Simulation of pellet model with multicomponent mass diffusion closure using least squares spectral element solution method

Abstract

Mass‐ and mole‐based pellet models have been solved using least‐squares formulation to describe the evolution of species composition, pressure, velocity, total concentration and mass diffusion fluxes in porous pellets for the steam methane reforming (SMR) process. The objective of this work has been to compare the mass‐ and mole‐based diffusion flux models, convection and fluid velocity for the SMR process using the least‐squares spectral element method (LS‐SEM). The diffusion–reaction problems are computationally intensive, requiring efficient numerical methods for dealing with them. This paper presents formulation and algorithm of LS‐SEM for solving multicomponent mass diffusion pellet models. The mass diffusion flux is described according to the rigorous Maxwell Stefan model. This flux may be defined either with molar‐ or mass‐averaged velocities. The effectiveness factors have been calculated for the SMR process and compared with the literature data. The model evaluations revealed that: ‐ The least‐squares method is well‐suited for solving the multicomponent mass diffusion pellet models for the SMR process, achieving exponential convergence. ‐ Molar‐ and mass‐based pellet models do not give fully identical results for the SMR process, since the pellet model is not completely consistent.

A numerical study of fixed bed reactor modelling for steam methane reforming process

“…For the case of pseudo‐homogeneous and simplified heterogeneous reactor models, effectiveness factors are used for the reforming and water‐gas shift reactions to account for internal diffusion resistance. The effectiveness factors are found from simulations of one catalyst pellet with typical bulk gas phase compositions as boundary conditions, and were found to be 0.01, 0.07, and 0.008 for the reactions in Reactions (I–III), respectively . These values are used in the simulation of pseudo‐homogeneous and simplified heterogeneous reactor models.…”

“…The effectiveness factors are found from simulations of one catalyst pellet with typical bulk gas phase compositions as boundary conditions, and were found to be 0.01, 0.07, and 0.008 for the reactions in Reactions (I-III), respectively. [67] These values are used in the simulation of pseudo-homogeneous and simplified heterogeneous reactor models. Figure 5 shows the comparison of the pseudo-homogeneous, conventional, and simplified heterogeneous reactor models for the SMR process.…”

Evaluation of spectral, spectral‐element and finite‐element methods for the solution of the pellet equation

“…Recently, Rout and Jakobsen [60] aimed to check the suitability of the least-squares spectral-element method on a diffusion-reaction problem. The authors state that "the least-squares method is wellsuited for the solution of the pellet model equations, achieving exponential convergence in the method order".…”

“…The idea of piecewise polynomial trial functions has been used before for pellet problems. [1,60,67] Jiang [15] summarize some important features of the element framework:…”

“…The large gradients in the solution of the latter case test the numerical methods on a nearly discontinuous problem. For example, the steam-methane reforming process is known to be strongly diffusion limited, [60] whereas considerable less diffusion limitation is present in pellet of the methanol synthesis. [18] Symmetry is commonly assumed in simulation of spherical pellets; [1,18,35] hence, the considered diffusion-reaction problem in the present study is one-dimensional.…”