. IMECE2017-72712, pp. V008T10A087; 6 pages
doi:10.1115/IMECE2017-72712
From:
  • ASME 2017 International Mechanical Engineering Congress and Exposition
  • Volume 8: Heat Transfer and Thermal Engineering
  • Tampa, Florida, USA, November 3–9, 2017
  • Conference Sponsors: ASME
  • ISBN: 978-0-7918-5843-1
  • Copyright © 2017 by ASME

abstract

All undergraduate heat transfer textbooks available today give an overriding role to the method of separation of variables to the point of excluding other frameworks and views for solving the underlying governing PDEs. However, the transition from a linear to a nonlinear heat transfer model not only makes the method of separation of variables inapplicable, but also introduces additional mathematical and computational difficulties that must be studied further and overcome. Yet, none of these textbooks discuss integral methods for solving the governing PDEs in heat transfer which are at least as good as the common separation of variables and finite difference techniques taught in the classroom. In this paper, we extend our new methodology from the linear to the nonlinear heat conduction problems by bringing such powerful methods to the undergraduate heat transfer classroom with no prior student experience with PDEs [1]. Integral methods of Von Kármán together with Ritz and Kantorovich methods are used to show our students in the undergraduate heat transfer course how to find approximate analytic solutions to nonlinear multidimensional steady and unsteady conduction problems involving surface radiation and temperature-dependent thermo-physical properties under distinct temperature profiles. The approach has a certain elegance in the sense that it expresses the complete physical effect of the system in terms of a single integral representing the first law of thermodynamics; moreover, the implications of using integral methods in this undergraduate course show the value of mathematical simplification in reducing the order of the governing PDEs and/or the number of associated independent variables. No knowledge of separation of variables or transform methods is needed to obtain an approximate analytic solution to such nonlinear multidimensional steady or unsteady problems with accuracy acceptable by most engineering standards.

Copyright © 2017 by ASME

Figures

Tables

Interactive Graphics

Video

Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature

Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal

NOTE:
Citing articles are presented as examples only. In non-demo SCM6 implementation, integration with CrossRef’s "Cited By" API will populate this tab (http://www.crossref.org/citedby.html).

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In