Effect Of Axial Diffusion Research Articles

Mathematical analysis of affinity membrane chromatography

A mathematical model including convection, diffusion and Freundlich adsorption is developed. To examine the validity of the model, the affinity membranes were prepared by coating chitosan on the nylon membranes, a ligand of poly- l-lysine was bound to the chitoan-coating membranes, and the adsorption behavior of bilirubin through the stacked affinity membranes was investigated. The agreements between the theoretical and experimental results are exceptional. Using our new model, we show that: (1) As Pe increases, the breakthrough curves become sharper. For Pe greater than 30, the effect of axial diffusion is insignificant; (2) As m increases, the time of total saturation is delayed and the loading capacity at the point of breakthrough is increased; (3) As n decreases, the time of total saturation is delayed and the loading capacity at the point of breakthrough is increased; (4) As r increases, both the time of total saturation and the loading capacity at the point of breakthrough are increased; (5) adsorption rate influences the time of total saturation strongly but contributes little to the loading capacity.

A mathematical method for determining oxygen distribution in skeletal muscle with multiple capillaries and axial diffusion

A mathematical method is presented for analysing oxygen transport in skeletal muscle that includes both the interaction of very large number of capillaries and the effect of axial diffusion in the tissue. The analysis takes into account differences in the blood flow velocity among the capillaries and in the oxygen concentration and haematocrit of the blood entering the capillaries at the arterial end. First, the oxygen concentration is determined within a functional unit consisting of a single capillary surrounded by a region of tissue, in which a flux is prescribed on the outer boundary of the region. The method of matched asymptotic expansions is used to include the effect of axial diffusion in the tissue. The flux that is prescribed on the boundary of the functional unit is a result of the interaction among all the capillaries comprising the vascular bed. These fluxes are found by matching the oxygen concentration along the boundaries of adjacent functional units. This leads to a system of ordinary differential equations for the capillary concentrations coupled with a system of algebraic equations for the fluxes. Results are presented for a two-dimensional array of capillaries when each capillary has a different blood flow velocity. Axial diffusion causes the oxygen concentration to drop more rapidly at the arterial end and remain below the level determined when axial diffusion is neglected. This difference decreases toward the venous end, because the excess oxygen that left the capillaries at the arterial end and diffused axially is available to the tissue there. In some cases a single capillary model, which neglects interaction, incorrectly indicates that portions of the tissue become anoxic, whereas including interaction shows that all of the tissue is in fact supplied with oxygen.

Dispersion of Solute by Electrokinetic Flow through Post Arrays and Wavy-Walled Channels

In this work, we simulate electrokinetically driven transport of unretained solute bands in a variety of two-dimensional spatially periodic geometries (post arrays as well as sinuous/varicose channels), in the thin Debye layer limit. Potential flow fields are calculated using either an inverse method or a Schwarz-Christoffel transform, and transport is modeled using a Monte Carlo method in the transformed plane. In this way, spurious "numerical diffusion" transverse to streamlines is completely eliminated, and streamwise numerical diffusion is reduced to arbitrary precision. Late-time longitudinal dispersion coefficients are calculated for Peclet numbers from 0.1 to 3162. In most geometries, a Taylor-Aris-like scaling law for the dispersion coefficient D(L)/D(L0) = 1 + Pe2/alpha underpredicts dispersion when Pe approximately O(alpha1/2) (here D(L0) is the effective axial diffusion coefficient in the periodic geometry). A two-parameter correlation widely used in the porous media literature, D(L)/D(L0) = 1 + Pe(n)/alpha, agrees slightly better, but much better agreement can be obtained using a new quadratic form: D(L)/D(L0) = 1 + Pe/alpha1 + Pe2/alpha2. A quasi-universal relationship for stream-wise dispersion is offered that predicts 96% of the simulation data to within a factor of 2 in all geometries studied. Comparison with previous work shows that in circular post arrays, the dispersion coefficient for electrokinetic flow is a factor of 3-10 less (depending on Pe and relative post size) than for pressure-driven flow.

Modelling and Simulation of Ternary Liquid-Phase Adsorption onto Activated Carbon

A computational procedure is presented for solving the set of rigid hyperbolic and parabolic partial differential equations describing the simultaneous adsorption of a ternary system in a column packed with adsorbent particles using the non-linear adsorption isotherms of Fritz and Schluender. The model equations account for the effects of axial diffusion in the fluid and the film and internal diffusional mass-transfer resistances of the particles. Orthogonal collocation and Michelsen's modified third-order semi-implicit Runge-Kutta method combined with a step-size adjustment strategy were used to solve the general form of the resulting 6N coupled ordinary differential equations for the simultaneous adsorption of butan-2-ol, t-amyl alcohol and phenol in fixed beds. The simulated results obtained from this model were compared with experimental data. Excellent agreement between the simulated results and previously published experimental data was obtained. The breakthrough profiles were also indicative of the...

Modeling and Simulation of Liquid-Phase Ternary Adsorption in Activated Carbon Column

A computational procedure is presented for solving the set of stiff hyperbolic and parabolic partial differential equations describing the simultaneous adsorption of a ternary system in a column packed with adsorbent particles using nonlinear adsorption isotherms of Fritz and Schluender. The model equations account for the effects of axial diffusion in the fluid and in the film and internal diffusional mass-transfer resistances of the particles. Orthogonal collocation and Michelsen's modified third-order semiimplicit Runge−Kutta methods combined with a step-size adjustment strategy are used to solve the general form of the resulting 6N coupled ordinary differential equations for simultaneous adsorption of 2-butanol, tert-amyl alcohol, and phenol in fixed beds. Simulated results obtained from this model are compared with experimental data. Excellent agreement between simulated results and previously published experimental data is obtained in 0.41- and 0.82-m adsorbers when the film mass-transfer coefficien...

Ultra thick film open tubular traps with an increased inner diameter

In this paper, the concept of open tubular traps, coated with a very thick film of polydimethyldisiloxane for enrichment of trace volatile components has been further explored. From theoretical calculations as well as practical experiments it is demonstrated that it can be advantageous to increase the inner diameter of such traps. For a given sampling flow rate and phase ratio, the plate number of the traps is not dependent on the inner diameter, provided that the linear flow velocity remains sufficiently high to offset the effect of axial diffusion. It is shown that this is due to the basic fact that for a given sampling flow rate, the average linear flow velocity in the trap is inversely proportional to the square of the inner diameter of the trap. However, in contrast to chromatographic separations, the linear flow velocity is not important. Under conditions of a constant phase ratio, an increased inner diameter also increases the amount of sorbent in the trap, which is a key parameter for obtaining high breakthrough volumes. Open tubular traps with an expanded inner diameter have very low pressure drop characteristics, which provides the possibility to construct new, simplified sampling systems.

Model of the plasma discharge in a Hall thruster with heat conduction

The inclusion of heat conduction into a one-dimensional, macroscopic model of the plasma inside a Hall thruster and in the near plume is found to smooth the temperature profile of previous solutions with a nonconductive model. The spatial structure still consists of reverse-flow, ionization, and acceleration regions. Conductive energy flow, being of the same order of convective flow, has significant effects on the rear part of the channel where it can make impossible the establishment of a steady anode sheath. As a result, there is an upper bound on the plasma reverse flow for the existence of stationary solutions. The analysis of inertial effects on the electron dynamics concludes that the main contribution is the azimuthal electron motion, which can produce extra collisionality, mainly in the near plume. The different contributions to the effective axial diffusion of electrons and the ion temperature are evaluated. A parametric investigation yields the basic scaling laws of the thruster stationary performance.

Effect of turbulence on Taylor dispersion for oscillatory flows

Experiments were carried out to explore the effect of the turbulence on Taylor dispersion for oscillatory tube flow. We found from measured velocity profiles that, while the velocity fluctuations exist over the parameter range in the present investigation, periodic conditional turbulent flow occurs and the turbulence intensity increases rapidly when the Stokes layer-based Reynolds number exceeds 535. The effective axial mass diffusivity for oscillatory flow deviate from the theoretical predictions corresponding to laminar flow as Re δ value increases, and the deviation becomes noticeable as Re δ increases to 235. It was also found that an ratio of effective diffusivity to theoretical value for laminar flow as large as 100 is available during turbulent oscillatory flows.

A Melnikov function for the break-up of closed streamlines in steady Navier–Stokes flows

A Melnikov approach is proposed to study the existence/nonexistence of closed streamlines and asymptotic orbits in solutions of the steady Navier–Stokes equations. In certain limiting cases the method gives important insight into the physical processes that cause the break-up of closed and heteroclinic streamlines. The approach sheds light on the important issue of transport in three-dimensional Navier–Stokes flows. Specifically we show that the criterion for the existence of closed streamlines is that the line integral of the viscous force, induced from the perturbation of the flow along the unperturbed closed streamline, must vanish. We apply the approach to wavy Taylor vortex flow and show that the space averaged Melnikov function is related to the effective axial diffusivity.

Predicting chaotic dispersion with Eulerian symmetry measures: Wavy Taylor-vortex flow

In a recent investigation of particle transport in numerically computed wavy Taylor-vortex flow, Rudman estimated an effective axial diffusion coefficient, Dz, to characterize the enhanced mixing due to chaotic advection [AIChE J. 44, 1015 (1998)]. We find that Dz is proportional to the product of two measures of symmetry deviation. The first is a measure of the average deviation of the flow from rotational symmetry, and the second is a measure of the average deviation from flexion-free flow (a flow where the curl of the vorticity is zero). Because these quantities are obtained directly from the velocity field, we call them Eulerian symmetry measures. Thus, we show that the macroscopic transport behavior in a flow can be quantified directly in terms of the velocity field and its gradients, and hence provides a connection between Eulerian and Lagrangian pictures of transport—a problem of fundamental and widespread interest.

Chaotic advection in bounded Navier–Stokes flows

We discuss mixing and transport in three-dimensional, steady, Navier–Stokes flows with the no-slip condition at the boundaries. The advective flux is related to the dynamics of the Navier–Stokes equations and a prediction is made of the scaling of the advective flux with the Reynolds number: the flux is expected to decay as the Reynolds number goes to infinity. This prediction is made via a Melnikov-type calculation together with boundary layer concepts through which the flow is split into an integrable and a small non-integrable part. The rate of decay is related to the details of viscous flow in boundary layers. The Melnikov function is related to the Bernoulli integral of the underlying Euler flow. The effects of molecular diffusivity are discussed and the effective axial diffusivity scaling predicted as a function of Reynolds and Péclet numbers. Using these ideas, we study the mass transport in the wavy vortex flow in the Taylor–Couette apparatus as a particular example. We propose an explanation of the observed non-monotonic behaviour of flux with increasing Reynolds number that was not captured in any of the previous models. It is shown that there is a Reynolds number at which the axial flux in the wavy vortex flow is maximized. At the low range of Reynolds numbers for which the wavy vortex flow is stable the flux increases, while for large Reynolds numbers it decreases. We compare these predictions with the available experimental and numerical data on the wavy vortex flow.

Breakthrough Curve Performance Using Plate-and-Frame Affinity-Membrane Modules

The effects of inlet flow rate on the breakthrough curve performance of lysozyme using plate-and-frame dye-affinity-membrane modules were investigated in this work. For the nonadsorption results, it is found that the influences of flow rate on both permeate and retentate breakthrough curves mainly came from the flow maldistribution and extra-module mixing effects. On the other hand, the effects of axial and radial diffusion were negligible in this case. When single-protein adsorption occurred, the elution peak height in the permeate decreased with increasing flow rate, which resulted from the limitation of convective flow. Moreover, this study investigated three larger-scale designs: a multiple-membrane stack in one module, two modules connected in parallel, and two modules in tandem. Both of the two-module designs resulted in a greater amount of eluted protein, which is attributed to their lower flow rates in the modules than the multiple-membrane-stack design. As for the two-protein breakthrough curve performance, a simple separation of lysozyme from nonadsorbable bovine serum albumin (BSA) was observed but the existence of BSA declined the lysozyme adsorption. The degree of influence on lysozyme adsorption was the same in both one-module and two-module-in-tandem designs. Furthermore, the recovered percentages of lysozyme in both one-module and two-module designs were low, and this was mainly restricted by the limitation of flow rate for the membrane modules adopted in this study.

Tunable single phase-shifted and superstructure gratings using microfabricated on-fiber thin film heaters

Tunable optical filters have applications in wavelength-division multiplexing, optical sensing, and fiber lasers. Fiber gratings that are tuned by resistive Joule heating of a thin metal film deposited onto the fiber are attractive because of their compactness, low power consumption, and controllable optical characteristics. In particular, the small effective axial thermal diffusion lengths in these systems allow complex heating profiles to be defined simply by controlling the local film thickness. In this paper, we demonstrate devices with single and multiple thermally actuated phase-shifted elements. The measured reflection spectra of these devices compare favorably with simulations based on a one-dimensional model of heat flow and confirm the heat flow characteristics expected from theory.

Effect of axial conduction on the ice crystal growth in laminar falling films

A finite volume numerical code has been developed to numerically approximate the rate of ice crystal growth in a laminar falling film flowing down a cooled vertical plate. The governing energy equation contains the phase energy as the source term. Enhancement of heat transfer as a result of suspended ice crystals is accounted for in the use of effective values of thermal conductivity, viscosity, thermal diffusivity, and specific heat as function of volumetric concentration of ice crystals in the falling film. Nusselt number, overall heat transfer coefficients between the fluid and cooled plate, and ice crystal growth rate were calculated for different film thicknesses with and without axial diffusion. Nusselt number and ice crystal growth rates were found to be dependent on film thickness. Axial diffusion effects were found to be negligible for larger film thickness (large flowrate).

Simulation of an ultrafiltration process of bovine serum albumin in hollow-fiber membranes

This work presents a numerical simulation of an ultrafiltration process of bovine serum albumin in solution, using hollow-fiber membranes. Such membranes are constituted of tiny polymer cylinders disposed in a tube-and-shell arrangement. The concentrate flows through the interior of the fibers and the pure solvent is recovered in the shell, assuming perfect solute rejection. In modeling the process, the flow of concentrate inside the fibers was considered to be laminar, with constant density, viscosity and solute diffusivity. Axial diffusion and angular effects were ignored. The model combines the effect of concentration polarization and adsorption, which are the two main limiting phenomena in ultrafiltration processes. The pressure on the shell side was considered constant and inside the fibers a linear pressure profile, dependent on the axial position, was adopted. The solution of the problem was achieved with the method of orthogonal collocation, with adequate choice of the weight function in the radial direction. In the axial direction, a finite-difference method was used. The numerical results were compared with experimental data available in the literature.

The DMA Transfer Function with Brownian Motion a Trajectory/Monte-Carlo Approach

The transfer function for the Differential Mobility Analyzer (DMA) is derived based on particle trajectories for both nondiffusing particles and diffusing particles. The effect of particle diffusion is assessed by using a Monte-Carlo method for particles of sizes 1, 3, 10, 30, and 100 nm. This approach includes both the effect of wall losses and axial diffusion. The range of validity of the Stolzenburg analysis is assessed by comparing his transfer function, the peak of his transfer function, and its dimensionless width with similar calculations based on the Monte-Carlo. For particle sizes smaller than 10 nm, the Monte-Carlo method indicates large wall losses, which result in a reduction in the peak of the transfer function by as much as a factor of 10 to 30, sensitivity to the flow-field, and skewness of the transfer function. It is shown that Stolzenburg's approximate formula for the standard deviation of the width of the transfer function agrees with Monte-Carlo simulations for particle sizes of 3 nm a...

Augmentation of axial dispersion by intermittent oscillatory flow.

The efficiency of axial gas dispersion during ventilation with high-frequency oscillation (HFO) is improved by manipulating the oscillatory flow waveform such that intermittent oscillatory flow occurs. We therefore measured the velocity profiles and effective axial gas diffusivity during intermittent oscillatory flow in a straight tube to verify the intermittency augmentation effect on axial gas transfer. The effective diffusivity was dependent on the flow patterns and significantly increased with an increase in the duration of the stationary phase. It was also found that the ratio of effective diffusivity to molecular diffusivity is two times greater than that in sinusoidal oscillatory flow. Moreover, turbulence during deceleration or at the beginning of the stationary phase further augments axial dispersion, with the effective diffusivity being over three times as large, thereby proving that the use of intermittent oscillatory flow effectively augments axial dispersion for ventilation with HFO.

Automated Simulation of Electrode Processes: Quantitative Mechanistic Analysis via Working Surface Interpolation

An efficient automated simulation of the transport-limited electrolysis in a channel flow cell, based on the multigrid method, is developed and is used to compute the working curve for the transport-limited current within the Lévêque approximation. An accurate universal two-dimensional working surface (making neither the Lévêque approximation nor ignoring axial diffusion) is presented for the steady-state transport-limited current from which the simulated response may be interpolated. The multigrid simulation is extended to include homogenous ECE kinetics and a working surface is presented for analysis of this mechanism within the Lévêque approximation. This is then modified using a Gauss−Newton method to solve the nonlinear EC2E system for which an analogous working surface is generated. The effect of axial diffusion on the measurement of rate constants for these mechanisms is discussed. The merits of bilinear, cubic convolution, bicubic spline, and artificial neural network methods are assessed as interpolation methods for working surfaces. Working surface interpolation is shown to be a viable cost-effective alternative to repeated simulation methods, especially for problems of more than one dimension. A free data-processing service using the working surface data together with a suitable interpolation method is available via the World Wide Web (http://physchem.ox.ac.uk:8000/wwwda).

Voltammetry under High Mass Transport Conditions. The High-Speed Channel Electrode and Transient Measurements

The application of the high-speed channel microband electrode to potential step transient measurements is reported. Potential step experiments, from conditions of no current flows up to a potential value where the mass transport limited current is reached, using the one-electron oxidation of N,N,N‘,N‘-tetramethyl-1,4-phenylendiamine and the one-electron reduction of p-bromonitrobenzene and p-chloronitrobenzene are reported. The applicability of a model in which the dominant mass transport form is by diffusion normal to the electrode and by convection axially through the channel is demonstrated so that axial diffusion effects are negligible. This permits the use of the backward implicit procedure to analyze the experimental transient responses incorporating any kinetic complications, as appropriate. It is shown that homogeneous rate constants of at least 1 order of magnitude higher are accessible by potential step transient measurements for an ECE mechanism as compared to steady-state experiments.

Hydrodynamic microelectrodes. The microdisk electrode: Theory and experiment

AbstractA computationally efficient fully implicit approach to the simulation of linear sweep and potential step voltammetry at a channel microdisk electrode is derived. The generality of the method is illustrated with reference to the simple case of a reversible one electron transfer reaction. The influence of flow rate and the effects of axial, lateral and normal diffusion upon the electrolysis current are quantified. Scan rate and electrode geometry dependences are established and the requirements for measurement of true steady state hydrodynamic voltammograms defined.