Heat Transfer in a Bar with Rectangular Cross Section

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Consider a bar with a rectangular cross section subject to both temperature and heat flux boundary conditions.

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The governing equation and boundary conditions in dimensionless form are:

, with and ,

(continuity of heat flux at the boundary),

(constant temperature at the boundary),

(same constant temperature at the boundary), and

(different constant temperature at the boundary),

with , , and .

The Chebyshev orthogonal collocation method implemented in Mathematica with collocation points gives the temperature distribution in the bar represented in the snapshots for user-set values of and .

The analytical solution of the differential equation obtained using the separation of variables technique [1] is given by:

,

where .

We have found excellent agreement between our numerical solution and the analytical solution.

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Contributed by: Housam Binous and Ahmed Bellagi (August 2015)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Reference

[1] T. Bennett, Transport by Advection and Diffusion, New York: Wiley, 2012.



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