Dispersion hydrodynamic

Hydrodynamic dispersion represents the combined action of flow and molecular self-diffusion, resulting in superimposed displacements of the molecules. The situation is characterized by the Peclet number [41] defined by 

The tortuosity of flow streamlines causes an additional source of incoherence that must be accounted for at large Peclet numbers. 

For displacements shorter than the mean pore dimension, (z2) a, where flow velocities tend to be spatially constant and homogeneously distributed, Brownian diffusion is the only incoherent transport phenomenon that contributes to the hydrodynamic dispersion coefficient. As a direct consequence, the dispersion coefficient approaches the ordinary Brownian diffusion coefficient, 

Time intervals permitting displacement values in the scaling window a ) l are related with the tortuous flow as a result of random positions of the obstacles in the percolation model [4]. Hydrodynamic dispersion then becomes effective. For random percolation clusters, an anomalous, i.e., time dependent dispersion coefficient is expected according to 

The exponent turned out to be x 1. This finding demonstrates that coherent flow determines transport in the mechanical dispersion regime and that diffusion is negligible under such conditions. For a discussion also see Ref. [43]. 

Mass transfer of a solute dissolved in a fluid is not only the fundamental mechanism of mixing processes, it also determines the residence-time distribution in micro fluidic systems. As mentioned in Section 1.4, in many applications it is desir- 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation  

The analysis of Taylor and Aris was extended to arbitrary time values by Gill and Sankarasubramanian [151] for the dispersion of an initially plug-like profile, i.e. 

While the previous studies refer to straight channels exceptionally, microfluidic devices often comprise channels with a curvature. It is therefore helpful to know how hydrodynamic dispersion is modified in a curved channel geometry. This aspect was investigated by Daskopoulos and Lenhoff [155] for ducts of circular cross- 

As mentioned earlier, in curved channels a secondary flow pattern of two counter-rotating vortices is formed. Similarly to the situation depicted in Figrue 2.43, these vortices redistribute fluid volumes in a plane perpendicular to the main flow direction. Such a transversal mass transfer reduces the dispersion, a fact reflected in the dependence in Eq. (108) at large Dean numbers. For small Dean numbers, the secondary flow is negligible, and the dispersion in curved ducts equals the Taylor-Aris dispersion of straight ducts. 

In this expression, c denotes the concentration averaged over the cross section of the tube, u the average velocity and De an effective diffusivity, also denoted dispersion coefficient, which is given by 

In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary fiow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form 

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Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Hydrodynamic Dispersion Macroscopic dispersion is produced in a capillar) even in tlie absence of molecular diffusion because of the velocity profile produced by the adherence of the fluid to tlie wall. Tlris causes fluid particles at different radial positions to move relative to one anotlier, witli tlie result tliat a series of mixing-cup samples at tlie end of tlie capillary e.xhibits dispersion. 

Column efficiency (number of theoretical plates) As in batch chromatography, one needs to determine the efficiency of the column in order to evaluate the dispersion of the fronts due to hydrodynamics dispersion or kinetics limitations. The relationship of N proportional to L can be expressed in terms of the equation for height equivalent to a theoretical plate (HETP) as ... 

Optimum flowrates, resulting in high productivity and low eluent consumption, are estimated first for an ideal system , which means that kinetic and hydrodynamic dispersive effects are assumed to be negligible [46]. This procedure has recently been improved [57]. 

Fig. 2.9.7 Hahn spin-echo rf pulse sequence combined with bipolar magnetic field gradient pulses for hydrodynamic-dispersion mapping experiments. The lower left box indicates field-gradient pulses for the attenuation of spin coherences by incoherent displacements while phase shifts due to coherent displacements on the time scale of the experiment are compensated. The box on the right-hand side represents the usual gradient pulses for ordinary two-dimensional imaging. The latter is equivalent to the sequence shown in Figure 2.9.2(a).

In order to avoid flow artifacts it may be advisable to replace the spatial encoding pulses (right-hand box) by velocity compensated pulses such as shown in Figure 2.9.4(e) for phase encoding. The amplitude of the Hahn spin-echo is attenuated by hydrodynamic dispersion. Evaluation of the echo attenuation curve for fixed intervals but varying preparation gradients (left box) permits the allocation of a hydrodynamic dispersion coefficient to each voxel, so that maps of this parameter can be rendered. 

In most instances, hydrodynamic dispersion is not great enough to require detailed consideration in hydrogeologic modeling for fate assessment of deep-well-injected wastes. Flowever, regional variations (such as the presence of an USDW in the same aquifer as the injection zone, as is the case in parts of Florida) should be evaluated before a decision is made to exclude it. 

Chemical mass is redistributed within a groundwater flow regime as a result of three principal transport processes advection, hydrodynamic dispersion, and molecular diffusion (e.g., Bear, 1972 Freeze and Cherry, 1979). Collectively, they are referred to as mass transport. The nature of these processes and how each can be accommodated within a transport model for a multicomponent chemical system are described in the following sections. 

Hydrodynamic dispersion is in many cases taken to be a Fickian process, one whose transport law takes the form of Fick s law of molecular diffusion. If flow is along x only, so that vx = v and vy = 0, the dispersive fluxes (mol cm-2 s-1) along x and y for a component i are given by,... 

Molecular diffusion (or self-diffusion) is the process by which molecules show a net migration, most commonly from areas of high to low concentration, as a result of their thermal vibration, or Brownian motion. The majority of reactive transport models are designed to simulate the distribution of reactions in groundwater flows and, as such, the accounting for molecular diffusion is lumped with hydrodynamic dispersion, in the definition of the dispersivity. 

The accounting for diffusion in these models, in fact, is in many cases a formality. This is because, as can be seen from Equations 20.19 and 20.21, the contribution of the diffusion coefficient D to the coefficient of hydrodynamic dispersion D is likely to be small, compared to the effect of dispersion. If we assume a dispersivity a of 100 cm, for example, then the product av representing dispersion will be larger than a diffusion coefficient of 10-7-10-6 cm2 s-1 wherever groundwater velocity v exceeds 10 9-10-8 cm s 1, or just 0.03-0.3 cm yr-1. 

We might properly refer to this value as the apparent Peclet number, because by many formal definitions the Peclet number accounts for the relative importance of advection and molecular diffusion, without mention of hydrodynamic dispersion. 

Truncation error arises from approximating each of the various space and time derivatives in the transport equation. The error resulting from the derivative in the advection term is especially notable and has its own name. It is known as numerical dispersion because it manifests itself in the calculation results in the same way as hydrodynamic dispersion. 

To see why numerical dispersion arises, consider solute passing into a nodal block, across its upstream face. Over a time step, the solute might traverse only a fraction of the block s length. In the numerical solution, however, solute is distributed evenly within the block. At the end of the time step, some of it has in effect flowed across the entire nodal block and is in position to be carried into the next block downstream, in the subsequent time step. In this way, the numerical procedure advances some of the solute relative to the mean groundwater flow, much as hydrodynamic dispersion does. 

The physical transport of dissolved organic compounds through the subsurface occurs by three processes advection, hydrodynamic dispersion, and molecular diffusion. Together, these three cause the spread of dissolved chemicals into the familiar plume distribution. Advection is the most important dissolved chemical migration process active in the subsurface, and reflects the migration of dissolved chemicals... 

Hydrodynamic dispersion refers to the tendency of a solute or chemical dissolved in the fluid, to spread out over time (i.e., to become dispersed in the subsurface). The mechanical component of dispersion results from the differential flow of the fluid through pore spaces that are not the same size or shape, and from different flow velocities and the fluid near the walls of the pore where the drag is greatest vs. the fluid in the center of the pore (Figure 5.4). 

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