Reaction-Diffusion in a Two-Dimensional Catalyst Pellet

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Consider the reaction-diffusion in a two-dimensional catalyst pellet with governing equations and boundary condition:




at and ,

where is the Thiele modulus, is the adiabatic temperature rise (the Prater temperature), is the activation energy, and is the Lewis number.

The steady-state temperature and reaction conversion along are plotted in magenta and blue, respectively. You can vary the parameters , , and as well as the number of Chebyshev collocation points, .


Contributed by: Housam Binousand Brian G. Higgins (May 2013)
Open content licensed under CC BY-NC-SA



In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extrema of the Chebyshev polynomials of the first kind, .

The Chebyshev derivative matrix at the quadrature points is an matrix given by

, , for , and for , , and ,

where for and .

The discrete Laplacian is given by where is the identity matrix, is the Kronecker product operator, , and is without the first row and first column.


[1] P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.

[2] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.

[3] T. R. Marchant and M. I. Nelson,"Semi-analytical Solutions for One- and Two-Dimensional Pellet Problems," Proceedings of the Royal Society A, 460(2048), 2004 pp. 2381–2394. doi:10.1098/rspa.2004.1286.

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