Linearized non-equilibrium and non-isothermal two-dimensional model of liquid chromatography for studying thermal effects in cylindrical columns

A non-equilibrium and non-isothermal two-dimensional lumped kinetic model (2?D-LKM) is formulated and analytically solved to study the influence of temperature variations along the axial and radial coordinates of a liquid chromatographic column. The model includes convection-diffusion partial differential equations for mass and energy balances in the mobile phase coupled with differential equations for mass and energy in the stationary phase. The solutions are derived analytically through sequential implementation of finite Hankel and Laplace transformations using the Dirichlet inlet boundary conditions. The coupling between the thermal waves and concentration fronts is demonstrated through numerical simulations and important parameters are recognized that influence the column performance. For a more comprehensive study of the considered model, numerical temporal moments are obtained from the derived solutions. Several case studies are conducted and validity ranges of the derived analytical solutions are identified. The current analytical results will play a major role in the improvements of non-equilibrium and non-isothermal liquid chromatographic processes.     

Numerical approximation of a nonisothermal two-dimensional general rate model of reactive 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.     

Using a quasi-heat-pulse method to determine heat and moisture transfer properties for porous orthotropic wood products or cellular solid materials

Understanding heat and moisture transfer in a wood specimen as used in the K-tester has led to an unconventional numerical solution and intriguing protocol to deriving the transfer properties. Laplace transform solutions of Luikov's differential equations are derived for one-dimensional heat and moisture transfer in porous hygroscopic orthotropic materials and for a gradual finite heat pulse applied to both surfaces of a flat slab. The K-tester 637 (Lasercomp) supplies a quasi-heat-pulse to both sides of a 2-ft-square specimen and records precise signals as function of time from surface thermocouples and heat flux thermopiles. We obtained transfer properties for moist and oven-dried redwood lumber flooring.     

Prediction of vapor–liquid and liquid–liquid equilibria for polymer systems: Comparison of activity coefficient models

A large number of equations of state and activity coefficient models capable of describing phase equilibria in polymer solutions are available today, but only a few of these models have been applied to different systems. It is therefore useful to investigate the performance of existing thermodynamic models for complex polymer solutions which have not yet been widely studied. The present work studies the application of several activity coefficient models [P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, New York, NY, 1953; T. Oishi, J.M. Prausnitz, Estimation of solvent activities in polymer solutions using a group-contribution method, Ind. Eng. Chem. Process Design Dev. 17 (1978) 333; H.S. Elbro, A. Fredenslund, P. Rasmussen, A new simple equation for the prediction of solvent activities in polymer solutions, Macromolecules 23 (1990) 4707; G.M. Kontogeorgis, A. Fredenslund, D. Tassios, Simple activity coefficient model for the prediction of solvent activities in polymer solutions, Ind. Eng. Chem. Res. 32 (1993) 362; C. Chen, A segment-based local composition model for the Gibbs energy of polymer solutions, Fluid Phase Equilib. 83 (1993) 301; A. Vetere, Rules for predicting vapor–liquid equilibria of amorphous polymer solutions using a modified Flory–Huggins equation, Fluid Phase Equilib. 97 (1994) 43; C. Qian, S.J. Mumby, B.E. Eichinger, Phase diagrams of binary polymer solutions and blends, Macromolecules 24 (1991) 1655; Y.C. Bae, J.J. Shim, D.S. Soane, J.M. Prausnitz, Representation of vapor–liquid and liquid–liquid equilibria for binary systems containing polymers: applicability of an extended Flory–Huggins equation, J. Appl. Polym. Sci. 47 (1993) 1193; G. Bogdanic, J. Vidal, A segmental interaction model for liquid–liquid equilibrium calculations for polymer solutions, Fluid Phase Equilibria 173 (2000) 241] and activity coefficient from equations of state [F. Chen, A. Fredenslund, P. Rasmussen, Group-contribution Flory equation of state for vapor–liquid equilibria en mixtures with polymers, Ind. Eng. Chem. Res. 29 (1990) 875; M.S. High, R.P. Danner, Application of the group contribution lattice—fluids EOS to polymer solutions, AIChE J. 36 (1990) 1625]. The evaluation of these models was carried out both at infinite dilution and at finite concentrations and the results compared to experimental data. Furthermore, liquid–liquid equilibrium predictions for binary polymer solutions using six activity coefficient models are compared in this work. The parameters were estimated for all the models to achieve the best possible representation of the reported experimental equilibrium behavior.     

A discontinuous Galerkin method to solve chromatographic models

This article proposes a discontinuous Galerkin method for solving model equations describing isothermal non-reactive and reactive chromatography. The models contain a system of convection-diffusion-reaction partial differential equations with dominated convective terms. The suggested method has capability to capture sharp discontinuities and narrow peaks of the elution profiles. The accuracy of the method can be improved by introducing additional nodes in the same solution element and, hence, avoids the expansion of mesh stencils normally encountered in the high order finite volume schemes. Thus, the method can be uniformly applied up to boundary cells without loosing accuracy. The method is robust and well suited for large-scale time-dependent simulations of chromatographic processes where accuracy is highly demanding. Several test problems of isothermal non-reactive and reactive chromatographic processes are presented. The results of the current method are validated against flux-limiting finite volume schemes. The numerical results verify the efficiency and accuracy of the investigated method. The proposed scheme gives more resolved solutions than the high resolution finite volume schemes.     

Numerical solution of nonlinear and non-isothermal general rate model of reactive liquid chromatography

Abstract

A nonlinear general rate model (GRM) of liquid chromatography is formulated to analyze the influence of temperature variations on the dynamics of multi-component mixtures in a thermally insulated liquid chromatographic reactor. The mathematical model is formed by a system of nonlinear convection–diffusion reaction partial differential equations (PDEs) coupled with nonlinear algebraic equations for reactions and isotherms. The model equations are solved numerically by applying a semi-discrete high-resolution finite volume scheme (HR-FVS). Several numerical case studies are conducted for two different types of reactions to demonstrate the influence of heat transfer on the retention time, separation, and reaction. It was found that the enthalpies of adsorption and reaction significantly influence the reactor performance. The ratio of density time heat capacity of solid and liquid phases significantly influences the magnitude and velocity of concentration and thermal waves. The results obtained could be very helpful for further developments in non-isothermal reactive chromatography and provide a deeper insight into the sensitivity of chromatographic reactor operating under non-isothermal conditions.     

Numerical Simulation of Electrochemical Systems by Coupling Analytical Calculation of the Diffuse Double Layer and Finite Element Description of Solution Bulk

Numerical modelling of electrochemical systems covers length scales from the nanometers up to the macroscopic scale. With finite element methods, the mesh must be extremely fine to describe the diffuse double layer, thus increasing the needed computational resources. We propose a method to describe the diffuse double layer by analytical equations, expressed as boundary conditions for the partial differential equations describing the solution bulk. We apply the method to a one-dimensional system, i. e. to a cell with plane parallel electrodes, in the presence of a redox couple and a supporting electrolyte. We provide evidence of the precision of the method.     

Numerical approximation of a two-dimensional model of non-isothermal reactive liquid chromatography involving slow rates of adsorption–desorption kinetics

Abstract

A two-dimensional general rate model of non-isothermal reactive column chromatography is formulated considering homogenous and heterogeneous reaction rates, slow rates of adsorption–desorption kinetics, and enthalpies of adsorption and reaction. The model is expressed by a system of six nonlinear partial differential equations (PDEs) coupled with algebraic expressions for the adsorption and reaction rates. The nonlinearity of adsorption isotherm and reaction term hinders the derivation of analytical solutions. For that reason, a flux-limiting high-resolution finite volume scheme is suggested to numerically approximate the model equations. The effects of several kinetic and thermodynamic parameters are rigorously analyzed on the reactant conversion and components separation.     

Simulation of nonisothermal reactive liquid chromatography using two-dimensional lumped kinetic model

A nonisothermal two-dimensional lumped kinetic model of reactive liquid chromatography is formulated and applied to simulate the separation of multicomponent mixtures in a fixed-bed cylindrical column operating under nonisothermal condition. The axial and radial variations of concentration and temperature as well as reversibility of the chemical reactions are incorporated in the model equations. The model comprises a system of convection-diffusion-reaction partial differential equations coupled with algebraic and differential equations. Due to the nonlinearity of adsorption and reaction kinetics, it is required to apply an accurate numerical scheme for solving the model equations. In this study, an efficient and accurate high-resolution flux-limiting finite-volume scheme is proposed to solve the model equations. A number of stoichiometrical reactions are numerically simulated to determine the level of coupling between the temperature and concentration profiles. Moreover, the effects of various critical parameters on the process performance are examined. The results obtained are beneficial for understanding reaction and separation processes inside a liquid chromatographic reactor and to improve its performance.     

Photokinetic methods: A mathematical analysis of the rate equations in photochromic systems

In this article the analytical integration of kinetic equations describing the dynamic behavior of reversible photoreactions has been undertaken. Photochemical systems giving rise to competing thermal and photochemical steps are kinetically analyzed; the rate law is set up and analytical solutions are obtained under precisely defined boundary conditions. More complicated kinetic systems, where several species interact both thermally and photochemically, are also investigated. The kinetic treatment leads to a system of coupled differential equations which are amenable to analytical solutions, under the appropriate experimental and boundary conditions. The usefulness of the equations developed is illustrated by their application to some spirooxazine and chromene photochromic systems: two examples are described in detail. © 1999 John Wiley & Sons, Inc., Int J Chem Kinet 31: 303–313, 1999     

Multiple discrete energy levels and the bistable state of weak anchoring NLC cells

The weak anchoring nematic liquid crystal (NLC) cell is investigated with regard to energy. Because the Gibbs free energy of liquid crystal system used in theory does not include temperature and entropy, and because the equations and boundary conditions for δG=0 are also the mechanical equilibrium conditions of the continuum, the Gibbs free energy G is equivalent to the energy E of the liquid crystal continuum. There are multiple solutions which satisfy these equations and boundary conditions, each solution corresponding to a certain energy value. We call these discrete energy values and energy levels. Adopting a simple liquid crystal cell model, the energy levels are calculated in detail by means of analytical and numerical methods. The results show that there are three energy levels (or more in certain cases). The values and sequence of the energy levels are related to the external field and anchoring parameters. The relationships between the energy level structure and the bistable. Fréedericksz transition are disussed, together with their influence on the response time. The physical condition for the existence of more than three energy levels is also given.     

On the structure of the single-phase flow field in hollow fiber membrane modules during filtration

Whereas the structure of the flow field generated in the hollow fiber membrane modules operating as liquid–liquid or gas–liquid contactors has been studied extensively theoretically, this is not the case for the much more complex flow field generated during the filtration operation of the same type of modules. The present work is a first approach to analyze and understand the particular flow field. The mathematical problem is formulated and then geometrical and physical simplifications are introduced in order to decompose the full problem in a series of simpler sub-problems. In addition to the presentation of the general structure and the features of the problem, the focus of the present work is to derive all possible analytical solutions of the sub-problems. Several results based on analytical or simple numerical approaches are also presented.     

Rheology of dilute polyelectrolyte solutions in narrow channels

The shear flow of dilute polyelectrolyte solutions bounded by either neutral or repulsive walls is modeled using a nonlinear dumbbell with conformation-dependent friction. Assuming that the configurational probability density function depends on the internal coordinates (r) and the distance of the center of mass of the molecule to the walls, coupled differential equations for the tensor moments <rr> are obtained. Coulombic repulsion between beads is considered to simulate the charge repulsion between ionized sites distributed along the backbone of a real polyelectrolyte. The repulsive interaction between the polyelectrolyte molecule and the charged walls is that of the DLVO model and the molecule is considered to be a charged sphere. Numerical solutions for the components of the tensor <rr> are worked out with the preaverage approach, and only when neutral walls considered are exact solutions obtained. Viscosity results show that in the limit of very wide channels, the corresponding viscosity in the bulk is obtained. The wall repulsion on the charged molecules produces migration of molecules towards the center of the channel resulting in a depleted layer with lower viscosity next to the walls. The calculated slip phenomenon using the method employed by Grisafi and Brunn is dependent on the beads repulsion and the shear rate. The slip velocity obtained with the Mooney method shows similarities with available experimental results for polyelectrolyte solutions. Birefringence calculations are performed in narrow and wide channels for different bead repulsions, with interesting results for both flexible and rigid molecules. Received: 26 September 1998 Accepted in revised form: 11 March 1999     

Application of power expansion in the thermodynamics of compressible Markovian copolymers

The problem of constructing phase diagrams for a compressible melt of a binary Markovian copolymer is reduced to a set of nonlinear differential equations in partial derivatives with transcendental relationships. Using power expansions, the closed set of nonlinear differential equations is derived. This set allows its further analytical study. Eigenvalues of a linearized system are analyzed, and the boundaries of the thermodynamic stability of melts are defined. Nonlinear equations in normal coordinates are obtained; for symmetric melts, these equations are reduced to a single equation by adiabatic elimination of small-scale variables. Binodal curves are calculated for such solutions of this equation, which correspond to the free energy minimum of melts. Corrections reflecting the effect of melt nonsymmetry are found. The results are applied for copolymers, whose composition is similar to that of homopolymers, diblock copolymers, and random and regularly alternating copolymers. Spinodals and binodals corresponding to microphase separation are constructed.     

SULFONATION IN A FALLING FILM REACTOR: COMPARATION OF EXPERIMENTAL RESULTS WITH MATHEMATICAL MODEL PREDICTIONS

A mathematical model is developed to simulate a falling film reactor for sulfonation/sulfation. In the model, the reaction rate is considered to be controlled by the mass transfer in the gas phase or in the liquid phase. The gas phase mass and heat transfers are calculated by empiric equations; in the liquid phase, they are calculated by solving with numerical methods the partial differential equations which describe the system. In these equations, and eddy diffusion is considered, following the Levich's theories

The model results are compared with the experimental results obtained by the authors in a pilot plant, for the dodecylbenzene sulfonation.     

Single-center method of calculation for clusters and molecules other than hydrides

A new single-center method is proposed for solving the one-particle Schrödinger equation for molecules other than hydrides and for clusters, based on the method of associated differential and integral equations. The higher terms of the expansion of the wave function of the electron are replaced by linear combinations of analytical functions. This reduces the system of integro-differential equations to a system of differential and algebraic equations, for which stable numerical solutions have been worked out. Calculations are given of the energy and functions of the 2s state of an oxygen atom with a displaced center.Translated from Teoreticheskaya i Éksperimental'naya Khimiya,Vol. 25, No. 1, pp. 12–20, January–February, 1989.The authors are grateful to A. G. Kochur for making available the program for the SC expansion of atomic functions, and also to V. L. Sukhorukov for discussing the results.     

Conversion of Parameters Among Variants of Scatchard’s Neutral-Electrolyte Model for Electrolyte Mixtures that Have Different Numbers of Mixing Terms

Various model equations are available for representing the excess Gibbs energy properties (osmotic and activity coefficients) of aqueous and other liquid mixed-electrolyte solutions. Scatchard’s neutral-electrolyte model is among the simplest of these equations for ternary systems and contains terms that represent both symmetrical and asymmetrical deviations from ideal mixing behavior when two single-electrolyte solutions are mixed in different proportions at constant ionic strengths. The usual form of this model allows from zero to six mixing parameters. In this report we present an analytical method for transforming the mixing parameters of neutral-electrolyte-type models with larger numbers of mixing parameters directly to those of models with fewer mixing parameters, without recourse to the source data used for evaluation of the original model parameters. The equations for this parameter conversion are based on an extension to ternary systems of the methodology of Rard and Wijesinghe (J. Chem. Thermodyn. 35:439–473, 2003) and Wijesinghe and Rard (J. Chem. Thermodyn. 37:1196–1218, 2005) that was applied by them to binary systems. It was found that the use of this approach with a constant ionic-strength cutoff of I≤6.2 mol⋅kg⁻¹ (the NaCl solubility limit) yielded parameters for the NaCl+SrCl₂+H₂O and NaCl+MgCl₂+H₂O systems that predicted osmotic coefficients φ in excellent agreement with those calculated using the same sets of parameters whose values were evaluated directly from the source data by least-squares, with root-mean-square differences of RMSE(φ)=0.00006 to 0.00062 for the first system and RMSE(φ)=0.00014 to 0.00042 for the second. If, however, the directly evaluated parameters were based on experimental data where the ionic strength cutoff varied with the ionic-strength fraction, i.e., because they were constrained by isopiestic ionic strengths (MgCl₂+MgSO₄+H₂O) or solubility/oversaturation ionic strengths (NaCl+SrCl₂+H₂O and NaCl+MgCl₂+H₂O), then parameters converted by this approach assuming a constant ionic-strength cutoff yield RMSE(φ) differences about an order of magnitude larger than the previous case. This indicates that for an accurate conversion of model parameters when the source model is constrained with variable ionic strength cutoffs, an extension of the parameter conversion method described herein will be required. However, when the source model parameters are evaluated at a constant ionic strength cuttoff, such as when source isopiestic data are restricted to ionic strengths at or below the solubility limit of the less soluble component, or are Emf measurements that are commonly made at constant ionic strengths, then our method yields accurate converted models. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.     

Relations between kinetic parameters of different column models for liquid chromatography applying core-shell particles

This article presents analytical solutions of the general rate model (GRM), the lumped kinetic model (LKM), and the simpler equilibrium dispersive model (EDM) for core-shell particles and linear adsorption isotherms. The solutions in the Laplace domain are applied to derive analytical expressions for the temporal moments of these models. The results provide relations between the model specific kinetic parameters by matching one or more of the temporal moments. Several case studies are considered for illustration. The results show that simpler models are in many cases as good as the most detailed GRM if their kinetic parameters fulfill the matching relations. Thus, it is possible to reliably predict elution profiles using the simpler models. The derived analytical expressions can also be utilized to efficiently estimate model parameters from experimentally observed elution profiles to further optimize core-shell particles and to identify suitable column sizes and operating conditions.     

Computational Modelling of the Behaviour of Potentiometric Membrane Biosensors

This paper presents a mathematical model of a potentiometric biosensor based on a potentiometric electrode covered with an enzyme membrane. The model is based on the reaction–diffusion equations containing a non-linear term related to theMichaelis–Menten kinetics of the enzymatic reaction. Using computer simulation the influence of the thickness of the enzyme membrane on the biosensor response was investigated. The digital simulation was performed using the finite difference technique. Results of the numerical simulation were compared with known analytical solutions.