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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 contemplated case studies incorporate the elution process of liquid chromatography with an increasing number of components. In particular, single component, two component, and three component mixtures are considered for the assessment of process performance. The formulated numerical algorithm provide an efficacious mechanism for investigating the retention behavior and the influence of mass transfer kinetics on the shapes of elution profiles. [ABSTRACT FROM AUTHOR]

A nonlinear lumped kinetic model of liquid chromatography is formulated and solved numerically to theoretically investigate the effect of column overloading on gradient elution. Linear solvent strength (LSS) model is utilized for Henry's constant, non linearity coefficient and axial dispersion coefficient. A semi-discrete high-resolution finite volume scheme is extended and applied to obtain the approximate solutions of the governing model equations. The effects of changing modulator concentration are examined on the single and two-component elution. The benefits of gradient elution over isocratic elution are thoroughly discussed. The influences of optimizing free-parameters available in gradient chromatography are analyzed on the efficiency of the column and on the production of targeted components. For instance, the results obtained are used to study the effects of gradient slope, modulator concentration, solvent strength, nonlinearity coefficient, mass transfer coefficient, and axial dispersion coefficient on the concentration profiles. [ABSTRACT FROM AUTHOR]

A two-component model of reactive liquid chromatography is presented considering M → N type reaction. The model incorporates surface and pore diffusions in the adsorbates, axial dispersion, interfacial mass transfer, first order chemical reactions in the liquid and particle phases, and two sets of boundary conditions. The model contains a system of four coupled PDE describing the dynamics of reactants and products in both phases. The Laplace transformation and eigen-decomposition technique are jointly applied to solve the model equations analytically. An efficient and accurate numerical Laplace inversion technique is utilized to retrieve back solutions in the original time domain. The developed semi-analytical results are verified against the numerical results of a high resolution finite volume scheme. A good agreement between the solutions not only confirms the accuracy of semi-analytical results but also validates the accuracy of proposed numerical scheme. This study extends and generalizes our previous analysis on heterogeneous reactions in the liquid chromatography. In order to analyze the behavior of a chromatographic reactor, different case studies are presented showing the effects of model parameters on the process. [ABSTRACT FROM AUTHOR]

A nonlinear general rate model is numerically approximated to simulate fixed-bed chromatographic columns packed with core-shell particles. The model incorporates explicitly the effect of finite rates of adsorption and desorption at the adsorption sites, typically assumed to be very fast compared to the rates of the various transport processes. Using core-shell particles as a stationary phase can have advantages over applying a fully-porous stationary phase, such as higher efficiencies and better resolution of the sample components. A high resolution finite volume scheme is extended and applied to approximate the model equations. Ranges of the kinetic parameters in which limited rates of the intrinsic adsorption and desorption steps needs to be taken into account are estimated. A few case studies of predicting the elution of single-component and two-component mixtures are considered to evaluate the effects of adsorption and desorption rate constants, core-radius fraction, axial-dispersion coefficient, film mass transfer, and intraparticle diffusion on the elution profiles. Furthermore, it is demonstrated that optimum values of the inert core radius can be obtained by evaluating a typical criterion for process performance. [ABSTRACT FROM AUTHOR]

This work is concerned with the analytical solutions and moment analysis of a linear two-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 non-perfect 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. [ABSTRACT FROM AUTHOR]

A set of analytical solutions are presented for a model describing the transport of a solute in a fixedbed reactor of cylindrical geometry subjected to the first (Dirichlet) and third (Danckwerts) type inlet boundary conditions. Linear sorption kinetic process and first-order decay are considered. Cylindrical geometry allows the use of large columns to investigate dispersion, adsorption/desorption and reaction kinetic mechanisms. The finite Hankel and Laplace transform techniques are adopted to solve the model equations. For further analysis, statistical temporal moments are derived from the Laplace-transformed solutions. The developed analytical solutions are compared with the numerical solutions of high-resolution finite volume scheme. Different case studies are presented and discussed for a series of numerical values corresponding to a wide range of mass transfer and reaction kinetics. A good agreement was observed in the analytical and numerical concentration profiles and moments. The developed solutions are efficient tools for analyzing numerical algorithms, sensitivity analysis and simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment. [ABSTRACT FROM AUTHOR]

A linear general rate model of two-component liquid chromatography is analyzed considering heterogenous reactions of types A → B and A ⇄ B. The model equations incorporate axial dispersion, external and intra particle pore diffusions, interfacial mass transfer, linear sorption kinetics, and first order heterogeneous chemical reactions. The solution methodology successively employs the Laplace transform and linear transformation steps to uncouple the governing set of coupled differential equations. The resulting system of uncoupled ODEs is solved by applying an elementary solution technique. The numerical Laplace inversion is employed to transform back the solutions in the actual time domain. The current solutions extend and generalize the recent solutions of nonreactive general rate model for single-solute transport. For validation, a high resolution finite volume scheme is implemented to obtain the numerical solutions. Different case studies are considered to verify the correctness of semi-analytical solutions and the accuracy of the numerical scheme. To further study the behavior of a chromatographic reactor, numerical temporal moments of the elution profiles are presented for both reactant and product. The derived semi-analytical solutions are useful tools to study the influence of solid phase reaction rate constant, interfacial mass transfer rate, intra particle pore diffusion, and reactant adsorption affinity on the concentration profiles. [ABSTRACT FROM AUTHOR]

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 of cylindrical geometry. In contrast to previous studies, the work includes joint analysis of deviations from equilibrium and the possibility that radial concentration profiles can develop. The considered radial gradients are typically ignored, which can be problematic in the case of non perfect injections. The model consists of nonlinear convection-diffusion partial differential equations coupled with some differential and algebraic equations. A high resolution finite volume scheme is applied to solve the model equations numerically. The considered case studies include single-component, two-component and three-component elution on fixed (non-movable) beds of liquid chromatography. The developed numerical algorithm is an efficient tool to study the effects of mass transfer kinetics on the elution profiles. [ABSTRACT FROM AUTHOR]

Core–shell particles allow highly efficient and fast separation of complex samples. They provide advantages over fully porous particles, such as highly efficient separation with fast flow rate due to shorter diffusional path length in particle macropores. On the other hand, capacities are reduced due to the inert core. This work is focused on the numerical approximation of a nonlinear general rate model for fixed-beds packed with core–shell particles. The model equations consider axial dispersion, interfacial mass transfer, intraparticle diffusion, and multi-component Langmuir isotherm. A semi-discrete high resolution flux-limiting finite volume scheme is proposed to accurately and efficiently solve the model equations. The scheme is second order accurate in axial and radial coordinates. The resulting system of ordinary differential equations (ODEs) are solved by using a second-order TVD Runge–Kutta method. For illustration, a few selected scenarios of single solute and multi-component elution bands are generated to study theoretically the effects of the core radius fractions on the course of elution curves. Typically applied performance criteria are evaluated for identifying ranges of optimum values of the core radius fraction. [ABSTRACT FROM AUTHOR]

Due to their proven performance and improved availability, core–shell particles are increasingly applied for chromatographic separations. This paper presents semi-analytical solutions and a moment analysis of a detailed mathematical model for fixed-beds packed with core–shell particles. The model considers axial dispersion, interfacial mass transfer, intraparticle diffusion, linear adsorption, and the injection of rectangular pulses. The Laplace transformation is used as a basic tool to derive semi-analytical solutions. In addition the first three statistical temporal moments are derived from solutions in the Laplace domain. The numerical Laplace inversion is applied for back transformation of the solution in the actual time domain. In order to demonstrate their potential, different scenarios are considered to quantify the effects of the relative core size, axial dispersion, film mass transfer resistance and intraparticle diffusion resistance in the porous layer on the elution profiles. An important new result is the derivation of a plate height equation for fully porous and core–shell particles respecting the Danckwerts boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents the analytical and numerical investigations of two established models for simulating liquid chromatographic processes namely the equilibrium dispersive and lumped kinetic models. The models are analyzed using Dirichlet and Robin boundary conditions. The Laplace transformation is applied to solve these models analytically for single component adsorption under linear conditions. Statistical moments of step responses are calculated and compared with the numerical predictions for both types of boundary conditions. The discontinuous Galerkin finite element method is proposed to numerically approximate the more general lumped kinetic model. The scheme achieves high order accuracy on coarse grids, resolves sharp discontinuities, and avoids numerical diffusion and dispersion. For validation, the results of the suggested method are compared with some flux-limiting finite volume schemes available in the literature. A good agreement of the numerical and analytical solutions for simplified cases verifies the robustness and accuracy of the proposed method. The method is also capable to solve chromatographic models also for non-linear and competitive adsorption equilibrium isotherms. [Copyright &y& Elsevier]

• A transport model is used to describe gradient elution in liquid chromatography. • The Laplace transform approach is applied to solve the model analytically. • The analytical expressions of first three temporal moments are derived. • The moment solutions derived are seen valuable to estimate model parameters. • A finite volume scheme is applied for validation to solve the model numerically. A transport model is considered to describe gradient elution in liquid chromatography in packed beds with the linear isotherm dependent on the mobile phase modulator. By applying a coordinate transformation, the model is solved analytically using the Laplace transform approach. The moment generating property of the Laplace domain solution is used to derive analytical expressions for the first three moments of the response to rectangular injections. These moments are instructive for analyzing the retention time, band broadening and asymmetry of elution profiles. Compared to isocratic elution, the derivation of analytical solutions and moments for gradient elution is more complicated, because the retention behavior of the solutes depends on the varying mobile phase modulator. Several case studies are evaluated theoretically. To gain confidence on the derived analytical results, a high-resolution finite volume scheme is also applied to solve the same model equations numerically. The analytical solutions and moments provided are utilized to predict the effects of starting and ending times of gradient, magnitude of modulator concentration variation, gradient slopes, and mass transfer coefficient on retention and peak shape. The analytical moment expressions derived can be used to determine retention and mass transfer parameters from experimental peaks and to predict elution behaviors if these parameters are known. [ABSTRACT FROM AUTHOR]

[Display omitted] • A two-dimensional non-equilibrium model of liquid chromatography is considered. • Bi-Langmuir isotherm is considered to describe non-linear adsorption conditions. • The HR-FVM is extended and applied to solve the model equations. • Elution profiles over nonlinear chromatographic conditions are investigated. • The solutions quantify the influence of solute transport in radial direction. A two-dimensional non-equilibrium and non-linear lumped kinetic model of liquid chromatography is formulated and numerically approximated to simulate the separation of multi-component mixtures in a packed fixed bed cylindrical column operating under isothermal conditions. The model equations incorporate the bi-Langmuir adsorption thermo-dynamics as well as the radial and axial variations of concentration. By introducing distinct regions of injection at the column inlet, radial concentration gradients are generated to intensify the effect of mass transfer rate in the radial-direction, inside the column. The mathematical model is developed by a system of non-linear convection-diffusion partial differential equations for mass balance in the mobile phase, coupled with differential equation for mass balance in the stationary phase and algebraic equations for adsorption isotherm. In this study, a high-resolution, semi-discrete, finite-volume technique is formulated and applied to gain the numerical solution of the governing non-linear-model equations. A few numerical case studies are performed to investigate the effects of the various critical parameters on the process performance. The developed numerical algorithm provide an efficacious mechanism for investigating the retention behavior, systematic monitoring and efficient operation of non-equilibrium, liquid chromatographic processes. [ABSTRACT FROM AUTHOR]