On Relaxation Phenomena in Field-Flow Fractionation

Abstract

The two-dimensional unsteady convective diffusion equation satisfied by the local concentration of the colloid introduced in a field-flow fractionation (FFF) column is solved by the method of finite differences. The alternating direction implicit (ADI) method proposed by Peaceman and Rachford is used. The axial convection term is approximated by a backward difference approximation to obtain a stable and convergent scheme.

Numerical results are obtained for various values of the transverse Peclet number for the case of steady laminar flow and a slug input. The numerical results from the ADI method are validated by comparison with numerical solutions obtained using an explicit scheme as well as by internal consistency checks.

The results of this work show that the transverse concentration profiles depend in a complex fashion on axial position along the cloud during relaxation. In the presence of a field, asymptoticity in the transverse profiles is approached first in the rear of the colloid cloud, and progresses gradually through the axial extent of the cloud. Ultimately, at a sufficiently large value of time, almost all of the colloid relaxes to asymptotic exponential distributions in the transverse coordinate as predicted from theory. The local concentration of colloid in the system is observed to reach a global maximum value at intermediate values of time during relaxation. The area average concentration distribution is observed to exhibit strong asymmetry when plotted against the axial coordinate at intermediate times both in the presence and in the absence of a field. This asymmetry is in accord with pure convection theory. In contrast, truncated two-term dispersion equations only predict symmetric distributions for symmetric initial conditions. Thus there may be a need to retain higher order terms in the application of generalized dispersion theory in order to predict the observed results.     

Peak Capacity in Field-Flow Fractionation

Abstract

Field-flow fractionation (FFF) peak capacity values have been computed with only two major assumptions: first, the plate height is supposed the sum of only two contributions, axial molecular diffusion and transversal nonequilibrium, and second, the steric effect has been neglected in the equations of retention and peak broadening.

Several reduced parameters have been defined to generalize the equations and limit the number of variable parameters. It appears that among the already implemented FFF subtechniques for which the elution spectrum is an explicit function of the principal dimension, or mass, of the retained sample (which excludes electrical FFF), sedimentation FFF has some peculiar characteristics due to the fact that the field-induced velocity depends on a particular sample, while in thermal and flow FFF it is the same for all samples of a given type under fixed experimental conditions. For example, in sedimentation FFF, the axial diffusion contribution to the plate height persists at a much larger reduced eluant velocity than for the other techniques.

The effect on the peak capacity of the retention volume, the channel length, the eluant velocity as well as the influence of detection limit and analysis time have been studied. Simple relationships between peak capacity and these parameters are established in the high retention and negligible axial diffusion limits which previal in most experimental situations, and deviations from these limits are discussed. It is shown that for all three     

Thermogravitational Field-Flow Fractionation: An Elution Thermogravitational Column

Abstract

The major operating characteristics of thermal field-flow fractionation (thermal FFF) and of thermogravitational columns are compared, and it is shown that the two approaches can be advantageously combined in a method we call thermogravitational FFF. The theory of this technique is developed, with primary attention given to a change in the velocity profile under different flow conditions and its effect on component retention, column efficiency, resolution, and selectivity. Experimental results are shown to be in good overall accord with theory. It is shown that the potential of thermogravitational FFF lies in the fractionation of low molecular weight polymers or of other species having weak thermal diffusion.     

Measurement of polydispersity of ultra-narrow polymer fractions by thermal field-flow fractionation

In this paper we demonstrate that the polydispersity µ = M?_w/M?_N of narrow polymer fractions can be readily obtained by measuring band broadening and its velocity dependence in a thermal field–flow fractionation (thermal FFF) system. The thermal FFF method is shown to be more accurate than size exclusion chromatography for the determination of polymer polydispersities due to the simpler band dispersion function and the higher selectivity inherent to the technique. The polydispersities of a series of four narrow polystyrene samples prepared by anionic polymerization were consequently determined by thermal FFF and found to be much smaller (1.003–1.006) than the ceiling values (1.06) suggested by the suppliers. As part of this investigation, an experimental study of band dispersion in thermal FFF is used to examine current theory. The data show nonequilibrium to be the dominant factor, whereas relaxation effects are insignificant at lower flow rates and can be subdued at higher flow rates. A high correlation between nonequilibrium theory and experiment allows for the estimation of diffusion coefficients from plate height–velocity data.     

A Comparison of Mobile-Phase Peak Dispersion in Gas and Liquid Chromatography

Abstract

The field of axial and radial dispersion of unsorbed bands in columns or beds packed with spherical particles is reviewed and it is shown that there is broad agreement between various workers: at low reduced velocities both axial and radial dispersion occur by obstructed molecular diffusion. At higher velocities the processes are more complex but at very high velocities and at Reynolds numbers in excess of about 10 the reduced plate height becomes independent of velocity and has a value for axial dispersion of about 2 and for radial dispersion of about 0.2. In the intervening region the dispersion process is complex and shows dependence upon the column-to-particle-diameter ratio. The most inefficient columns appear to be formed when this ratio is between 10 and 30. It is therefore suggested that efforts be made to design and construct columns with greater trans-column uniformity. When trans-column packing inequalities are unimportant, the reduced plate height in the high-velocity region is only slightly affected by fluid velocity, in strong contrast to the situation in open tubes. With gases the reduced plate height does not rise much above 2 for well-constructed columns, whereas with liquids it rises to about 4 before turbulence becomes important and again limits the dispersion, so that it falls to about 2.     

Chromatography in a Bed of Spheres

Abstract

A computationally efficient method is presented for calculating the combined effects of intraparticle diffusion, interphase mass-transfer resistance, and fluid-phase axial dispersion for chromatography in a column of uniform regularly packed spheres. The method uses an extension of the generalized Sturm-Liouville theory of Ramkrishna and Amundson to adapt the Taylor-Gill-Subramanian dispersion analysis to two-phase systems. The primary utility of the analysis is to determine the importance of diffusional transients and to obtain asymptotic limiting behavior for very long columns. Simple closed-form solutions are given for this limiting condition. Our analysis suggests that the transients neglected in presently used lumped-parameter analyses are in fact often small, especially for small packing diameter and low flow rates. In addition, the Glueckauf and Coates approximation for internal diffusional resistance is found to be a valid asymptotic limit. However, conditions do arise in practice where transients should be considered.     

Removal of Refractory Organics by Aeration. III. A Fast Algorithm for Modeling Solvent Sublation Columns

Abstract

A fast algorithm is developed for modeling the operation of batch and continuous flow solvent sublation columns. Mass transfer kinetics and axial dispersion are taken into account, and a Langmuir isotherm is used. Results are presented illustrating the effect on column performance of influent flow rate, axial dispersion, and bubble radius.     

HYDRODYNAMIC CHROMATOGRAPHY: THREE DIMENSIONAL LAMINAR DISPERSION IN RECTANGULAR CONDUITS WITH TRANSVERSE FLOW

Experimental observations^1,9 indicate much poorer separations than are predicted by two dimensional theory. The purpose of this work is to explain these differences and suggest ways in which system performance can be improved.

The large effect of span-wise variation in axial velocity caused by side walls on hydrodynamic separations carried out in rectangular conduits with transverse flow is studied theoretically. As the aspect ratio increases, the steady stale retentivity (convection coefficient) approaches an asymptotic value obtained by neglecting side wall effects. However, the dispersion coefficient does not reduce to that for a flow with no side walls. Indeed, the asymptotic steady state dispersion coefficient is at least six times larger than that obtained by two dimensional theory which neglects side wall effects. As the transverse Peclet number increases, the effect of side walls on the dispersion coefficient becomes much larger.

The present three dimensional theoretical predictions, in contrast to two dimensional ones, are in good agreement with the experimental data of Caldwell, et al.⁹ and Kesner, et al.¹ on electrical field flow fractionation. The results indicate that side wall effects may be of major importance in hydrodynamic chromatography even when the aspect ratio is 70 or more.

The adverse effect of side walls may be avoided by having the membranes enclose thin annular regions rather than rectangular conduits. This should improve performance significantly.     

DISPERSION IN TURBULENT FILM FLOW

Dispersion of miscible solutes from non-uniformly distributed instantaneous sources in two dimensional smooth open channel turbulent flows is described satisfactorily by using generalized dispersion theory wherein the coefficients of the dispersion model depend on time. The truncation of the generalized dispersion model after the first two terms is justified by comparison of results with experimental data. The behavior of the convective coefficient, k₁, (t) and the dispersion coefficient, k₂(t), for turbulent flow is studied in detail for all values of time, k₁(t) and k₂(t) reach asymptotic constant values at t 0.5 y² _m,/ε at which time the generalized dispersion model reduces to the Taylor constant coefficient model.

Area mean concentration distributions predicted by the generalized dispersion theory agree with the experimental data of Fischer (1966) rather closely and the theoretical results agree with the data much better than those from Taylor's model.     

Reliability of binary diffusion coefficients determined from tailing response curves measured by the Taylor dispersion method in the near critical region

Two kinds of tracer response measurements, by the Taylor dispersion method using a stainless-steel diffusion column and the chromatographic impulse response (CIR) method using a polymer-coated diffusion column, were carried out by using the curve fitting to determine infinite-dilution binary diffusion coefficients of vitamin K₃ (2-methylnaphthalene-1,4-dione), as a medium polar compound, in supercritical CO₂ at 308.2 K and pressures from 7.75 to 31.00 MPa. The response curves in the Taylor dispersion method showed more significantly tailing closer to the critical point, whereas those did not at conditions away from the critical point. However, the CIR method provided the response curves almost without tailing over an entire range of pressure studied. Consequently, in the CIR method the response curves were accurately reproduced with the determined values of two parameters, binary diffusion coefficient and retention factor. The determined values of diffusion coefficients showed a little slow-down in the near critical region. In the Taylor dispersion method, however, the response curves with tailing were poorly reproduced with the determined values of diffusion coefficients, which apparently led to a steep-down in the near critical region.     

Diffusion Coefficients and Molecular Weight Distributions of Humic and Fulvic Acids Determined by Flow Field-Flow Fractionation

Abstract

The ability to characterize molecules whose physical and chemical properties are intimately linked to their diffusion coefficients and molecular weight is important to further understanding of chemical transport in the environment. Flow field-flow fractionation (flow FFF) was used to obtain separations of water-soluble macromolecules of varying molecular weight, including polystyrene sulfonates and humic substances. The separation occurs due to differing diffusion rates for chemical species of differing molecular weight in aqueous solution. Flow FFF uses fluid flow as the mechanism of separation. A model that yields liquid phase diffusion coefficients as a function of molecular weight was utilized to determine molecular weights from degree of separation. Separations of polystyrene sulfonates, a humic acid, and two fulvic acids of known molecular weight were accomplished using flow FFF. The separations obtained were used to develop a relationship between flow FFF separation and species molecular weight. Separations were obtained for humic and fulvic acids of unknown molecular weight.     

Surface Barriers for Retention Enhancement in Field-Flow Fractionation

Abstract

It is proposed to modify the surface of a field-flow fractionation (FFF) channel by introducing small barriers perpendicular to flow. Possible advantages include increased retention, sample capacity, and selectivity. It is shown that this approach brings FFF into a closer relationship with chromatography and countercurrent distribution. Approximate theories are developed for retention, plate height, and selectivity, and sources of departure from theory are discussed. Two experimental thermal FFF systems are described, one with barriers established by cutting grooves in a Mylar sheet and another with grooves cut in a copper bar. Despite an observed deviation from the assumed rectangular groove shape, retention enhancement was considerable, and was in reasonable agreement with theory. Plate height, however, greatly exceeded the values observed for nongrooved systems.     

Feasibility Study of Dielectrical Field-Flow Fractionation

Abstract

The possible use of dielectrophoretic forces for the development of a new subtechnique of field-flow fractionation (FFF) termed dielectrical FFF is examined. Dielectrical FFF is based on the dielectrophoresis of neutral particles in the nonuniform electric field of an annular channel (or charged coaxial capacitor). The feasibility of the subtechnique is assessed by estimating the magnitudes of retention ratio R predicted from theory for select species representative of several classes of particle/fluid mixtures. Minimum attainable R values are calculated using estimates of the maximum electric field strengths applicable to the mixtures. Calculations show that. the dielectrophoretic force is strong enough to retain and separate ultrahigh-molecular-weight polymers and submicron-diameter particles dissolved or suspended in organic liquids of high dielectric constant Evidence suggests that pearl-chain formation may impose a fundamental limitation on particle retention at the inner cylinder of the annular channel, especially in aqueous suspensions.     

Nonequilibrium Theory for Field-Flow Fractionation in Annular Channels

Abstract

The principles of field-flow fractionation (FFF) and reasons for extending the FFF methodology from parallel-plate channels to annular channels (ANNCs) are briefly reviewed. A theory for the nonequilibrium plate height H of FFF zones in ANNCs is developed by extending the nonequilibrium theory of FFF to polar coordinates. The principal assumption in the theory is that component zones are localized near the ANNC walls by the general force F = A/rⁿ , where A and n are constants and r is the radial coordinate. Equations for H are developed as functions of n, the inner-to-outer radius ratio of the ANNC, and the fundamental FFF parameter, γ. A closed-form analytical solution to H is obtained when n = 1; the n ≠ 1 solution must generally be expressed as a ratio of the integrals involved. These integrals can be approximated analytically, however, when γ ? 1. The functions for H are compared to their parallel-plate counterpart, and differences are rationalized.     

Dispersion in pulsed systems—II: Theoretical developments for passive dispersion in porous media

A general theory is developed that allows one to calculate the dispersion tensor for pulsed systems in spatially periodic porous media. The analysis has its origin in the original work of Taylor, and the results quite naturally reduce to the Taylor-Aris result for passive dispersion when applied to the case of laminar flow in a tube. In Part III the theory is compared with axial dispersion data for packed beds.     

Dispersion models of unsteady tubular reactors

Axial dispersion in time-variable laminar flow in a tubular reactor is analyzed using an exact procedure for the case of a homogenous first-order reaction. For the first time since the Taylor Dispersion model was originally introduced for the modeling of reactors, its validity is examined over a wide range of the reaction rate parameter by comparison against an exact analysis. It is shown that a constant coefficient dispersion model can be obtained from first principles for large values of time only for initial distribution problems; however, this simple approximate model also is reasonably good for describing concentration distributions for the present inlet distribution problem for slow reactions and for axial locations sufficiently far away from the inlet. For rapid reactions, while the dispersion model is inaccurate in describing axial concentration distributions, it is surprisingly good for predicting the reactor length required for complete conversion. In contrast to the conclusion of a recent article, it will be shown that the dispersion coefficient is independent of the reaction rate constant.     

Diffusion models in chromatographic columns packed with fully and superficially porous particles

Most theoretical analyzes of molecular diffusion in chromatographic columns are based on more or less approximate models. Macroscopic sample diffusion along packed columns results from a complex combination of the local sample diffusivity in the external porosity of the bed (D_m) and in the porous particles (ΩD_m). A further complication arise from the use of the new superficially porous (or core–shell) particles. The obstruction to axial diffusion caused by the presence of their impermeable core has to be quantified.Two original models of longitudinal diffusion in packed beds are derived for these ternary composite materials. They account for the actual 3D micro-structure of the packed column bed. The micro-structure results from the presence of (1) the impermeable spherical cores; (2) the porous shells surrounding these cores; and (3) the eluent within which the particles are randomly dispersed. The theoretical approach is based on the combination of the effective medium theory of Garnett for core–shell spherical inclusions and of the probabilistic theory of Torquato for randomly dispersed spheres in a continuous matrix. The impacts of the core to the shell diameter, ρ, and of the porous shell to the bulk diffusivity, Ω, on the longitudinal diffusion B coefficient of chromatographic columns packed with core–shell particles are analyzed from a theoretical point of view.     

Gas-Solid Chromatography: Longitudinal and Intraparticle Diffusion of Acetylene in Activated Carbon

Abstract

The time-dependent transmission of 100 ppm acetylene in helium at 1 atm through columns of Columbia 4LXC 12/28 activated carbon at 25°C was measured at several flow rates in the range from about 1 to 7 cm³/s. Transmission is the gas-phase concentration at the column outlet divided by the input concentration. The gas-and adsorbed-phase diffusion coefficients were calculated from a homogeneous-solid diffusion model by means of the method of moments. The gas-phase diffusion coefficient was constant in this range of flow rates. Fitting the experimental transmission curves to a theoretical transmission expression for a model with only gas-phase diffusion indicated that solid-phase diffusion in this carbon cannot be neglected at high flow rates.     

Shear Field-Flow Fractionation: Theoretical Basis of a New,Highly Selective Technique

Abstract

Shear field-flow fractionation (shear FFF) is described as an FFF system in which shear forces are responsible for migration perpendicular to flow. It is shown that a desirable configuration for shear FFF is a concentric cylinder system with one cylinder rotating. After providing the relevant theoretical framework of FFF, the equations of Shafer et al. describing shear migration are simplified and applied to the limiting case of very thin annular spaces to get tractable retention expressions. On this basis the maximum selectivity is predicted to be 3 or greater, a value considerably higher than that for any other macromolecular separation technique. This high selectivity is confirmed using an alternate shear migration theory developed by Tirrell et al. However, it is shown that shear FFF is only applicable to macromolecules of high molecular weight, perhaps ～10⁷ and above. It may also be applicable to globular particles.