Analytical and Numerical Solutions of Two-Dimensional Brusselator System by Modified Variational Iteration Method (original) (raw)

Numerical approximation of nonlinear and non-equilibrium two-dimensional model of chromatography

Computers & Chemical Engineering, 2016

This article is concerned with the numerical approximation of a nonlinear model describing the two-dimensional non-equilibrium transport of multi-component mixtures in a chromatographic column. The model consists of nonlinear convection-diffusion partial differential equations coupled with some differential and algebraic equations. Due to the unavailability of analytical solutions for nonlinear models, numerical solution techniques are the only tools to get accurate solutions. A semi-discrete high resolution finite volume scheme is extended to solve the model equations numerically. The scheme is second order accurate in axial and radial coordinates. The accuracy of the scheme is guaranteed by applying a second order accurate Runge-Kutta method to solve the resulting system of ordinary differential equation. The considered radial gradients were typically ignored in pervious studies. They can be relevant in particular in the case of non-perfect injections. The effects of possible rate limitations of the mass transfer in the radial direction are studied assuming hypothetical injections in outer or inner sections of the column inlet cross-section. The case studies consider single-component, two-component and three-component elution. The developed numerical algorithm is an efficient tool to study the effects of mass transfer kinetics on the shape of elution profiles.

Analysis of linear two-dimensional general rate model for chromatographic columns of cylindrical geometry

Journal of Chromatography A, 2017

This work is concerned with the analytical solutions and moment analysis of a linear two-9 dimensional general rate model (2D-GRM) describing the transport of a solute through a chromatographic column of cylindrical geometry. Analytical solutions are derived through successive implementation of finite Hankel and Laplace transformations for two different sets of boundary conditions. The process is further analyzed by deriving analytical temporal moments from the Laplace domain solutions. Radial gradients are typically neglected in liquid chromatography studies which are particularly important in the case of nonperfect injections. Several test problems of single-solute transport are considered. The derived analytical results are validated against the numerical solutions of a high resolution finite volume scheme. The derived analytical results can play an important role in further development of liquid chromatography.

Analytical solutions and moment analysis of general rate model for linear liquid chromatography

Chemical Engineering Science

The general rate model (GRM) is considered to be a comprehensive and reliable mathematical model for describing the separation and mass transfer processes of solutes in chromatographic columns. However, the numerical solution of model equations is complicated and time consuming. This paper presents analytical solutions of the GRM for linear adsorption isotherms and different sets of boundary conditions at the column inlet and outlet. The analytical solutions are obtained by means of the Laplace transformation. Numerical Laplace inversion is used to transform back the solution in the time domain because analytical inversion cannot be obtained. The first four temporal moments are derived analytically using the Laplace domain solutions. The moments of GRM are utilized to analyze the retention times, band broadenings, front asymmetries and kurtosis of the elution profiles. Relationships are derived among the kinetic parameters to match the first four moments of GRM and the simpler lumpe...

Particle transport method for simulation of multicomponent chromatography problems

Journal of Chromatography A, 2008

A novel numerical technique, called the particle transport method (PTM), is applied in simulation of multicomponent chromatography problems with non-linear isotherms. The method is based on the operator-splitting approach, and combines the Lagrangian method of characteristics and the method of lines. Instead of a moving mesh, a set of numerical points (called particles) which can be interpolated to any fixed grid with a fast monotone projection procedure is used. The technique includes a special spatial adaptivity to simulate the propagation of steep fronts (shock waves) that are often encountered in nonlinear chromatography. The application of PTM for solving ideal and nonideal models of chromatography is demonstrated with two-component systems following the competitive Langmuir and Fowler isotherms. The accuracy and calculation time of PTM for various cases are compared with those of classical methods such as OCFEM, MOL and the Rouchon algorithm. The numerical experiments show that the proposed approach exhibits good mass conservation while being a computationally cheap scheme.

Analytical solution of non-isothermal two-dimensional general rate model of liquid chromatography

Adsorption, 2019

A non-isothermal two-dimensional general rate model is formulated and analytically solved to analyze the effects of temperature changes inside liquid chromatographic columns of cylindrical geometry. The model equations form a system of convection-diffusion partial differential equations. The finite-Hankel transformation, the Laplace transformation, the eigendecomposition technique and a conventional solution technique of ordinary differential equations are used to solve the equations of the model. The coupling between concentration and temperature fronts is demonstrated and important parameters that affect the performance of the column are evaluated. To find the ranges of validity of our analytical results, a semi-discrete high resolution finite volume method is applied to solve the same system of equations for both linear and nonlinear isotherms. The results of this contribution can be helpful to optimize non-isothermal liquid chromatographic processes in which both radial and axial gradients occur.

Numerical approximation of non-linear chromatographic models considering Bi-Langmuir isotherm

Thermal Science, 2020

In this research article, two standard models of liquid chromatograophy, namely the dispersive equilibrium model and the kinetic lumped model are approximated numerically. We studied the transport of multi components in a single column of chromatography considering non-linear adsorption thermodynamics. The models are analyzed for standard bi-Langmuir and generalized bi-Langmuir types adsorption equilibrium isotherms using Danckwert boundary conditions. Mathematically, the model equations form a non-linear system of PDE accounting for the phenomena of advection and diffusion, paired with an algebraic equation or a differential equation for adsorption isotherm. An extended semi-discrete high resolution finite volume scheme is employed to obtain the approximate solutions of the governing model equations. The method has second to third order accuracy. Several test case studies are conducted to examine the influence of various critical parameters on the process performance. The contempla...

Analytical and Numerical Solutions of Two-Dimensional Problems of Asymmetric Elasticity Theory

This work is concerned with the analytical and numerical solutions of linear and nonlinear two-dimensional general rate models (2D-GRMs) describing the transport of single-solute and multi-component mixtures through chromatographic columns of cylindrical geometry packed with core-shell particles. The finite Hankel and Laplace transformations are successively applied to derive analytical solutions for a single-solute model considering linear adsorption isotherms and two different sets of boundary conditions. Moreover, analytical temporal moments are derived from the Laplace domain solutions. The process is further analyzed by numerically approximating the nonlinear 2D-GRM for core-shell particles considering multi-component mixtures and nonlinear Langmuir isotherm. A high resolution finite volume scheme is extended to solve the considered 2D-model equations. Several case studies of single-solute and multi-component mixtures are considered. The derived analytical results are validated against the numerical solutions of a high resolution finite volume scheme. Typical performance criteria are utilized to analyze the performance of the chromatographic process. The results obtained are considered to be useful to support further development of liquid chromatography.

Numerical Approximation of a Two-Dimensional Nonlinear and Nonequilibrium Model of Reactive Chromatography

Industrial & Engineering Chemistry Research, 2016

A two-dimensional nonlinear and non-equilibrium model of liquid chromatography is numerically approximated to simulate the dynamics of multi-component mixtures in a fixed-bed isothermal liquid chromatographic reactor. The mathematical model is formed by a system of nonlinear convection-diffusion-reaction partial differential equations coupled with differential and algebraic equations. A semi-discrete high resolution finite volume scheme is applied to solve the model equations. The scheme is second order accurate in axial and radial coordinates. The resulting system of ordinary differential equations is solved by a second order accurate Runge-Kutta method. The proposed scheme capably captures narrow peaks and sharp discontinuities in the concentration profiles. Radial gradients were not considered in the pervious studies which are particularly important in the case of non-perfect injections. Several test problems of heterogeneously catalyzed reversible reactions are carried out. The considered case studies include three and four-component elution assuming hypothetical injections of the reactants in inner or outer sections of the column inlet cross-section. The developed numerical algorithm and results are useful tools for further improvements in reactive chromatography.

Efficient and accurate numerical simulation of nonlinear chromatographic processes

Computers & Chemical Engineering, 2011

Models for chromatographic processes consist of nonlinear convection-dominated partial differential equations (PDEs) coupled with some algebraic equations. A high resolution semi-discrete flux-limiting finite volume scheme is proposed for solving the nonlinear equilibrium dispersive model of chromatography. The suggested scheme is capable to suppress numerical oscillations and, hence, preserves the positivity of numerical solutions. Moreover, the scheme has capability to accurately capture sharp discontinuities of chromatographic fronts on coarse grids. The performance of the current scheme is validated against other flux-limiting schemes available in the literature. The case studies include single-component elution, two-component elution, and displacement chromatography on non-movable (fixed) and movable (counter-current) beds.

Modeling chromatographic columns: Non-equilibrium packed-bed adsorption with non-linear adsorption isotherms

Journal of Chromatography A, 2004

In this work a new mathematical model, based on non-equilibrium conditions, describing the dynamic adsorption of proteins in columns packed with spherical adsorbent particles is used to study the performance of chromatographic systems. Simulations of frontal chromatography, including axial dispersion, for non-equilibrium systems with non-linear adsorption isotherms are made and compared to those of the experimentally determined protein A affinity chromatography breakthrough curves of hIgG, gathered from the literature. The non-equilibrium model developed here combines external mass transfer and intra-particle transport by solid (surface) diffusion, and permits the prediction of (time and bed height dependent) interface and average solid concentrations, along with interface and bulk liquid concentrations. The present non-equilibrium approach significantly improved the model predictions of experimentally observed distended breakthrough fronts over local equilibrium based models, and can be used to evaluate the influence of system parameters on the performance of chromatographic packed-bed adsorption columns.