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A Nuclear Magnetic Resonance (NMR) spectroscopy method is presented that facilitates inline analysis of dynamic processes. Compared to other commonly used optical inline analysis methods, NMR spectroscopy has the advantage that it can resolve different species in complex multicomponent mixtures. An inline NMR spectroscopy method was optimized to enable analysis with high temporal resolution. The method was applied to study the dynamic behavior of a fixed-bed chromatographic reactor (FBCR) by monitoring the composition at the reactor outlet. The heterogeneously catalyzed hydrolysis reactions of methyl acetate and methyl formate were chosen as test systems. The influence of different process parameters such as the concentration of reactants, reactor temperature and flow rate of the mobile phase (water) were systematically studied with the presented method. The concentration profiles of the different reactants and products could be determined accurately even though the two hydrolysis reactions proceeded simultaneously in the FBCR and the concentration profiles of the different species overlapped strongly. The measured concentration profiles are in good agreement with additional RI measurements which, however, do not facilitate a component specific analysis. The accurate measurement of the concentration profiles enables to study the interaction of reaction and separation in the FBCR. At low concentrations of the reactants the measured concentration profiles agree well with predictions based on a model of the FBCR developed in previous works. At higher concentrations, however, the comparison of the predictions and experimental results reveals deficits in the model. The results demonstrate that the presented inline NMR spectroscopy method is a valuable tool to gain insights into complex dynamic processes and to gather accurate experimental data that is essential for the development of reliable process models. [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]

The effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated. The considered systems of partial differential equations (PDEs) are non-strictly hyperbolic and non-conservative due to the presence of non-conservative differential terms on the right hand side. The solutions of these model equations are very challenging for a numerical scheme. Thus, our primary goal is to introduce an improved numerical scheme which can handle the non-conservative differential terms efficiently and accurately. In this paper, the space–time conservation element and solution element (CESE) method is extended to approximate these model equations. The proposed scheme has capability to overcome all difficulties posed by this nonlinear system of PDEs. The performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central NT scheme. The accuracy of the scheme is verified qualitatively and quantitatively. [ABSTRACT FROM AUTHOR]

• Segmented temperature gradients were experimentally implemented. • A repetitive temperature switching strategy for mixture separation was developed. • Productivity of a target component could be increased. • Retention behaviour for temperature gradients was predictable. • Insights into applications and limitations of temperature gradients. The amount of substance adsorbed on solid surface depends on temperature. Therefore, the migration velocities of the solutes in a chromatographic column can be altered by introducing temperature gradients. Such gradients designed to change retention behaviours can be exploited to improve the separation performances in preparative chromatography. To describe key process features, we used analytical solutions of the equilibrium model with instant stepwise shift of temperature. To achieve a more realistic description, the equilibrium dispersion model was additionally applied to treat finite column efficiencies. The effect of temperature gradients was illustrated experimentally using two identical columns sequentially connected. Temperature of the second column was modulated by thermostats. Wide pulse injections of a single component led to instructive elution profiles in a preliminary investigation. The observations were found to be in qualitative agreement with predictions of the equilibrium dispersion model. Subsequently, the separation of a ternary model mixture was investigated considering a simple two-step temperature gradient. To support the quantitative analysis and to identify suitable switching and cycle times, the temperature dependencies of the Henry constants were determined by short pulse injections. A meaningful variation of the parameters of the temperature gradient is required for adjusting the cycle times, which is the time difference between two consecutive injections that needs to shorten. Decreasing this time is connected with a desirable increase in process productivity. The results achieved revealed that relatively simple to implement stepwise temperature gradients offer an option to improve and fine-tune the performance of repetitive batch chromatography. [Display omitted] [ABSTRACT FROM AUTHOR]

• This paper is about multi-component flow. • In this paper one- and two-dimensional homogenous multi-component flow models are numerically investigated by using a high resolution splitting scheme and this scheme is known as kinetic flux vector splitting scheme. • This scheme preserves positivity conditions and resolves shocks, rarefaction and contact discontinuity. • The scheme is based on splitting of flux functions. • Runge-Kutta time stepping technique with MUSCL-type initial reconstruction is used to guarantee higher order accurate solution. • The results obtained are compared with central scheme to verify the efficiency of studied scheme. This paper is about multi-component flow. There is no doubt that multi-component flow has a wide range of applications, specially in aerospace it plays a vital role during reentry of space ship into earth's atmosphere thats why it cannot be neglected for a proper vehicle design. In this paper one- and two-dimensional homogenous multi-component flow models are numerically investigated by using a high resolution splitting scheme and this scheme is known as Kinetic Flux Vector Splitting scheme. This scheme preserves positivity conditions and resolves shocks, rarefaction and contact discontinuity. The scheme is based on splitting of flux functions. Moreover Runge-Kutta time stepping technique with MUSCL-type initial reconstruction is used to guarantee higher order accurate solution. This work is first done by Qamar and Warnecke (2004) for the homogeneous multi-component flow equations using central scheme, here we investigate the same work using kinetic flux vector splitting scheme (KFVS) and compared the results with central scheme to verify the efficiency of studied scheme. [ABSTRACT FROM AUTHOR]

A two-component model of gradient elution chromatography is investigated to theoretically study the effects of simultaneous variations in temperature and solvent strength on the retention behaviors of elution profiles in thermally insulated liquid chromatographic columns. The gradient elution technique is based on the gradual increase or decrease in eluent strength during the chromatographic operation by varying the composition of the mobile phase. The enthalpy of adsorption is primarily responsible for internal temperature variations inside the column, as heat adsorbs during the adsorption process and releases in the desorption phase. Both types of variations change the propagation speeds of moving pulses inside the column which can lead to better separation of the components and a reduction in the recycling time for the next injection. The equilibrium dispersive model (EDM) coupled with the energy balance equation for temperature and transport equation for the volume fraction of the solvent is utilized to simulate this complex process. The resulting nonlinear model equations are approximated by applying a semi-discrete second-order finite volume scheme. The numerical solutions are used to study the impact of a gradient starting and ending times, volume-fraction of the solvent, solvent strength parameter, the slope of gradient, enthalpy of adsorption, injection temperature, and the ratio of specific heats on the concentration profiles., (© The Author(s) 2023. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com.)

In this paper, a rigorous theoretical study is conducted to analyze the influence of varying solvent compositions on the retention characteristics of elution profiles within a fixed-bed liquid chromatographic column. In gradient chromatography, the propagation speed of elution profiles is manipulated through a progressive variation in the mobile-phase composition. Consequently, enhanced separation of the mixture components can be achieved together with a reduction in the requisite recycling times for subsequent injections. In other words, both the efficiency and the selectivity of the column can be enhanced. The lumped kinetic model coupled with the convection-diffusion equation for the volume fraction of the solvent is applied to simulate the process. The resulting nonlinear model equations are numerically solved by applying a semidiscrete second-order finite-volume method. The numerical solutions are utilized to quantify the effects of gradient starting and ending times, solvent composition, solvent strength parameters, and gradient slope on the concentration profiles. Additionally, temporal numerical moments are plotted versus the starting and ending times of the gradient, and standard performance criteria are presented for evaluating the process performance. The outcomes of this investigation will contribute to further enhancements in gradient elution chromatography., Competing Interests: The authors declare no competing financial interest., (© 2024 The Authors. Published by American Chemical Society.)

Core-shell particles enable the separation of intricate mixtures in a highly efficient and rapid manner. The porous shell particles increased the separation efficiency with expedited flow rates due to an abatement in the pore volume accessible for longitudinal diffusion and a decrease in diffusion path length. This study focuses on the numerical approximation of a nonlinear isothermal general rate model applied to stationary bed columns that are replete with inert core adsorbents featuring double adsorption sites. The transport of solute in heterogeneous porous media can be modeled by a nonlinear convection acquiescent partial differential equation system together with a specific nonlinear algebraic relation: the bi-Langmuir adsorption isotherm. Therefore, it is important to develop accurate and reliable numerical techniques that can perform accurate numerical simulations of these models. We extended and implemented a second-order, semidiscrete, high-resolution finite volume method to simulate the governing equations of the model. Single solute flow and multi component mixture flows are assessed through a series of numerical experiments to theoretically illustrate the repercussions of intraparticle diffusion, film mass resistance, axial dispersion, and the size of the inert core radius upon simulated elution curves. Standard performance criteria are assessed to determine the optimal core radius fraction range for optimizing the separation performance., Competing Interests: The authors declare no competing financial interest., (© 2023 The Authors. Published by American Chemical Society.)

This paper presents the numerical approximation of a nonlinear equilibrium-dispersive (ED) model of multicomponent mixtures for simulating single-column chromatographic processes. Using Danckwerts boundary conditions (DBCs), the ED is studied for both generalized and standard bi-Langmuir adsorption isotherms. Advection-diffusion partial differential equations are used to represent fixed-bed chromatographic processes. As the diffusion term is significantly weaker than the advection term, sophisticated numerical techniques must be applied for solving such model equations. In this study, the model equations are numerically solved by using the Runge-Kutta discontinuous Galerkin (RKDG) finite element method. The technique is designed to handle sudden changes (sharp discontinuities) in solutions and to produce highly accurate results. The method is tested with several case studies considering different parameters, and its results are compared with the high-resolution finite volume scheme. One-, two-, and three-component liquid chromatography elutions on fixed beds are among the case studies being considered. The dynamic model and its accompanying numerical case studies provide the initial step toward continuous monitoring, troubleshooting, and effectively controlling the chromatographic processes., Competing Interests: The authors declare no competing financial interest., (© 2023 The Authors. Published by American Chemical Society.)

The current study investigates a nonequilibrium and nonlinear two-dimensional lumped kinetic transport model of nonisothermal reactive liquid chromatography, considering the Bi-Langmuir adsorption isotherm, heterogeneous reaction rates, radial and axial concentration variations, and the adsorption and reaction enthalpies. The mathematical models of packed bed chromatographic processes are expressed by a highly nonlinear system of coupled partial differential algebraic equations connecting the phenomena of convection, diffusion, and reaction, for mass and energy balance, the differential algebraic equations for mass balance in the solid phase, and the algebraical expressions for the adsorption isotherms and for the reaction rates. The nonlinearity of the reaction term and the adsorption isotherm preclude the derivation of an analytical solution for the model equations. For this reason, a semidiscrete, high-resolution, finite-volume technique is extended and employed in this study to obtain the numerical solution. Several consistency checks are performed to evaluate the model predictions and analyze the precision of the proposed numerical scheme. A number of heterogeneously catalyzed stoichiometric reactions are numerically simulated to examine reactor performance under the influence of temperature and Bi-Langmuir adsorption dynamics, the level of coupling between mass and energy fronts, and to study the effects of various critical parameters. The numerical results obtained are beneficial for optimal predictive control and process optimization during production and the development of methods for systematic design and fault detection of nonisothermal liquid chromatographic reactors, and hence constitute the first step to provide deeper insight into the overall evaluation of integrated reaction and separation processes., Competing Interests: The authors declare no competing financial interest., (© 2023 The Authors. Published by American Chemical Society.)

In both linear and nonlinear chromatography, the lumped kinetic model is a suitable model for predicting elution bands when appropriate equilibrium functions and mass transfer coefficients are accessible. This model also works well in the case of gradient elution chromatography if variations in the equilibrium functions due to changes in the mobile phase composition are known. The rational selection of an optimum gradient is explored in this study from three different perspectives using the lumped kinetic model. Elution profiles generated by using (a) linear solvent strength, (b) quadratic solvent strength, and (c) power law are investigated. The effectiveness and reliability of the suggested numerical approach, utilizing the flux-limiting finite volume method, are demonstrated through numerical simulations. The impacts of axial dispersion, nonlinearity coefficient, Henry's constant, mass transfer coefficient, and gradient parameters are studied on single and two-component elution profiles., Competing Interests: The authors declare no competing financial interest., (© 2022 The Authors. Published by American Chemical Society.)

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., (Copyright © 2017 Elsevier B.V. All rights reserved.)

This work is concerned with the analysis of models for linear reactive chromatography describing irreversible A→B and reversible A↔B reactions. In contrast to previously published results rectangular reactant pulses are injected into initially empty or pre-equilibrated columns assuming both Dirichlet and Danckwerts boundary conditions. The models consist of two partial differential equations, accounting for convection, longitudinal dispersion and first order chemical reactions. Due to the effect of involved mechanisms on solute transport, analytical and numerical solutions of the models could be helpful to understand, design and optimize chromatographic reactors. The Laplace transformation is applied to solve the model equations analytically for linear adsorption isotherms. Statistical temporal moments are derived from solutions in the Laplace domain. Analytical results are compared with numerical predictions generated using a high-resolution finite volume scheme for two sets of boundary conditions. Several case studies are carried out to analyze reactive liquid chromatographic processes for a wide range of mass transfer and reaction kinetics. Good agreements in the results validate the correctness of the analytical solutions and accuracy of the proposed numerical algorithm., (Copyright © 2015 Elsevier B.V. All rights reserved.)

This work focuses on the analysis of two standard liquid chromatographic models, namely the lumped kinetic model and the equilibrium dispersive model. Analytical solutions, obtained by means of Laplace transformation, are derived for rectangular single solute concentration pulses of finite length and breakthrough curves injected under linear conditions. In order to analyze the solute transport behavior by means of the two models, the temporal moments up to fourth order are calculated from the Laplace-transformed solutions. The limiting cases of continuous injection and negligible mass transfer limitations are evaluated. For validation, the analytical solutions are compared with the numerical solutions of models using the discontinuous Galerkin finite element method. Results of different case studies are discussed for linear and nonlinear adsorption isotherms. The discontinuous Galerkin method is employed to obtain moments for both linear and nonlinear models numerically. Analytically and numerically determined concentration profiles and moments were found to be in good agreement., (Copyright © 2013 Elsevier B.V. All rights reserved.)