Analysis of two-dimensional models for liquid chromatographic reactors of cylindrical geometry

This work presents the analytical solutions of two-dimensional isothermal reactive general rate models for liquid chromatographic reactors of cylindrical geometry. Both irreversible and reversible reactions are considered. The model equations form a linear system of convection-diffusion-reaction partial differential equations coupled with algebraic equations for isotherms. Analytical solutions are derived by integrated implementation of finite Hankel transform, Laplace transform, eigen-decomposition technique, and conventional ordinary differential equations solution technique. To verify the analytical results, a high-resolution finite volume scheme is also applied to numerically approximate the model equations. The current results can be very useful to optimize and upgrade the liquid chromatographic reactors.     

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.     

A combined linear ion trap for mass spectrometry

We propose a novel ion cyclotron resonance ion trap capable of confining ions even at high pressure. The trap consists of three capacitively coupled axial sections, each composed of four circular cross-section rods parallel to the magnetic field axis. Ion confinement along the magnetic field direction is provided by applying the same static voltage to each set of “endcap” rods. As for a two-dimensional quadrupole mass filter, a sufficiently high rf frequency (several MHz) leads to an “effective” electrostatic “pseudopotential” well with a minimum on the trap central axis. Ions are confined radially by the combination of an applied axial static magnetic field and a radially inward-directed electric field resulting from differential rf voltages applied to each set of four rods. Ion confinement properties are revealed from a Paul traplike “stability diagram,” whereas ion trajectories are analyzed in terms of Penning-type ion cyclotron rotation, magnetron rotation, and axial oscillation motional modes. Ion cyclotron frequency increases with the strength of the rf trapping field. Ion magnetron motion becomes stable if the rf voltage is high enough. Therefore, ion trajectories can be stable even in the presence of ion-neutral collisions. Adding an ac potential to a Penning trap should dramatically increase the upper mass detection limit.     

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.     

Modelling the temporal intensity distribution in laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS) using scanning and drilling mode

Signal equations basing on dispersion functions describing the measured temporal intensity distribution for laser ablation inductively coupled plasma-mass spectrometry (LA-ICP-MS) used in scanning and drilling mode are developed. Variable ablation rates due to either varying focussing conditions typical for drilling mode and due to the changes of physical and chemical properties in inhomogeneous samples as typically investigated in scanning mode are considered for. The model accounts for intermixing of the sample aerosol within the sampling chamber, the influence of transport in a cylindrical transport channel and the fact that normally not the entire vapour generated and transported to the ICP can be observed. The absolute signal response is influenced by the actually ablated, transported and observed analyte mass. The dispersion functions describing the relative signal response depend on sample chamber volume, the volume of the transport channel, the laser shot frequency, the carrier gas flow rate and the part of observable cross-section at the MS interface compared to the entire cross-section filled by the vapour. All these parameters depend on the experimental set-up and the selected operating conditions only. Using the signal equation the influence of all mentioned parameters on signal course is shown both theoretically and experimentally. The signal equation can be used for calculation of optimal experimental conditions.

On this basis, an algorithm is proposed providing the relative temporal distribution of any analyte with significantly higher temporal resolution than the measured temporal intensity distribution itself. Furthermore, usage of dispersion functions for investigation of a given transport system, for explanation of typical signal deviations, for the proof of homogeneous regions in a heterogeneous sample, for examination of changes in ablation rate and for investigation of fractionation effects is shown.     

A simplified compression test for the estimation of the Poisson's ratio of viscoelastic foams

This paper describes a simplified procedure for determining the Poisson's ratio of homogeneous and isotropic viscoelastic materials. A cylindrical shaped material is axially excited by an electromagnetic shaker and consequent displacement waves are investigated. Using a frequency sweep as an excitation signal, the frequency domain displacement response is measured upstream and sideways of the sample itself. A plane cross-section analytical model of the experimental setup is used to estimate Poisson's ratio through a minimisation-based procedure, applied to radial displacement once the complex modulus has been directly determined under the assumption of spring-like behaviour of the axial displacement. The results are presented and discussed for different materials and compared to well-established quasi-static and finite element simulations.     

Thermophysical Characterization of Paraffin Wax Based on Mass-Accommodation Methods Applied to a Cylindrical Thermal Energy-Storage Unit

Two mass-accommodation methods are proposed to describe the melting of paraffin wax used as a phase-change material in a centrally heated annular region. The two methods are presented as models where volume changes produced during the phase transition are incorporated through total mass conservation. The mass of the phase-change material is imposed as a constant, which brings an additional equation of motion. Volume changes in a cylindrical unit are pictured in two different ways. On the one hand, volume changes in the radial direction are proposed through an equation of motion where the outer radius of the cylindrical unit is promoted as a dynamical variable of motion. On the other hand, volume changes along the axial symmetry axis of the cylindrical unit are proposed through an equation of motion, where the excess volume of liquid constitutes the dynamical variable. The energy–mass balance at the liquid–solid interface is obtained according to each method of conceiving volume changes. The resulting energy–mass balance at the interface constitutes an equation of motion for the radius of the region delimited by the liquid–solid interface. Subtle differences are found between the equations of motion for the interface. The differences are consistent with mass conservation and local mass balance at the interface. Stationary states for volume changes and the radius of the region delimited by the liquid–solid interface are obtained for each mass-accommodation method. We show that the relationship between these steady states is proportional to the relationship between liquid and solid densities when the system is close to the high melting regime. Experimental tests are performed in a vertical annular region occupied by a paraffin wax. The boundary conditions used in the experimental tests produce a thin liquid layer during a melting process. The experimental results are used to characterize the phase-change material through the proposed models in this work. Finally, the thermodynamic properties of the paraffin wax are estimated by minimizing the quadratic error between the temperature readings within the phase-change material and the temperature field predicted by the proposed model.     

Freedericksz transitions in smectic liquid crystals in annular geometries

This paper investigates the stability of simple static orientation patterns in a sample of smectic liquid crystal confined to a cylindrical annulus, when a magnetic field is applied. Four different arrangements are considered, covering cases where the layer normal is everywhere either radial or axial, and the (orthogonal) magnetic field is either radial, azimuthal or axial. A classification is given of the threshold radii for mechanical instabilities, and of the threshold magnetic fields for Freedericksz transitions for these cases, with strong anchoring at the boundaries.     

Electroosmotic flows with random zeta potential

The hydrodynamic problem of electroosmotic flow in a cylindrical capillary with random zeta potential is solved in the limit of small Deybe length and low Reynolds number. Averages are defined over multiple experiments and the mean axial velocity is found to be a plug flow. The variance of the velocity exhibits parabolic-like variation across the capillary. Average concentrations of samples transported by the flow are approximated by defining an effective diffusivity coefficient. Theoretical formulas for the average concentration are supported by numerical experiments.     

Prediction of coupled menisci shapes by Young-Laplace equation and the resultant variability in capillary retention

This paper shows how 2 coupled Young-Laplace equations can be solved to predict the shapes of two coupled menisci formed in a capillary system. Experiments are performed, which demonstrate that the equilibrium volume of liquid retained in a vertical capillary, can be variable, even when all the properties of the system are invariant. This variability in liquid retention also leads to different equilibrium shapes of the top and bottom menisci. A coupled form of the Young-Laplace equation is solved to predict the two coupled menisci shapes. The curvature of the top meniscus is fitted to the experimentally recorded meniscus shape. The coupled Young-Laplace equation solution is used to predict the shape of the bottom meniscus. The shape of the bottom meniscus thus obtained, is shown to match the experimentally recorded bottom meniscus shape reasonably well. This observed coupling of the menisci has a significant impact on some porosimetric techniques which are based on liquid extrusion and explains why the volume of liquid that can be retained in a capillary can vary, under invariant conditions. Retention of liquids in capillaries is of interest in several applications like fabric wash.     

A capacitively coupled microwave plasma atomic emission spectrometer for the determination of trace metals in microsamples

A capacitively coupled microwave plasma operating at 450 to 850 W is used for atomic emission spectroscopy. The laboratory-constructed system contains a tungsten cup electrode capable of holding a volume of up to 30 μl. Microwaves are used to dry the sample, while at higher powers the plasma is ignited for sample vaporization and excitation. The entire analysis can be carried out in less than 5 min. A mixture of helium and hydrogen is used as the plasma gas. A spherical or cylindrical shaped plasma can be formed depending upon the gas flow rate and the microwave power selected. The effects of experimental parameters, such as gas flow rate, atomization power, electrode position and plasma shape are examined. Detection limits for Cd, Mg and Zn are in the low picogram range for a 10 μl sample; the relative standard deviation is less than 10%.     

An Insoluble Surfactant Model for a Vertical Draining Free Film

A mathematical model is constructed to study the evolution of a vertically oriented thin liquid film draining under gravity when there is an insoluble surfactant with finite surface viscosity on its free surface. Lubrication theory for this free film results in three coupled nonlinear partial differential equations describing the free surface shape, the surface velocity, and the surfactant transport at leading order. We will show that in the limit of large surface viscosity, the evolution of the free surface is that obtained for the tangentially immobile case. For mobile films with small surface viscosity, transition from a mobile to an essentially immobile film is observed for large Marangoni effects. It is verified that increasing surface viscosity and the Marangoni effect retard drainage, thereby enhancing film stability. The theoretical results are compared with experiment; the purpose of both is to act as a model problem to evaluate the effectiveness of surfactants for potential use in foam-fabrication processes. Copyright 2000 Academic Press.     

Numerical simulations of viscoelastic particle migration in a microchannel with triangular cross-section

The migration of a spherical particle immersed in a viscoelastic liquid flowing in a microchannel with a triangular cross-section is investigated by direct numerical simulations under inertialess conditions. The viscoelastic fluid is modeled through two constitutive equations to investigate the effect of the second normal stress difference and the resulting secondary flows on the migration phenomenon. The results are presented in terms of trajectories followed by the particles released at different initial positions over the channel cross-section in a wide range of Weissenberg numbers and confinement ratios. Particles suspended in a fluid with a negligible second normal stress difference migrate toward the channel centerline or the closest wall, depending on their initial position. A much more complex dynamics is found for particles suspended in a fluid with a relevant second normal stress difference due to the appearance of secondary flows that compete with the migration phenomenon. Depending on the Weissenberg number and confinement ratio, additional equilibrium positions (points or closed orbits) may appear. In this case, the channel centerline becomes unstable and the particles are driven to the corners or “entrapped” in recirculation regions within the channel cross-section. The inversion of the centerline stability can be exploited to design efficient size-based separation devices.     

Coupled axial–torsional behavior of cylindrical elastomer bushings

Elastomer bushings are important components in a vehicle suspension system to isolate vibration. The purpose of the work is to present experimental and computational studies of cylindrical bushings subjected to axial, torsional and coupled axial-torsional deformation modes. The experimental results show that the relationship between the forces and moments and their corresponding displacements and rotations are nonlinear due to the nature of the elastomer material. When an axial deformation is superimposed on a torsional deformation, a nonlinear coupled response exists; the torsional moment initially decreases and then increases with increasing axial deformation. This nonlinear coupling effect for small superimposed deformations is not well predicted computationally when modeling the bushing using common constitutive material laws in a commercial finite element package.     

Ringlike cores of cylindrically confined nematic point defects

Nematic liquid crystals confined in a cylindrical capillary and subjected to strong homeotropic anchoring conditions is a long-studied fundamental problem that uniquely incorporates nonlinearity, topological stability, defects, and texture physics. The observed and predicted textures that continue to be investigated include escape radial, radial with a line defect, planar polar with two line defects, and periodic array of point defects. This paper presents theory and multiscale simulations of global and fine scale textures of nematic point defects, based on the Landau-de Gennes tensor order parameter equations. The aim of this paper is to further investigate the ringlike nature of point defect cores and its importance on texture transformation mechanisms and stability. The paper shows that the ringlike cores can be oriented either along the cylinder axis or along the radial direction. Axial rings can partially expand but are constrained by the capillary sidewalls. Radial rings can deform into elliptical structures whose major axis is along the capillary axis. The transformation between several families of textures under capillary confinement as well as their stability is discussed in terms of defect ring distortions. A unified view of nematic textures found in the cylindrical cavities is provided.     

Computer modeling of enclosed inductively coupled plasma discharges

A mathematical model for an enclosed inductively coupled plasma (EICP) is developed for configurations important for spectrochemical analysis. The model, based on steady state, laminar flow, axial symmetry and local thermodynamic equilibrium assumptions, accounts for natural convection and radiative transfer effects. The computer simulations based in this model provide information about the temperature, flow, and electromagnetic fields for spherical and cylindrical discharge configurations.     

Electroosmotic flow in a rectangular channel with variable wall zeta-potential: comparison of numerical simulation with asymptotic theory

Electroosmotic flow in a straight micro-channel of rectangular cross-section is computed numerically for several situations where the wall zeta-potential is not constant but has a specified spatial variation. The results of the computation are compared with an earlier published asymptotic theory based on the lubrication approximation: the assumption that any axial variations take place on a long length scale compared to a characteristic channel width. The computational results are found to be in excellent agreement with the theory even when the scale of axial variations is comparable to the channel width. In the opposite limit when the wavelength of fluctuations is much shorter than the channel width, the lubrication theory fails to describe the solution either qualitatively or quantitatively. In this short wave limit the solution is well described by Ajdari's theory for electroosmotic flow between infinite parallel plates (Ajdari, A., Phys. Rev. E 1996, 53, 4996-5005.) The infinitely thin electric double layer limit is assumed in the theory as well as in the simulation.     

Electron Kinetics in Nonisothermal Plasmas with Spatially Two-Dimensional Structures

A method for the study of the steady-state nonlocal electron kinetics in radially and axially inhomogeneous cylindrical plasmas is presented. The method is based on the solution of the relevant space-dependent kinetic equation for the electron velocity distribution using the two-term expansion. A three-dimensional initial boundary value problem for the isotropic distribution component with the total energy as the evolution variable of the kinetic problem has to be treated. The technique is applied to analyze the electron kinetics in a cylindrical DC column plasma under impact of a space-independent axial field and a radially increasing radial field. Particularly, the spatial relaxation of the electron gas in axial direction in response to a disturbance of the axial homogeneity is investigated. A detailed analysis of the spatially resolved energy balance is given. Considerable modification of the results with respect to those obtained by earlier studies of axially homogeneous DC column plasmas, as well as of relaxation processes in one spatial dimension, has been found.     

A liquid chromatography-tandem mass spectrometry method for fluazifop residue analysis in crops

An LC-MS-MS assay is described for fluazifop residue analysis in crops. The residues are extracted with acidified organic solvent, the esters and conjugates are hydrolysed with 6 M hydrochloric acid, then the extracts are cleaned-up by solid phase extraction using C2(EC) and Si cartridges in tandem. Quantitative analysis is performed by gradient liquid chromatography coupled to triple quadrupole mass spectrometer using atmospheric pressure chemical ionisation. All fluazifop-P-butyl, free fluazifop-P and any conjugates are quantified as fluazifop-P. The limit of quantification is 0.01-0.05 mg/kg depending on crop matrices. The clean-up method is also suitable for LC-UV analysis with a compromise in higher limit of quantification 0.05-0.2 mg/kg.     

Solution of the excluded volume problem for biaxial particles

We report the solution of the excluded volume problem for a pair of biaxial hard molecules; namely, sphero-platelets. As an application of this result we study the isotropic to nematic liquid crystal transition for a fluid composed of these particles in the Onsager limit (length δ breadth or width). We show that the range of stability of the isotropic phase decreases with increasing particle biaxiality.