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Integral Methods in Solving Governing PDEs in the Undergraduate Heat Transfer Course

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2016 ASEE Annual Conference & Exposition


New Orleans, Louisiana

Publication Date

June 26, 2016

Start Date

June 26, 2016

End Date

August 28, 2016





Conference Session

Thermodynamics, Fluids, and Heat Transfer II

Tagged Division

Mechanical Engineering

Tagged Topic


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Paper Authors


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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Salim M. Haidar Grand Valley State University Orcid 16x16

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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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Undergraduate mechanical engineering students at our school take a first course in ordinary differential equations, but are not obliged to take any courses in partial differential equations. On the other hand, all undergraduate heat transfer textbooks available today never discuss integral methods for solving the governing PDEs in heat transfer, which are superior to the common finite difference method approach taught in the classroom. In this paper, we present our methodology in bringing such powerful methods to the undergraduate heat transfer classroom with no prior student experience with PDEs. Integral equations together with Ritz and Kantorovich methods are used to show our students in the undergraduate heat transfer course how to find approximate solutions to multidimensional steady and unsteady conduction problems, with better accuracy to that found by finite difference methods under distinct temperature profiles. The approach emphasizes the physical concepts by writing the first law of thermodynamics in integral form and choosing approximate temperature profiles satisfying boundary and initial conditions. In addition, the mathematical implications of using integral methods in this undergraduate course show students the value 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 a more reliable approximate solution to such multidimensional steady or unsteady problems. We believe that this new approach and strategy are ideal for teaching these senior undergraduates the fundamentals of how to use integral methods to better understand and solve heat transfer problems.

Mohammadzadeh, A. R., & Haidar, S. M. (2016, June), Integral Methods in Solving Governing PDEs in the Undergraduate Heat Transfer Course Paper presented at 2016 ASEE Annual Conference & Exposition, New Orleans, Louisiana. 10.18260/p.25753

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