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.
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.
A nonlinear and nonisothermal two‐dimensional general rate model is formulated and approximated numerically to allow quantitatively analyzing the effects of temperature variations on the separations and reactions in liquid chromatographic reactors of cylindrical geometry. The model equations form a nonlinear system of convection‐diffusion‐reaction partial differential equations coupled with algebraic equations for isotherms and reactions. A semidiscrete high‐resolution finite volume method is modified to approximate the system of partial differential equations. The coupling between the thermal waves and concentration fronts is demonstrated through numerical simulations, and important parameters are pointed out that influence the reactor performance. To evaluate the precision of the model predictions, consistency checks are successfully carried out proving the accuracy of the predictions. The results allow to quantify the influence of thermal effects on the performance of the fixed beds for different typical values of enthalpies of adsorption and reaction and axial and radial Peclet numbers for mass and heat transfer. Furthermore, they provide useful insight into the sensitivity of nonisothermal chromatographic reactor operation.
A high-resolution flux-limiting semi-discrete finite volume scheme (HR-FVS) is applied in this study to numerically approximate the nonlinear and non-isothermal flow of one-dimensional lumped kinetic model (1D-LKM), for a fixed-bed column loaded with core-shell particles. The developed model comprise a system of convection-dominated partial differential for mass and energy balances in the mobile phases coupled with differential equation and algebraic equation in the stationary phase. The solution of the model equations is obtained by utilizing a HR-FVS, the scheme has second-order accuracy even on the grid coarse and its explicit nature has the potential to resolve the arisen sharp discontinuities in the solution profiles. A second-order total variation diminishing (TVD) Runge-Kutta technique is used to solve the system of ODEs in time. Several forms of a single-solute mixture are produced to investigate the influences of the fractions of core radius on thermal waves and concentration fronts. Moreover, a particular criterion is introduced for analyzing the performance of the underlying process and to identify the optimal parameter values of the fraction of core radius.
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