Elsevier

Journal of Chromatography A

Volume 1218, Issue 40, 7 October 2011, Pages 7137-7146
Journal of Chromatography A

A discontinuous Galerkin method to solve chromatographic models

https://doi.org/10.1016/j.chroma.2011.08.005Get rights and content

Abstract

This article proposes a discontinuous Galerkin method for solving model equations describing isothermal non-reactive and reactive chromatography. The models contain a system of convection–diffusion–reaction partial differential equations with dominated convective terms. The suggested method has capability to capture sharp discontinuities and narrow peaks of the elution profiles. The accuracy of the method can be improved by introducing additional nodes in the same solution element and, hence, avoids the expansion of mesh stencils normally encountered in the high order finite volume schemes. Thus, the method can be uniformly applied up to boundary cells without loosing accuracy. The method is robust and well suited for large-scale time-dependent simulations of chromatographic processes where accuracy is highly demanding. Several test problems of isothermal non-reactive and reactive chromatographic processes are presented. The results of the current method are validated against flux-limiting finite volume schemes. The numerical results verify the efficiency and accuracy of the investigated method. The proposed scheme gives more resolved solutions than the high resolution finite volume schemes.

Introduction

Chromatography is widely used chemical technique to separate mixtures or components in chemical plants, hospitals, law enforcement, and environmental agencies, see for example Guiochon et al. [1]. It is used for separating mixtures into components in order to analyze, identify, purify, quantify, and optimize. Chromatographic separation processes are based on the differential adsorption of the components of a mixture. In the conventional liquid chromatography, the separation is achieved by injecting a pulse of the solute mixture to the chromatographic column. The components move with different velocities in the column due to their different affinity with the solid phase. Thus, the low retained component exits the column earlier than the more retained one and, hence, the separation is achieved. Chromatography technology has gained immense industrial popularity in the past few decades and further advances in chromatography processes constantly improved their performance for the separation of more complex mixtures.

Reactive chromatography can be very attractive technique to effectively reduce the number of units and enhancing the performance of the process. In reactive chromatography, chemical reactions and chromatographic separation of the product take place simultaneously in the column. This principle is comparable to reactive distillation, reactive extraction or reactive absorption. The concept is particularly advantageous to perform equilibrium limited reversible reactions due to the shifting of chemical equilibrium allowing to improve conversion, yield and separation efficiency. The reactive chromatography reduces capital investment, energy and operating cost, equipment size, waste and pollution, as well as improves selectivity, purity, and productivity. Despite several theoretical investigations of reactive chromatography, accurate database and models are still lacking for commercialization. There are several book chapters available regarding chromatographic reactor, e.g. Ganetsos and Barker [2], Sardin et al. [3], Villermaux et al. [4], Fricke et al. [5], and Borren et al. [6]. Much research is motivated in an effort to provide profound insights into all aspects to scale up the process for industrial applications.

The simulation of multicomponent chromatographic processes under non-linear conditions and reaction kinetics require fast and accurate numerical methods. A fundamental feature of the first order nonlinear hyperbolic partial differential equations is that discontinuities in the solutions can develop even for smooth initial and boundary data that cause major difficulties for the numerical schemes. Thus, much effort has been invested to develop appropriate numerical schemes for producing physically realistic solutions, see for example Guiochon et al. [1], Thiele et al. [7], Von Lieres and Andersson [8], and Shipilova et al. [9]. An additional feature is the local conservativity of the numerical fluxes, that is, the numerical flux should be conserved from one discretization cell to its neighbours. Finite volume methods (FVMs), which preserve such properties, were already applied to chromatographic models, see for example Webley and He [10], Cruz et al. [11], Von Lieres and Andersson [8], Javeed et al. [12], and Medi and Amanullah [13].

The Discontinuous Galerkin (DG) finite element method was initially introduced by Reed and Hill [14] for solving neutron transport equations. Afterwards, various DG methods were developed and formulated by Cockburn and Shu for nonlinear hyperbolic system in the series of papers, see for example Cockburn et al. [15], [16], [17], [18], [19]. DG-methods are being applied in the main stream of computational fluid dynamic models, see for example Chen et al. [20], Bassi and Rebay [21], Bahhar et al. [22], Aizinger et al. [23], Cockburn and Dawson[24], and Holik [25]. The DG methods are versatile, flexible, and have intrinsic stability making them suitable for convection dominated problems. The stability is an intrinsic property of the method to keep the solution bounded, i.e. numerical errors (roundoff due to finite precision of computers) which are generated during the solution of discretized equations should not be magnified. The numerical solution itself should remain uniformly bounded. DG-methods can be efficiently applied to partial differential equations (PDEs) of all kinds, including equations whose type changes within the computational domain. They were not applied to chromatographic models up to now.

DG-methods belong to the class of finite element method (FEM) which have several advantages over finite difference methods (FDMs) and finite volume methods (FVMs). For instance, they inherit geometric flexibility of FVMs and FEMs, retain the conservation properties of FVMs, and possess high-order properties of FEMs. Therefore, DG-methods are locally conservative, stable, and high order accurate. These methods satisfy the total variation bounded (TVB) property that guarantees the positivity of the schemes, see e.g. Cockburn et al. [15], [16], [18]. Positivity is the most common and fundamental mathematical requirement in physical models. In our case, concentrations are non-negative by their nature and their approximations should be non-negative as well. However, numerical solutions of scientific models often generate negative meaningless values. This may happen even when the numerical method is stable and highly accurate. In fact, the tendency to produce negative values may, paradoxically, increase with the order of accuracy of the numerical discretization. Loss of positivity may cause a computation to fail or produce meaningless results, especially conservation of mass can not be achieved. In contrast to high order FDMs and FVMs, DG-methods require a simple treatment of the boundary conditions in order to achieve high order accuracy uniformly. Moreover, DG methods allow discontinuous approximations and produce block-diagonal mass matrices that can be easily inverted through algorithms of low computational cost. These methods incorporate the idea of numerical fluxes and slope limiters in a very natural way to avoid spurious oscillations (wiggles), which usually occur due to shocks, discontinuities or sharp changes in the solution.

In this paper, the total variation bounded (TVB) Runge–Kutta DG-scheme of Cockburn et al. [15], [16], [17], [18] is implemented for solving chromatographic models. The scheme employs a DG-method in the axial-coordinate that converts the given PDE to a system of ordinary differential equations (ODEs). The resulting ODE-system is then solved by using explicit and nonlinearly stable high order Runge–Kutta method. The TVB property guarantees the positivity of the scheme, for example the non-negativity of the mixture concentrations in the present case. The numerical test problems of this manuscript verify the accuracy and efficiency of the DG-scheme for solving chromatographic models. For validation, the numerical results of the suggested scheme are compared with the high resolution finite volume scheme of Koren [26].

The structure of the article is as follows: The fixed-bed chromatographic reactor (FBCR) model is described in Section 2. Section 3 presents a derivation of the discontinuous Galerkin method for the FBCR model. Numerical test problems are presented in Section 4. Concluding remarks and future recommendations are given in Section 5.

Section snippets

The fixed-bed chromatographic reactor model

Chromatographic reactors were patented in the early 1960s, see e.g. Dinwinddie and Morgan [27], Magee [28], and Gaziev et al. [29]. A FBCR is defined by Langer et al. [30] as a “chromatographic column in which a solute or several solutes are intentionally converted, either partially or totally, to products during their residence in the column. The solute reactant or reactant mixture is injected into the chromatographic reactor as a pulse. Both conversion to product and separation take place in

The discontinuous Galerkin formulation for FBCR

In this section, the TVB Runge–Kutta DG-scheme of Cockburn and Shu [15] and Qiu et al.[32] is implemented for solving the FBCR model given in Eq. (1). Firstly, we suitably rewrite the original system as a degenerate (transformed) first-order system to obtain the weak formation for deriving the numerical scheme. Then, the TVB Runge–Kutta DG-scheme is applied in axial-coordinate that converts the given PDE to an ODE-system. The resulting ODE-system is approximated by using the TVB Runge–Kutta

Numerical test problems

In order to validate the DG scheme, several numerical test problems are investigated. For quantitative analysis, the results of the DG-scheme are compared with flux-limiting finite volume schemes investigated by Javeed et al. [12]. In all numerical test problems, linear basis functions are used in each cell, giving a second order accurate DG-scheme in axial-coordinate. The ODE-system was solved by a third-order Runge–Kutta method given by Eq. (39). The program is written in C-language under a

Conclusions

In this paper, a discontinuous Galerkin finite element method was implemented for solving isothermal non-reactive and reactive chromatographic models. The presented scheme satisfies the TVB property and gives second order accuracy. The scheme can be easily extended to higher orders by using high order basis functions and by employing better slope limiters, for example reconstruction with the WENO limiters of Qiu et al. [41]. This method incorporate the ideas of numerical fluxes and slope

Acknowledgements

The authors gratefully acknowledge the International Max Planck Research School Magdeburg (IMPRS) and Higher Education Commission (HEC) of Pakistan through Grant no. 1550 for financial support.

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