The reduced trans-column (or long-range eddy dispersion) height equivalent to a theoretical plate (HETP) of short and wide 4.6 mm × 30 mm columns packed with 2.6 μm Accucore-C18 superficially porous particles was measured under conventional (no split flow) and parallel segmented (outlet and inlet) flow chromatography. The overall reduced HETP was derived from the true moments of the recorded concentration profiles. The longitudinal diffusion HETP term was measured at a very small flow rate (0.05 mL/min). The solid–liquid mass transfer resistance was derived from the shell diffusivity, using the composite Garnett–Torquato model of effective diffusion in a heterogeneous system made of a dense packing of core–shell particles immersed in a continuous matrix (the eluent). The trans-channel and short-range interchannel eddy dispersion HETP terms were assumed to be equal to the calculated h data after solving the Navier–Stokes equation and simulating the advection–diffusion transport process. Experimental results confirmed that the optimum efficiency of these short columns was increased by a factor of about two. The ratio of the detection sensitivities on the PSFC stream and on a regular stream increased from 1 to 1.45 when the retention factor decreases from about 10 to 0.5. These phenomena are due to a strong reduction of the trans-column eddy dispersion HETP term. The system loses about 60% of the sample mass when only outlet skimming is carried out when the flow rate ratio of 55% is applied, as was done in this work. It loses about 50% of the sample when inlet/outlet segmentation is applied. In gradient elution, the peak capacity is increased by only 15%, due to post-column band spreading, which should imperatively be minimized when the outlet flow is split.

The adsorption isotherms of the naproxen enantiomers were measured on a Kromasil Whelk-O1 column, eluted with mixtures of supercritical carbon dioxide and methanol or ethanol. Five chromatographic methods were used: frontal analysis, frontal analysis by characteristic points, elution by characteristic points, retention time method and the inverse method. In all methods, the effects of the two modifiers were compared. The use of these methods suffers from limitations due to supercritical fluid chromatographic instruments. These problems and drawbacks are discussed. The accurate and precise determination of the isotherm parameters was not possible with the instrument currently used. In contrast, the different methods allow to show qualitatively that the adsorption of the naproxen enantiomers show heterogeneous mechanism on the adsorbent surface common in chiral chromatography. Finally, the experimental high concentration elution band profiles of naproxen are compared with those calculated from the isotherm parameters provided by the five different methods.

The pressure- and the density-drops along a chromatographic column eluted with supercritical fluid carbon dioxide were mapped as a function of the outlet column pressure and the temperature on the P-T diagram of neat CO(2). At low densities, the viscosity of CO(2) is low, which is expected to result into a low pressure drop along the column. However, at these low densities, the volumetric flow rates of the mobile phase at constant mass flow rates are high, which might result into a high pressure drop along the column. These conflicting effects of an adjustment in the mobile phase density on the pressure drop of the mobile phase along the column makes it nearly impossible to develop a simple intuitive understanding of the relationships between the net pressure drops and the operating temperatures and pressures. The development of a similar understanding of their relationships with the density drop along the column is even more complex, because this density drop depends also on the compressibility of the mobile phase, itself a function of the operating pressures and temperatures. Numerical calculations of the pressure and density drops along columns packed with particles of different sizes, under different operating conditions (temperature, outlet pressure, and flow rate), provide important insights regarding the extent of the pressure and density drops under these operating conditions.

The peak capacity per unit time or chromatographic productivity, Pct, was calculated and optimized for four different sample mixtures (small molecules, peptides from protein digests, and mixtures of small and large intact proteins) analyzed on 4.6 mm I.D. columns packed with three different brands of core–shell particles. The calculations were based on the column permeabilities and on their van Deemter curves measured under isocratic conditions. The 3D plots of the peak capacity are provided as a function of the gradient steepness and the column length at three values of the inlet pressure and two temperatures. The maximum peak capacities obtained per unit time were 103 min−1 for small molecules, 111 min−1 for peptides, 63 min−1 for small proteins, and 22 min−1 for large proteins. They are all achieved for the shortest column lengths run at the maximum achievable flow rate (5 mL/min) and for an optimum gradient steepness, which had to be calculated. Remarkably, the best separation productivity of these columns cannot be achieved with modern UPLC instruments that deliver flow rates no larger than 5 mL/min. With instruments able to deliver a 15 mL/min flow rate and acquire data at 320 Hz, 4.6 mm × 50 mm columns packed with shell particles operated at 600 bar and a temperature of 60 °C, would give peak capacities per unit time of 256 min−1 for small molecules, 237 min−1 for peptides, 186 min−1 for small proteins, and 59 min−1 for large proteins.

One dimensional solutions for the classic critical upward seepage gradient/quick condition and the time rate of consolidation problems are obtained using coupled routines for the finite volume method (FVM) and discrete element method (DEM), and the results compared with the analytical solutions. The two phase flow in a system composed of fluid and solid is simulated with the fluid phase modeled by solving the averaged Navier–Stokes equation using the FVM and the solid phase is modeled using the DEM. A framework is described for the coupling of two open source computer codes: YADE-OpenDEM for the discrete element method and OpenFOAM for the computational fluid dynamics. The particle–fluid interaction is quantified using a semi-empirical relationship proposed by Ergun [12]. The two classical verification problems are used to explore issues encountered when using coupled flow DEM codes, namely, the appropriate time step size for both the fluid and mechanical solution processes, the choice of the viscous damping coefficient, and the number of solid particles per finite fluid volume.

Chromatographic columns packed with shell particles are now nearly twice more efficient than columns packed with conventional, fully porous particles. Shell particles are made of a solid core surrounded by a porous shell of constant thickness. Diffusion through the bed of packed columns is complex due to their heterogeneity. It involves diffusion through the external and the internal fluid, and surface diffusion. Six diffusion models are compared that combine these diffusion mechanisms. They involve the external porosity of the bed (ϵ(e)), the ratio of the core to the particle diameters (ρ), and the ratio of the shell diffusivity to the bulk diffusion coefficient (Ω). Four different theoretical approaches were considered. They are based on (1) the additivity of the mass flux densities modulated by the obstruction factors caused by non-porous spherical inclusions; (2) the effective medium theory of Landauer; (3) the effective medium theory of Garnett for spherical inclusions; and (4) the probabilistic theory of Torquato (for binary composite materials only). The two Landauer models fail because they cannot account for the obstruction factor imposed by the presence of non-porous spherical inclusions. The ternary Garnett model (3) provides an excellent approximation of the actual diffusion mechanism but the most physically relevant model seems to be the one derived from a combination of the Garnett model for a binary core-shell particle and of the Torquato model for random dispersion of contacting spheres in a matrix. Accurate measurements of axial dispersion coefficients are needed to validate or reject the semi-empirical parallel diffusion models and to select the most appropriate one. The results of such measurements made with the peak parking method for various compounds are reported in the companion paper.

Mass transfer kinetics in a particular column packed with 2.6μm Kinetex-C18 particles was measured using a series of non-invasive methods. According to the manufacturer, this column was the most efficient one in a lot of 133 columns of the same dimensions, packed with particles of the same batch, with an efficiency about 6% larger than the lot average. The total eddy diffusion term of this column was determined by subtracting from the experimental HETP data (1) the longitudinal diffusion term (given by the peak parking method); (2) the trans-particle solid–liquid mass transfer resistance term (given by the peak parking method and an axial diffusion model); and (3) the external film mass transfer resistance term (derived from the Wilson and Geankoplis correlation). The results are analyzed based on the general coupling theory of eddy diffusion of Giddings and are compared to those previously obtained with a column packed with 2.7μm Halo-C18 column, which also has a low eddy diffusion term. The particular column investigated shows no trans-column velocity bias and a smaller reduced short-range inter-channel eddy diffusion term than the Halo column. This result explains the unusual efficiency of this exceptionally well packed column.

The heights equivalent to a theoretical plate of a weakly and strongly retained compounds were measured on two packed columns having different average mesopore sizes. The measurements were carried out in two different cases, with access to the mesopores by the sample molecules blocked (filled with n-nonane) or not. The experimental results demonstrate that the eddy dispersion terms of both columns are significantly smaller for porous than for nonporous particles. Two simultaneous phenomena explain this observation. First, packed columns are radially heterogeneous which causes significant trans-column velocity biases warping the bands. Second, radial dispersion contributes to mass transfer across the column, relaxing the radial concentration gradients that are caused by these velocity biases. The impact of these biases is minimized when the pores of the particles are not blocked; it decreases with increasing residence time and radial dispersion coefficient of the solutes. 2009 American Institute of Chemical Engineers AIChE J, 56: 1495-1509, 2010

This paper investigates the seepage flow through idealized particle assemblies comprised of spherical elements. Comparisons between an analytical solution and numerical approximation using the Discrete Element Method (DEM) with coupled hydromechanical interaction between the particles and the pore fluid are provided. In the coupled flow DEM, the fluid motion is treated at the macroscopic scale by solving the local averaged Navier-Stokes equation, while the particle interaction is treated at the microscopic scale. The numerical solution is compared with the classical analytical solution in terms of quantity of flow and the pore pressure distribution.

The isosteric heat of adsorption of a pure compound dissolved in a solution in equilibrium with a solid adsorbent was determined, based on the Gibbs surface excess model. The exact isosteric heat of adsorption differs from the usual value derived from the variation of the Henry’s constant with the reciprocal temperature because this procedure assumes ideal behavior of the bulk liquid solution, which, in most cases, is only approximately so. An experimental protocol, based on the determination of the excess adsorption isotherms by combining frontal analysis (for strongly adsorbed components) and spectrometric tracer pulse chromatography (for weakly adsorbed compounds) is proposed. It allows the determination of the exact isosteric heat of adsorption provided that the activity coefficient of the compound in the bulk solution can be explicitly expressed as a function of the bulk liquid composition.

The different operating conditions of an online two-dimensional liquid chromatographic separation (2D-LC), such as the length of the column, the linear velocity and the composition of the mobile phase used in the second dimension, its initial organic content if this separation is carried out in gradient elution, the number of fractions of the first column eluent collected, and the analysis time of the first dimension all affect the achievable separation power of 2D-LC online systems. The influences of these factors on the separation performance were investigated, and an equation was derived for the calculation of the achievable peak capacity in online 2D-LC assuming (1) that the option of undersampling the first-dimension separation is acceptable, (2) that the solutes follow linear-solvent-strength behavior, and (3) that all the separations are made in gradient elution. This theoretical discussion shows that (1) highly efficient separations made with online 2D-LC require the second-dimension peaks to be very narrow, (2) the separation power of 2D-LC systems is maximum for an optimum number of fractions collected in the first dimension, (3) higher peak capacities can be achieved by using shorter second-dimension columns and collecting a relatively large number of fractions, (4) the achievable 2D peak capacity is maximum for a certain eluent flow rate and column length of the second-dimension column, and (5) the maximum achievable peak capacity increases with decreasing velocity and initial organic content of the second-dimension eluent. As a consequence, due to the time restriction of the second-dimension gradient time, online 2D-LC schemes cannot realistically afford peak capacities exceeding 10,000, even if they are implemented with exceptionally efficient columns and if long analysis times are accepted.

Overloaded band profiles of aniline (25 microL injection of a 30.5mM solution) eluted with a methanol-aqueous buffer solution (30/70, v/v) were recorded at the exit of a 150 mm x 4.6mm column packed with 3.5 microm XBridge-C(18)porous particles. The SWpHs of the mobile phase was adjusted with phosphate (SWpHs approximately 2.7 and SWpHs approximately 7.5) or acetate buffers (SWpHs approximately 5.3) of different concentrations (11, 56, and 278 mM). The elution times and profiles of the bands observed at low ionic strength were successfully accounted for using the extended Debye-Hückel theory to estimate the activity coefficients of the ions in the bulk phase and a simple non-competitive Langmuir adsorption model, the adsorption of pure aniline or pure anilinium onto XBridge-C(18) being described by a Langmuir isotherm. The band profiles were calculated using this adsorption model and the equilibrium-dispersive model of chromatography. The calculated and the experimental band profiles are in excellent agreement at all buffer pH and ionic strength. This demonstrates that the elution times and the band profiles are controlled by the chromatographic dilution process and by the reaction of aniline with the buffer solution.

The theoretical advantages and drawbacks of using a multiple-, parallel column approach in on-line multidimensional liquid chromatography systems were investigated. Much time or peak capacity can be gained with the use of multiple parallel columns at the second-dimension while the aggregate time of separation increases only by the increment of the gradient time of the second-dimension. Multidimensional chromatographic systems are now used to perform many tasks ranging from routine, fast analyses to specialized, arduous separations. In this work, we focus on the advantages of a multiple, parallel columns approach to on-line multidimensional liquid chromatography systems. Calculations of the achievable peak capacities were made as functions of the number of columns operated in parallel. Increasing the number of second-dimension columns from one to two or three causes the largest increase in peak capacity with only a slight increase of aggregate time. We also present some practical aspects to consider when attempting multidimensional separations with multiple columns operated in parallel.

The use of graded rock fill is a cost-effective means of sinkhole repair, providing a stable platform for infrastructure while allowing controlled flow of surface water into the sinkhole throat. Typically, the soil around the sinkhole is excavated down to the rock throat, and large diameter (0.25 to 0.5 m) shot rock aggregate is dumped into the excavation. The shot rock forms a stable bridge over the throat, even though the largest rock particle is often smaller than the throat opening, and additional free draining fill is placed up to grade. This paper investigates the stability of the repair for a range of rock particle diameter (relative to the sinkhole throat diameter). The paper has two goals: a) describe the three-dimensional modeling of the placement of shot rock using the Discrete Element Method, where it is demonstrated that for a given throat radius, there exists an intermediate range of particles sizes for which stability depends upon how the particles arrange during the filling procedure, and b) present a statistical description of this intermediate range of particle sizes for which the throat is semi-stable, using logistic regression to describe the gradual transition from unstable to stable behavior.

We measured overloaded band profiles for a series of nine compounds (phenol, caffeine, 3-phenyl 1-propanol, 2-phenylbutyric acid, amphetamine, aniline, benzylamine, p-toluidine, and procainamidium chloride) on columns packed with four different C18-bonded packing materials: XTerra-C18, Gemini-C18, Luna-C18(2), and Halo-C18, using buffered methanol–water mobile phases. The pHWS of the mobile phase was increased from 2.6 to 11.3. The buffer concentration (either phosphate, acetate, or carbonate buffers) was set constant at values below the maximum concentration of the sample in the band. The influence of the surface chemistry of the packing material on the retention and the shape of the peaks was investigated. Adsorbents having a hybrid inorganic/organic structure tend to give peaks exhibiting moderate or little tailing. The retention and the shape of the band profiles can easily be interpreted at pHsWS that are well above or well below the pKWSa of the compound studied. In contrast, the peak shapes in the intermediary pH range (i.e., close to the compound pKWSa) have rarely been studied. These shapes reveal the complexity of the competitive adsorption behavior of couples of acido-basic conjugated compounds at pHsWS that are close to their pKWSa. They also reveal the role of the buffer capacity on the resulting peak shape. With increasing pHWS, the overloaded profiles are first langmuirian (isotherm type I) at low pHsWS, they become S-shaped (isotherm type II), then anti-langmuirian (isotherm type III), S-shaped again at intermediate pHsWS, and finally return to a langmuirian shape at high pHsWS. A new general adsorption isotherm model that takes into account the dissociation equilibrium of conjugated acidic and basic species in the bulk mobile phase accounts for these transient band shapes. An excellent agreement was achieved between experimental profiles and those calculated with a two-sites adsorption isotherm model at all pHsWS. The neutral species adsorbs strongly on a first type of sites that have a high density while the ionic species adsorb preferentially on a second type of sites that have a very low density. The evolution of the peak shape when the pHWS changes from acidic to basic is well explained by the weak buffer capacity of the mobile phase used compared to the concentration of the eluted compounds.

The separation of the peptide digests of myoglobin and bovine serum albumin was performed with an off-line combination of two commercial, conventional HPLC columns. The first column was packed with a strong ion exchanger and eluted with a KCl gradient. The second column was packed with particles of C18-bonded silica and eluted with an acetonitrile gradient. The conditional peak capacities of the 2D separations achieved exceed 7000 under the experimental conditions investigated. This performance is achieved at the cost of an analysis time of the order of 28 hours. Possible improvements to the separation method described here are discussed.

Although the availability of discrete-element method (DEM) codes has improved, the need still exists to solve simple verification problems to obtain an understanding of these codes. Different DEM codes may have subtle differences in the manner in which the method is implemented, and the significance of these differences may be problem dependent. This paper investigates a series of simple, one and two-particle contact problems. These problems, which employ various types of damping, are shown to be equivalent to classical one-dimensional vibration problems. The solutions are discussed in the context of the DEM, and results from the DEM are shown to compare very well with the classical solutions. It is demonstrated that results from a well-known commercial two-dimensional code (PFC2D) and the open source three-dimensional code (YADE) yield identical solutions to these problems provided the problem solution process is manipulated properly. A discussion of the differences in how gravity and damping are implemented may be of interest to users of PFC2D.