Reaction-Diffusion in a Two-Dimensional Catalyst Pellet
Consider the reaction-diffusion in a two-dimensional catalyst pellet with governing equations and boundary condition:[more]
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, .[less]
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.
 P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.
 L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.
 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.