Effective Longitudinal Diffusion Research Articles

An upscaling model for simulation of geothermal processes in stratified formations

In stratified porous media, non-uniform velocity between layers combined with thermal conduction across layers causes spreading of the thermal front: thermal Taylor dispersion. Conventional upscaling not accounting for this heterogeneity within simulation grid blocks underestimates thermal dispersion, leading to overestimation of thermal breakthrough time. We derive a model for effective longitudinal thermal diffusivity in the direction of flow, αeff, to represent the effective thermal dispersion in two-layer media. αeff, accounting for thermal Taylor dispersion, is much greater than the thermal diffusivity of the rock itself. We define a dimensionless number, NTC, a ratio of times for longitudinal convection to transverse conduction, as an indicator of transverse thermal equilibration of the system during cold- or hot-water injection. For NTC > 5, thermal dispersion in the two-layer system closely approximates a single layer with αeff. This suggests a two-layer medium satisfying NTC > 5 can be combined into a single layer with an effective longitudinal thermal diffusivity αeff. In application to a geothermal reservoir, one can apply the model to perform upscaling in stages, i.e. combining two layers satisfying the NTC criterion in each stage. The αeff model accounting for the fine-scale heterogeneity within simulation grid blocks would enhance the prediction accuracy of thermal breakthrough time and thus thermal lifetime.

On the modelling of the effective longitudinal diffusion in bi-continuous chromatographic beds

We report on the possibility to extend to bi-continuous packings the two models for the effective longitudinal diffusion Deff, or B-term band broadening, recently proposed for discontinuous chromatographic beds. In bi-continuous packings, like monolithic columns, solutes experience a connected end-to-end pathway in both the mobile and stationary zones, as opposed to discontinuous packings, wherein the stationary adsorptive zone is distributed over a set of isolated elements. Since it is unclear whether a densely packed bed of spherical particles should be treated as a continuous or a bi-continuous medium, this extension is also crucial to fully understand the behaviour of packed particle beds. The proposed models for the effective longitudinal diffusion Deff originate from the adoption of the Two Zone Moment Analysis (TZMA) method by which Deff can be expressed as a linear combination of two essential quantities γm and γs, referred to as effective zone-diffusion factors. In the present work we propose two analytical models for γm and γs that now cover both the discontinuous and the bi-continuous case. To validate the theory, several bi-continuous packings are investigated, including the tetrahedral skeleton model (TSM), six different Triple Periodic Minimal Surface (TPMS) monoliths and randomly packed beds of spheres. For all of these, the models provide highly accurate results for Deff over a wide range of porosities and zone retention factors k″. The comparison with literature experimental data for both monolithic silica columns and columns packed with fully porous and porous-shell particles is also presented.

An alternative general model for the effective longitudinal diffusion in chromatographic beds filled with ordered porous particles

The two-zone moment-analysis method for the determination of the dispersion tensor in hierarchical retentive porous media has been adopted to compute and model the effective longitudinal diffusion Deff, or equivalently the B-term band broadening, in chromatographic beds filled with ordered porous particles. On the one hand, this approach offers accurate numerical results for Deff while keeping computational expenses low. On the other hand, it also gives direct insight for the analytical modelling, readily revealings the two main essential quantities (resp. referred to as the mobile-zone and stationary-zone effective diffusion factors γm and γs) that contribute to Deff. Modelling these two main parameters provided us with two new analytical models for Deff: a general one, valid for diluted and concentrated packings and accurate in the whole range of relevant intra-particle diffusion coefficient Dpz, and an approximate one, reliable for diluted packings and accurate also for concentrated packings with low to intermediate values of Dpz. The large advantage of both models is that they do not need any fitting parameter because all the required information is incorporated into the experimentally accessible geometric obstruction factor in the mobile phase originating from the tortuosity of the through-pore space (limiting case of fully solid particles without any retention). These models hence serve as an alternative to the Effective Medium Theory (EMT) models used so far in the literature. To validate the theory, five ordered geometries have been investigated. The accuracy of the general model proposed has been quantified and found to be comparable with that of the 3rd order approximate Torquato model for four geometries, even for macro-porosities close to the close-packing limit. The case of a 2-d triangular array of ellipsoidal particles with different elongations is also investigated to show the general validity and applicability of the models.

Two- and three-body attachment, electron transport and ionisation in water-air mixtures

Three-body electron attachment in the mixtures of H2O and dry air have been measured over a wide range of the density-reduced electric field, E/N, from 3–130 Td and gas pressures, for mixture combinations ranging from 1% to 50% of H2O. We have measured the regions of three-body attachment (3–30 Td) and two-body dissociative attachment (40–130 Td). Besides, the increasing amount of H2O in the mixture causes an increase in the three-body reaction rates of up to two orders of magnitude in comparison with that measured for dry air. On the other hand, the three-body attachment coefficients exceed the two-body ones (dissociative attachment) at high pressures. Good agreement has been found with previous measurements of H2O-dry air mixtures with H2O concentrations of up to 2%. We know of no previous work for higher H2O concentrations. Values of the effective ionisation coefficients and longitudinal diffusion coefficients derived from the same measurements are also presented.

Scanning drift tube measurements and kinetic studies of electron transport in CO

We present scanning drift tube measurements of electron swarm transport coefficients in CO as a function of the reduced electric field at room temperature under time-of-flight conditions. The measurements are compared to modeling results and other available experimental data on swarm transport over the broad range of from 2 Td to 1603 Td. The modeling results are obtained in Monte Carlo simulations and by solving the electron Boltzmann equation using a multi-term approach and the density gradient expansion procedure. We find generally good agreement between the measured and calculated transport coefficients. We propose a strategy to improve the cross-section set used to explain certain discrepancies at lower values. Measurements and calculations of electron transport coefficients under hydrodynamic conditions are complemented by Monte Carlo simulations of electron transport in an idealized steady-state Townsend (SST) setup. The ionization coefficient is calculated as a function of from the spatial density profiles of the electrons and compared to the corresponding values evaluated from the knowledge of the effective ionization frequency, drift velocity and longitudinal diffusion coefficient. Contrary to the traditional views, according to which the spatial relaxation of the mean energy and other transport properties for electrons in molecular gases is most commonly monotonic or quasi-monotonic, we find a ‘window’ of where the SST transport properties of the electrons exhibit oscillatory behavior as they relax towards the equilibrium state far downstream from the electron emitting boundary.

Measurement of the molecular diffusion coefficient and the effective longitudinal diffusion under supercritical fluid chromatography conditions in packed bed columns

The improvement of supercritical fluid chromatography (SFC) instrumentation enhanced its reliability and utility over the past decade. The further development of high speed and high resolution separations is however obstructed by the lack of accurate models for axial dispersion in SFC. This work is a first step to tackle this by developing more reliable methods to measure molecular (Dmol) and longitudinal diffusion (Deff) in SFC, as these affect all aspects of separation efficiency. In the present contribution, we report on an improved method, to enable more flexible, reliable and accurate measurements of Dmol in SFC using commercial instrumentation. A two-column variant of the stopped-flow experiment is proposed as an adapted set-up for measuring the effective longitudinal diffusion coefficient Deff in SFC-conditions. Using the set-ups for a number of test-compounds, it has been found that Deff, and the coefficients describing its constituent sub-processes (cf. particle diffusion Dpart and surface diffusion γsDs), all vary in a linearly proportional way with the bulk diffusion coefficient Dmol within a high degree of accuracy. It has also been found that Deff decreases much more sharply with increasing retention factor compared to LC. By applying the effective medium theory, it was found that the relative surface diffusion coefficient γsDs/Dmol decreases strongly with retention factor for the investigated solutes and column, in contrary to what is typically observed in reversed phase liquid chromatography. Results indicate that this might be related to a change in retention behavior of the analytes. Obviously, more analytes and conditions need to be explored to complete this picture and the extend range of applicability of these observations.

Measurement of the longitudinal diffusion of ionization electrons in the MicroBooNE detector

Accurate knowledge of electron transport properties is vital to understanding the information provided by liquid argon time projection chambers (LArTPCs). Ionization electron drift-lifetime, local electric field distortions caused by positive ion accumulation, and electron diffusion can all significantly impact the measured signal waveforms. This paper presents a measurement of the effective longitudinal electron diffusion coefficient, DL, in MicroBooNE at the nominal electric field strength of 273.9 V/cm. Historically, this measurement has been made in LArTPC prototype detectors. This represents the first measurement in a large-scale (85 tonne active volume) LArTPC operating in a neutrino beam. This is the largest dataset ever used for this measurement. Using a sample of ∼70,000 through-going cosmic ray muon tracks tagged with MicroBooNE's cosmic ray tagger system, we measure DL = 3.74+0.28 -0.29 cm2/s.

Shape matters: a Brownian microswimmer in a channel

Abstract

Anisotropic Diffusion of Proteins in the Sarcoplasmic Reticulum of Skeletal Muscle

The way proteins move inside crowded organelles is determined by both the connectivity of the organellar lumen and the mutual interactions between molecules. To probe diffusional movement inside the SR we used FRAP of three proteins expressed in mouse paw muscles: the biosensor D1ER and two fluorescently tagged forms of calsequestrin (Casq1). Longitudinal diffusion (parallel to the fiber axis) was evaluated by the rate of fluorescence recovery in a rectangular bleached region, 10 μm wide and long enough to span the fiber width (∼40 μm). Diffusion in the transversal direction was evaluated with a bleached region of the same dimensions, placed longitudinally, right at the myofiber axis. The FRAP rates measured thus for D4cpV-Casq1 in resting myofibers corresponded (by theory and simulations) to an effective longitudinal diffusion coefficient DL of 0,012 μm2/s, while the transversal coefficient DT was 0,026 μm2/s. The difference was statistically significant. Faster transversal diffusion was also found for EYFP-Casq1. Additionally, the measurements evinced the presence of a large fraction of immobile Casq1, presumably polymerized within terminal cisternae. As reported elsewhere, the immobile fraction of Casq1 decreased, while the diffusional mobility of Casq1 and D1ER increased upon Ca2+ depletion in the SR. Electron microscopic images of resting myofibers demonstrate a greater connectivity of SR cavities in the transversal direction (i.e. along terminal cisternae) than longitudinally, to an extent consistent with and probably explaining in full the observed diffusional anisotropy. Regardless of the SR Ca2+ content, the diffusion coefficient values found for these proteins are much lower than the lowest values reported for proteins inside the ER, which probably reflects the intricacy of the SR structure and the polymeric state of calsequestrin in resting skeletal muscle. Supported by NIH grants AR049184 and GM111254.

A theoretical study on the advantage of core-shell particles with radially-oriented mesopores

We report on a first-principles numerical study explaining the potential advantage of core-shell particles with strictly radially-oriented mesopores. Comparing the efficiency of these particles with fully porous and core-shell particles with a conventional (i.e., randomly oriented) mesopore network, the present numerical study shows a similar strong reduction in minimal reduced plate height (hmin) as was very recently observed in an experimental study by Wei et al. (respectively a hmin-reduction on the order of about 1 and 0.5 reduced plate height-units). As such, the present work provides a theoretical basis to understand and confirm their experimental findings and quantifies the general advantage of “radial-diffusion-only” particles. Determining the effective longitudinal diffusion (B-term contribution) in a series of dedicated, independent simulations, it was found that this contribution can be described by a very simple, yet fully exact mathematical expression for the case of “radial- diffusion-only” particles. Using this expression, the significant increase in efficiency of these particles can be fully attributed to their much smaller B-term band broadening, while their C-term band broadening (representing the mass transfer resistance) remains unaffected.

Numerical study of a cylinder model of the diffusion MRI signal for neuronal dendrite trees

We study numerically how the neuronal dendrite tree structure can affect the diffusion magnetic resonance imaging (dMRI) signal in brain tissue. For a large set of randomly generated dendrite trees, synthetic dMRI signals are computed and fitted to a cylinder model to estimate the effective longitudinal diffusivity DL in the direction of neurites. When the dendrite branches are short compared to the diffusion length, DL depends significantly on the ratio between the average branch length and the diffusion length. In turn, DL has very weak dependence on the distribution of branch lengths and orientations of a dendrite tree, and the number of branches per node. We conclude that the cylinder model which ignores the connectivity of the dendrite tree, can still be adapted to describe the apparent diffusion coefficient in brain tissue.

Theory of isobaric pressure exchanger for desalination

Abstract A theory is developed to predict the time of sustained operation of a rotary pressure exchanger used for energy recovery in seawater reverse osmosis system. Based on past experiments for oscillating pipe flows, it is found that the existing plug flow velocity in the ducts is not high enough to induce turbulence in the wall boundary layer. Modeling the time series of the flow velocity in the inviscid core as a periodic series of rectangular pulses, the structure of the laminar momentum boundary layer is first derived. The mass boundary layer induced by the oscillating velocity is then solved in order to obtain the slow diffusion of the averaged brine concentration along the duct. With the result the effective longitudinal diffusivity (dispersivity) is found explicitly for arbitrary Schmidt number. The dispersivity is found to be small due to the small viscosity and mass diffusivity in the very thin boundary layers, however it is still augmented to hundreds times of the molecular diffusivity. For a range of duct and rotor dimensions and rotor frequencies, the time needed for the mixing zone to spread to the ends of the duct is predicted for large Schmidt number appropriate for salt in water. After transient mixing is over, a certain amount of salt leaks steadily into the fresh seawater reentering the membrane. However the leakage is shown to be small due to the small dispersivity.

Nodal and modal strategies for longitudinal thermal diffusivity profile estimation: Application to the non destructive evaluation of SiC/SiC composites under uniaxial tensile tests

In this paper, infrared image sequences of a SiC/SiC composite excited with a non uniform heat pulse (also known as Flash) are processed using two different strategies. In a first approach, a modal strategy based on the decomposition of infrared sequences in an orthogonal basis is applied. A discrete version of the heat equation is solved in the transformed space using only a very limited number of modes for which the sensitivity to diffusivity is optimal. In a second time, a nodal approach that consists in solving locally a finite difference scheme is performed at each pixel. A new sensitivity analysis based on a local correlation study of the partial derivatives of the experimental data (transformed or not) allows validating the relevance of each approach, and determining the optimal space where to perform estimations. As a result, effective local longitudinal diffusivity profiles of a SiC/SiC composite under mechanical testing are provided and analyzed to detect the presence of microcracks.

The Transport of Nanoparticles in Blood Vessels: The Effect of Vessel Permeability and Blood Rheology

The longitudinal transport of nanoparticles in blood vessels has been analyzed with blood described as a Casson fluid. Starting from the celebrated Taylor and Aris theory, an explicit expression has been derived for the effective longitudinal diffusion (Deff) depending non-linearly on the rheological parameter xi(c), the ratio between the plug and the vessel radii; and on the permeability parameters pi and omega, related to the hydraulic conductivity and pressure drop across the vessel wall, respectively. An increase of xi(c) or pi has the effect of reducing Deff, and thus both the rheology of blood and the permeability of the vessels may constitute a physiological barrier to the intravascular delivery of nanoparticles.

Moonlets wandering on a leash-ring

Abstract Since the Voyager flybys, embedded moonlets have been proposed to explain some of the surprising structures observed in Saturn's narrow F ring. Experiments conducted with the Cassini spacecraft support this suggestion. Images of the F ring show bright compact spots, and seven occultations of stars by the F ring, monitored by ultraviolet and infrared experiments, revealed nine events of high optical depth. These results point to a large number of such objects, but it is not clear whether they are solid moonlets or rather loose particle aggregates. Subsequent images suggested an irregular motion of these objects so that a determination of their orbits consistent with the F ring failed. Some of these features seem to cross the whole ring. Here we show that these observations are explained by chaos in the F ring driven mainly by the ‘shepherd’ moons Prometheus and Pandora. It is characterized by a rather short Lyapunov time of about a few hundred orbital periods. Despite this chaotic diffusion, more than 93 per cent of the F-ring bodies remain confined within the F ring because of the shepherding, but also because of a weak radial mobility contrasted by an effective longitudinal diffusion. This chaotic stirring of all bodies involved prevents the formation of ‘propellers’ typical of moonlets, but their frequent ring crossings explain the multiple radial ‘streaks’ seen in the F ring. The related ‘thermal’ motion causes more frequent collisions between all bodies which steadily replenish F-ring dust and allow for ongoing fragmentation and re-accretion processes (ring recycling).

Numerical simulation of the effects of secondary flow and boundary retention on contaminant dispersion

The secondary flow and dispersion of a tracer substance in a fluid flowing through a curved tube are studied numerically. The system consists of an infinitely long conduit defined by two concentric curved circular pipes. Thus the two phases correspond to a flowing fluid phase and the annular wall comprised of a stationary homogenous medium. The solute is soluble in the annular region and is assumed to satisfy a linear equilibrium relationship at the interface. Flow in the fluid phase is calculated using the finite difference scheme as described by Collins and Dennis [9]. The scheme is of second order accuracy with respect to grid sizes over the entire laminar flow region and reveals the asymptotic nature of the solution for large values of Dean number D. Further, a central difference scheme is used to solve the convection diffusion equation in both the phases with the condition that the injective solute is highly soluble in the wall layer according to a linear equilibrium relationship at the interface. Results for small Dean number are first verified with our earlier results (Jayaraman et al. [17]) which were based on Dean's solution [12,13]. The numerical calculations are then extended for larger Dean number and Schmidt number. The results are consistent with experimental results that the influence of secondary flows on dispersion is reduced if the tracer is very soluble in the wall. It is found that the effective longitudinal diffusivity falls below its straight tube value by an amount, which depends on the absorption coefficient, Dean number and the diffusivity in the wall. For large Dean number dispersion becomes negligible.

The Effective Dispersion of Nanovectors Within the Tumor Microvasculature

The effective longitudinal diffusion of nanovectors along non-permeable and permeable capillaries has been studied considering the contribution of molecular and convective diffusion based on the Taylor's theory of shear dispersion. The problem is of importance in the transport of nanovectors used for the intravascular delivery of drugs and contrast agents. It has been shown that for a given capillary size and hemodynamic conditions a critical radius acr exists for which the effective longitudinal diffusion along the capillary has a minimum: Nanovectors with a < acr diffuse mainly by Brownian diffusion whereas nanovectors with a < acr diffuse mainly by convection and the effective diffusion coefficient grows with a. In permeable conduits, the effective diffusion reduces significantly compared to normal non-leaky vessels and it has been derived that acr grows almost linearly with the hydraulic permeability Lp of blood vessels. It has been shown that the blood conduits with the largest effective longitudinal diffusivity could be preferentially targeted by the circulating vectors. Based on these findings, the following strategies are proposed to increase the number of nanovectors targeting the tumor vessels: (i) The use of nanovectors with a critical radius for normal vessels, (ii) the injecting of bolus of nanovectors with different radii, and (iii) the normalization of the tumor vasculature. Finally, it has been emphasized that the size of the vector should be selected depending on the body district where the tumoral mass is developing and on the type, malignancy, and state of the tumor.

Molecular Dynamics Simulations of Glutamate Diffusion in Synaptic Cleft

Diffusion of transmitters in the synaptic cleft critically influences synaptic efficacy by affecting both the amplitude and the time course of quantal events, but the value of the diffusion constant is speculative. In this study, we use molecular dynamics simulations to determine how the spatial confinement and membrane charges affect the diffusion constants of glutamate- and water as well as general properties of their diffusion. The synaptic cleft is represented as the space enclosed by two single-wall carbon sheets. Both water and especially glutamate are concentrated near the pore wall, where the concentration of glutamate can reach 30-50 times the mean value and the concentration of water can reach 2-8 times the mean value. Such spatial profiles of glutamate contradict the classical notions of diffusion on which both continuous and Monte Carlo simulations are built. The layering of glutamate- and water molecules suggests that the interfacial glutamate-cleft wall (or water-cleft wall) interactions may critically regulate their diffusion in the cleft. Indeed, the effective longitudinal diffusion constant of glutamate is steeply dependent on the cleft width, but only when the cleft is very narrow (< 5 nm). Therefore, even for a cleft as narrow as at the glutamatergic synapse in the central nervous system, the effective diffusion constant of glutamate will not be much lower than free diffusion in the bulk solution due to confinement. The effective diffusion constant of water is considerably less sensitive to cleft width over the same range of cleft widths than is glutamate, but is also higher than that of glutamate. Finally, the layering of glutamate and water and their effective diffusion constants are largely independent of how the cleft wall is charged. In conclusion, in the confined space of the synaptic cleft, glutamate is layered near the wall. Consequently, its diffusion constant becomes dependent on the cleft width. However, the diffusion of glutamate is slower than its free diffusion in water only if the cleft is very narrow. If the width of the cleft is consistent with that determined by morphometric studies in the central nervous system, glutamate diffusion should not be slowed by confinement and is thus likely to be similar to that in free solution.

Effective diffusion coefficient for two-band systems

The effective longitudinal diffusion coefficient for two-band systems is determined. Emphasis is put on the nonlinear transport regime realized at high electric fields. Particular results are obtained for bipolar transport in two-band semiconductors and vertical transport in semiconductor superlattices. An additional contribution to the effective diffusion coefficient is identified, which results from different drift velocities in the respective bands (minibands).

Dispersion of solute in a fluid flowing through a curved tube with absorbing walls

The dispersion of solute in a fluid flowing through a curved tube with absorbing walls is studied using a mathematical model of an infinitely long conduit defined by two concentric curved circular pipes. The annular wall is comprised of a stationary homogeneous medium, and the inner cylinder is the flowing fluid phase. The solute is soluble in the annular region and is assumed to satisfy a linear equilibrium relationship at the interface. A series expansion is derived for the effective longitudinal diffusivity, D eff , valid when both the Dean number N 1 /2 and the product σ N (σ is the Schmidt number) are sufficiently small. The theory is extended numerically using a spectral finite-difference method to widen the validity of the results to more realistic problems in which σ N can take large values although N remains small. The results are consistent with the experimental findings of Kaye et al. that the influence of secondary flows on dispersion is reduced if the tracer is very soluble in the wall