Consider the reactiondiffusion in a twodimensional catalyst pellet with governing equations and boundary condition: , , where is the Thiele modulus, is the adiabatic temperature rise (the Prater temperature), is the activation energy, and is the Lewis number. The steadystate 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, .
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 , 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,"Semianalytical Solutions for One and TwoDimensional Pellet Problems," Proceedings of the Royal Society A, 460(2048), 2004 pp. 2381–2394. doi:10.1098/rspa.2004.1286.
