Board # 82 : Application of the Heat-balance Integral to Linear Transient Heat Conduction in a Semi-infinite Medium

Salim M. Haidar; Ali R. Mohammadzadeh

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Conference: 2017 ASEE Annual Conference & Exposition
Location: Columbus, Ohio
Publication Date: June 24, 2017
Start Date: June 24, 2017
End Date: June 28, 2017
Conference Session: Mechanical Engineering Division Poster Session
Tagged Division: Mechanical Engineering
Page Count: 9
DOI: 10.18260/1-2--27934
Permanent URL: https://peer.asee.org/27934
Download Count: 716

Paper Authors

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Salim M. Haidar Grand Valley State University orcid.org/0000-0002-9756-9396

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Salim M. Haidar holds a Ph.D in Applied Mathematics from Carnegie-Mellon University and is a Professor of Mathematics at Grand Valley State University. His research interests are: Nonlinear Elasticity (regularity of equilibria and material instabilities); Variational Calculus (field theory, regularity of minimizers, relaxed formulations); and PDEs (compensated-compactness and homogenization methods in solving nonlinear pde’s;)

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Ali R. Mohammadzadeh Grand Valley State University

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Ali Mohammadzadeh is associate professor of mechanical engineering at School of Engineering at Grand valley State University. He earned his PhD degree in mechanical engineering from the University of Michigan in Ann Arbor. His research interest is in the field of fluid solid interaction.

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Abstract

Although undergraduate students in mechanical engineering are required to take a course in ordinary differential equations (ODE), they are not obligated to acquire formal instruction in partial differential equations (PDE) before graduation. This may be why very popular and widely used undergraduate heat transfer textbooks such as those by Incropera2, Kreith3 use finite difference method as their main solution approach to solve heat conduction problems in transient and steady 2-dimensional cases and mostly under time-independent boundary conditions with regular geometries.. With a broad range of topics covered in a single undergraduate heat transfer course, the current authors are of the same belief that there would be no room and time to teach them separation of variables and transform techniques for solving the governing PDE equations. Yet, none of these textbooks discuss integral methods as an alternate framework for solving the governing PDEs in heat transfer which is at least as good as, if not superior to, the common separation of variables and finite difference techniques taught in the classroom. The integral method of solving 2-D steady and transient heat conduction, though is extensively covered in advanced conduction textbooks such as the ones by Arpaci4 and Ӧzışık5, yet this powerful and promising approach seems to be almost totally missing from undergraduate engineering education, except for a treatment of forced and natural heat convection problems. In this paper, we extend our new methodology for solving linear heat transfer problems from finite to infinite domains by bringing such powerful methods to the undergraduate heat transfer classroom with no prior student experience with PDEs1. 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 such transient conduction problems involving time-dependent surface fluxes under distinct temperature profiles. This approach has a certain elegance to it 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 students 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 approximate analytic solutions to heat transfer problems in general with accuracy acceptable by most engineering standards We believe that this new approach and strategy, not only fulfill ABET Criterion 3[a, d, e, g, i], but are ideal for teaching these senior undergraduates the fundamentals of how to use integral methods to better understand and solve heat transfer problems.

1. ---- ----- --- Proceedings of the 2016 American Society for Engineering Education Annual Conference & Exposition, June 2016

Citation
Format

Haidar, S. M., & Mohammadzadeh, A. R. (2017, June), Board # 82 : Application of the Heat-balance Integral to Linear Transient Heat Conduction in a Semi-infinite Medium Paper presented at 2017 ASEE Annual Conference & Exposition, Columbus, Ohio. 10.18260/1-2--27934

TY  - CPAPER
AB  - 
Although undergraduate students in mechanical engineering are required to take a course in ordinary differential equations (ODE), they are not obligated to acquire formal instruction in partial differential equations (PDE) before graduation. This may be why very popular and widely used undergraduate heat transfer textbooks such as those by Incropera2, Kreith3 use finite difference method as their main solution approach to solve heat conduction problems in transient and steady 2-dimensional cases and mostly under time-independent boundary conditions with regular geometries.. With a broad range of topics covered in a single undergraduate heat transfer course, the current authors are of the same belief that there would be no room and time to teach them separation of variables and transform techniques for solving the governing PDE equations. Yet, none of these textbooks discuss integral methods as an alternate framework for solving the governing PDEs in heat transfer which is at least as good as, if not superior to, the common separation of variables and finite difference techniques taught in the classroom. The integral method of solving 2-D steady and transient heat conduction, though is extensively covered in advanced conduction textbooks such as the ones by Arpaci4 and Ӧzışık5, yet this powerful and promising approach seems to be almost totally missing from undergraduate engineering education, except for a treatment of forced and natural heat convection problems.
In this paper, we extend our new methodology for solving linear heat transfer problems from finite to infinite domains by bringing such powerful methods to the undergraduate heat transfer classroom with no prior student experience with PDEs1. 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 such transient conduction problems involving time-dependent surface fluxes under distinct temperature profiles.  This approach has a certain elegance to it 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 students 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 approximate analytic solutions to heat transfer problems in general with accuracy acceptable by most engineering standards  We believe that this new approach and strategy, not only fulfill ABET Criterion 3[a, d, e, g, i], but are ideal for teaching these senior undergraduates the fundamentals of how to use integral methods to better understand and solve heat transfer problems.  



1. ---- ----- ---
     Proceedings of the 2016 American Society for Engineering Education Annual Conference &amp; Exposition, June 2016 
	

AU  - Salim M. Haidar
AU  - Ali R. Mohammadzadeh
CY  - Columbus, Ohio
DA  - 2017/06/24
PB  - ASEE Conferences
TI  - Board # 82 : Application of the Heat-balance Integral to Linear Transient Heat Conduction in a Semi-infinite Medium
UR  - https://peer.asee.org/27934
DO  - 10.18260/1-2--27934
ER  -