Keywords: pellet equations, weighted residual methods, orthogonal collocation, tau, Galerkin, least-squares.

The heterogeneous catalysis is of the most significance in many fields of gas conversion and processing in the chemical industries. Mathematical modelling of intra-particle mass and heat transport in porous pellets has been widely investigated by numerous researchers because intra-particle transport limitations often play an important role in the design of chemical reactors. Accurate modelling of intra-particle heat and mass transport are a prerequisite for the design of many industrial processes and the interpretation of experiments. In this study, the orthogonal collocation, Galerkin, tau and least-squares methods are adopted solving a complex pellet model in the spectral framework. The steam methane reforming (SMR) process and the methanol synthesis are adopted to investigate the properties of the weighted residual methods because the two processes cover various aspects of a chemical reaction; in particular, one of them is strongly diffusion limited, while the other is not.

During the last decade, the least-squares method has attracted increasing interest in the chemical engineering field and is considered as a good candidate for solving reactor modelling problems. Because the least-squares method is associated with the most complicated linear algebra theory and thus has complicated implementation issues, it is of interest to investigate the performance of the least-squares technique, relative to the more frequently used Galerkin, tau and orthogonal collocation methods, for solving the important pellet equations.

Considering the accuracy of the numerical techniques, the Galerkin and tau methods are recommended above the orthogonal collocation and least-squares methods. The accuracy of the orthogonal collocation method is similar to the least-squares method but with smaller condition numbers. On the other hand, considering implementation issues, the orthogonal collocation is associated with less theoretical complexities, whereas the Galerkin method is considered the most time consuming method to implement. Though the least-squares have been considered as a good candidate to solve problems in the chemical reactor engineering field, the properties of the least-squares method should be directly compared to the remaining weighted residual methods for several types of partial differential equations and reactor models holding different numerical challenges. Moreover, the evaluation of the pellet problem should be evaluated in the spectral-element framework.

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