Moment based weighted residual method—New numerical tool for a nonlinear multicomponent chromatographic general rate model

doi:10.1016/j.compchemeng.2013.02.008

Computers & Chemical Engineering

Volume 53, 11 June 2013, Pages 153-163

https://doi.org/10.1016/j.compchemeng.2013.02.008 Get rights and content

Highlights

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A new numerical method is proposed to solve a chromatographic general rate model.
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The method precisely conserves mass and predicts the separation characteristic values.
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The method reaches desired accuracy with less variables and fast calculation speed.
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The method is also suitable for e.g. reactor and leaching models with particle phase.
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One real experiment is simulated to verify the method.

Abstract

A new numerical method is proposed to solve a nonlinear general rate model, frequently used to describe chromatographic multicomponent separations. The method is based on minimization of errors in chromatographic column profile moments, and it belongs to the family of weighted residual methods. Compared to most traditional weighted residual methods, the present formulation has some clear advantages. Firstly, it is inherently mass conserving. Secondly, the separation characteristic values of the effluent curve (retention time, physical dispersion and skewness) are predicted with good accuracy. Thirdly, the boundary conditions are treated naturally as source terms. The method is inherently of high order, so it gives high accuracy with a relatively low number of variables. This is a remarkable benefit especially for model parameter fitting or process optimization, when the model has to be solved repeatedly.

Introduction

Mathematical modeling and numerical analysis of chromatographic operations have received considerable attention since the late 1960s. Various mathematical models have been proposed (Golshan-Shirazi & Guiochon, 1992). Basically, they can be classified into the following three major categories: (i) ideal model, (ii) equilibrium-dispersive model and (iii) general rate model. Among them, the general rate equation model is the most realistic model for all kinds of chromatographic processes (Guiochon & Golshan-Shirazi, 2006). The model formulation is based on the mass balances of each species in bulk fluid and particle phases. It considers axial dispersion, film mass transfer, intra-particle diffusion, multicomponent linear/nonlinear isotherms and sometimes even finite rate of the adsorption reaction. Due to the complexity of the model, analytical solution is difficult or impossible. Numerical computation can be very time consuming if high accuracy is requested. Hence, an efficient algorithm is needed.

Various numerical tools have been proposed and implemented to solve the general rate model or similar ones. Liapis and coworkers (Balzli et al., 1978, Liapis and Litchfield, 1980, Liapis and Rippin, 1978) used orthogonal collocation (OC) method to discretize both bulk and particle phase equations. Yu and Wang (1989) used OC on finite element for bulk-phase equations and OC method for particle phase equations. Mansour (1989) applied finite difference method for the whole numerical procedure and iteration for nonlinear isotherm. Gu, Tsai, and Tsao (1990) used finite element (FE) method for the bulk-phase equations and the OC method for the particle-phase equations. In addition, method of lines (MOL) was used by Turku and Sainio (2009) to discretize the PDEs. Recently Lieres and Andersson (2010) discretized the model equations spatially with finite volumes and weighted essentially nonoscillatory (WENO) method.

The moment based weighted residual method (written more concisely as “moment method” from here in this article) belongs to the class of weighted residual methods, together with the subdomain method, least squares method, orthogonal collocation method and Galerkin method (Finlayson, 1972, Finlayson, 1980, Villadsen and Michelsen, 1978). The moment method was first developed for dynamic plug-flow reactor models and for simple plug flow chromatographic models in Alopaeus, Laavi, and Aittamaa (2008). A successive paper presented the moment method for solving the models including axial dispersion in reactors, but without intra-particle effects (Roininen & Alopaeus, 2011). The present paper extends the moment method for solving the widely used chromatographic general rate model, where the particle phase model to describe the intra-particle effects is involved. This paper also extends the previous works to solve the chromatographic models, which are coupled by competitive isotherms. This requires a novel implementation of the method. Besides the benefits of using the moment method for a realistic chromatographic model are discussed in this work. The method is tested with a number of practical examples and the relevant numerical parameters are studied.

Section snippets

General rate model

The chromatographic process involves complex hydrodynamic, thermodynamic, and kinetic phenomena, which often interact. In equilibrium-dispersive model and lumped kinetic model (Golshan-Shirazi & Guiochon, 1992), considerable simplifications are made by focusing on the most important phenomena. In band broadening mechanisms, many influential contributions take place simultaneously during migration of the solute. It is often impossible to identify one single contribution as rate determining.

The

Algorithm verifications and experimental simulation

In this section, the moment method is verified with respect to mass balance and compared to an applicable analytical solution. In addition the moment method applicability to solve coupled complex PDEs is tested to simulate an adsorption experiment.

Effects of numerical parameters

The number of elements (N_E) and the degree of polynomials (N_D, N_DI) are the most important numerical parameters which can be varied. The total number of variables of the whole differential equation system is then given by Eq. (38). In bulk phase, there are three ways of improving the accuracy of numerical solutions: (1) increasing the number of moments to be conserved within each element and thus increasing the degree of the polynomials approximating the profiles; (2) dividing the domain into

Discussion

One of the most attractive features of the moment based weighted residual method is that the spatial PDE solution inherently conserves mass. This results naturally from the set of model equations, provided that 0th order moment (mass balance) is included into the set of equation, as it is in all our test cases. In some of our test cases, we observed small but observable deviation from this property (Table 1), where the mass error of the moment method is in the range of 0.1–0.2%. The reason for

Conclusion

A moment based high order weighted residual method was developed to solve numerically the general rate model for chromatographic columns. The method is very flexible and allows standard integration packages to be used for integration of the model. This moment method can handle any complex competitive isotherms without iteration. The moment method conserves mass inherently. The important characteristic values of the effluent curve (retention time, physical dispersion and skewness) can be

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Moment based weighted residual method—New numerical tool for a nonlinear multicomponent chromatographic general rate model

Highlights

Abstract

Introduction

Section snippets

General rate model

Algorithm verifications and experimental simulation

Effects of numerical parameters

Discussion

Conclusion

Computers and Chemical Engineering

Journal of Chromatography

Journal of Chromatography A

Journal of Chromatography A

Computers and Chemical Engineering

Chemical Engineering Science

Chemical Engineering Science

Journal of Chromatography A

Computers and Chemical Engineering

Journal of Chromatography A

Separation and Purification Technology

Computers and Chemical Engineering

Applications of mathematical modeling to the simulation of multi-component adsorption in activated carbon columns

Transactions of the Institution of Chemical Engineering

The method of weighted residuals and variational principles