Evaluation of weighted residual methods for the solution of the pellet equations: The orthogonal collocation, Galerkin, tau and least-squares methods

In recent years a number of publications have adopted the least-squares method for chemical reactor engineering problems such as the population balance equation, fixed bed reactors and pellet equations. Evaluation of the performance of the least-squares method compared to other weighted residual methods is therefore of interest. Thus, in the present study, numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau, and least-squares methods, have been adopted to solve a non-linear comprehensive and highly coupled pellet problem. The methanol synthesis and the steam methane reforming process have been adopted for this study.Based on the residual of the governing equations, the orthogonal collocation method obtains the same accuracy as the Galerkin and tau methods. Moreover, the orthogonal collocation method is associated with the simplest algebra theory and thus holds the simplest implementation. Another benefit of the orthogonal collocation method is that the technique is more computational efficient than the other methods evaluated. The least-squares method does not obtain the same accuracy as the other weighted residual methods. In particular, the least-squares method is not suitable for strongly diffusion limited systems that give rise to steep gradients in the pellet. On the other side, considering the spectral–element framework, the least-squares method is less sensitive to the placement of the element boundaries than observed for the orthogonal collocation, Galerkin and tau methods.The present paper outlines the algebra of the weighted residual methods and illustrate the numerical solution techniques through a simplified pellet problem.