Heat Transfer in a Bar with Rectangular Cross Section

Consider a bar with a rectangular cross section subject to both temperature and heat flux boundary conditions.
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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[1] T. Bennett, Transport by Advection and Diffusion, New York: Wiley, 2012.
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