Solving two‐dimensional chemical engineering problems using the chebyshev orthogonal collocation technique

Abstract

The present paper describes how to apply the Chebyshev orthogonal collocation technique to solve steady-state and unsteady-steady two-dimensional problems. All problems are solved using one single computer algebra, Mathematica ß . The problems include: (1) steady-state heat transfer in a rectangular bar, (2) steady-state flow in a rectangular duct, (2) steady-state heat transfer in a cooling cylindrical pin fin, (4) steady-state heat conduction in an annulus, (5) unsteady-state heat transfer in a rectangular bar, and finally (6) unsteady-state diffusion reaction system. Whenever possible the results obtained with orthogonal collocation are compared to the analytical solution in order to validate the applied numerical technique. ß 2015 Wiley Periodicals, Inc. Comput Appl Eng Educ 24:144-155, 2016; View this article online at wileyonlinelibrary.com/journal/cae;

Solution of six chemical engineering problems using the Chebyshev orthogonal collocation technique

“…When using the Chebyshev orthogonal collocation, the computation of the one‐dimensional derivative matrix, explicitly defined in Binous et al , is sufficient to treat 1‐D as well as 2‐D and 3‐D problems. Indeed, the derivatives matrices for 2‐D and 3‐D PDEs can be readily derived from the one‐dimensional matrix (i.e., D ) using the Kronecker product [3,4]. In Matlab, similar to the other mathematical software, such algebraic operations can be easily implemented using the predefined function kron .…”

“…Several studies using this technique [2,9,14,15], and at least three monographs [7,11,16] presenting the method in detail have been published. A recent set of three pedagogical papers by the authors of this paper described the application of the Chebyshev orthogonal collocation in one‐, two‐, and three‐dimensional domains . In this paper, the orthogonal collocation technique is applied to elucidate six graduate‐level chemical engineering problems.…”

“…In this paper, the orthogonal collocation technique is applied to elucidate six graduate‐level chemical engineering problems. Table lists these six case studies as well as the challenges and the novelty of the treatment employed in each example (indicated in italic and red in Table ) compared to the problems we have solved in our previous work . All case studies elucidate a chemical engineering problem involving the transfer of mass, heat, and/or momentum.…”

“…The approach based on the Chebyshev orthogonal collocation technique is far simpler to implement with mathematical software (e.g., Matlab ® and M ATHEMATICA © ) than with finite differences, finite volumes, and/or finite elements. This approach only requires the calculation of the derivative matrix, D , which is explained briefly in the next section and thoroughly in the published papers and monographs .…”

Numerical elucidation of three‐dimensional problems in the chemical engineering graduate curriculum

“…The usage of computer software such as M ATHEMATICA © , M APLE , M ATLAB ® , P OLYMATH , M ATHCAD , J AVA , A SPEN ‐HYSYS or E XCEL is common in chemical engineering pedagogical literature . A literature survey over last three decades shows that only two papers report on the application of the Chebyshev orthogonal collocation technique, implemented in Mathematica © , for educational purposes [12,13], restricted however to one‐ and two‐dimensional problems.…”