Dispersion, hydrodynamic coefficient

Note that Eq. 10.5 is written to allow the velocity to vary as a function of location typical application of the advection-dispersion equation assumes the velocity and the hydrodynamic coefficients to be constant. Moreover, the time dependence of these parameters arises when flow (infiltration) is unsteady or transient in these cases, the contact time between contaminants and the solid matrix (and any immobile water within it) is too short to allow an equilibrium to be reached. 

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... 

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. 

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< )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 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. 

Dielectric dispersion measurements also provide a means of determining rotational diffusion coefficients or mean rotational relaxation times of solute molecules. In principle, data for these hydrodynamic quantities can be used for a... 

Moreover, the influence of the motions of the particles on each other (i.e., when the motion of a particle affects those of the others because of communication of stress through the suspending fluid) can also influence the measured diffusion coefficients. Such effects are called hydrodynamic interactions and must be accounted for in dispersions deviating from the dilute limit. Corrections need to be applied to the above expressions for D and Dm when particles interact hydrodynamically. These are beyond the scope of this book, but are discussed in Pecora (1985), Schmitz (1990), and Brown (1993). 

The most convenient of these methods is viscosity measurement of a liquid in which particles coated with a polymer are dispersed, or measurement of the flow rate of a liquid through a capillary coated with a polymer. Measurement of diffusion coefficients by photon correlation spectroscopy as well as measurement of sedimentation velocity have also been used. Hydrodynamically estimated thicknesses are usually considered to represent the correct thicknesses of the adsorbed polymer layers, but it is worth noting that recent theoretical calculations52, have shown that the hydrodynamic thickness is much greater than the average thickness of loops. 

Dispersion arises from the fact that, even in a relatively homogenous porous medium, small-scale heterogeneities exist which cause airflow to proceed along various channels at different rates. Barometric pumping causes a significant increase in the coefficient of hydrodynamic dispersion over a pure diffusion-based transport model, thus increasing the overall transport rate. 

For dilute dispersions of spherical particles, the diffusion coefficient can be related to the hydrodynamic diameter of the particles by the Stokes-Einstein equation... 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

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