Dispersion, hydrodynamic microscopic

Hydrodynamic Dispersion. Hydrodynamic dispersion is a movement process of solutes to even out the concentration or composition of the solutes throughout the solution, which results from the microscopic non-uniformity of flow velocity in the matrix conducting pores. Hydrodynamic dispersion is a process that differs from diffusion in its mechanism but which tends to produce an analogous or synergetic effect to diffusion ... 

Hydrodynamic Dispersion atthe Microscopic Pore Scale... 

The dispersion relations of the two modes are depicted in Fig. 2. The reactive mode is one of the kinetic modes existing beside the hydrodynamic modes such as the diffusive mode. Here also, we may wonder if these modes can be justified from the microscopic dynamics. 

Effective rates of sorption, especially in subsurface systems, are frequently controlled by rates of solute transport rather than by intrinsic sorption reactions perse. In general, mass transport and transfer processes operative in subsurface environments may be categorized as either macroscopic or microscopic. Macroscopic transport refers to movement of solute controlled by movement of bulk solvent, either by advection or hydrodynamic (mechanical) dispersion. For distinction, microscopic mass transfer refers to movement of solute under the influence of its own molecular or mass distribution (Weber et al., 1991). 

Membrane Formation. In earlier work. 2.) it was found that fumed silica particles could be dispersed in aqueous suspension with the aid of ultrasonic sound. Observations under the electron microscope showed that the dispersion contained disc-like particles, approximately 150-200 1 in diameter and 70-80 1 in height. Filtration experiments carried out in the "dead-end" mode (i.e., zero crossflow velocity) on 0.2 urn membrane support showed typical Class II cake formation kinetics, i.e., the permeation rate decreased according to equation (12). However, as may be seen from Figure 7, the decrease in the permeation rate observed during formation in the crossflow module is only t 1, considerably slower than the t 5 dependence predicted and observed earlier. This difference may be expected due to the presence of lift forces created by turbulence in the crossflow device, and models for the hydrodynamics in such cases have been proposed. 

Mixing due to the microscopic distribution of pore water velocities, known as hydrodynamic dispersion, is illustrated in Fig. 3-2. Figure 3-2A shows the pore water velocity profile, v(y) for Poiseuille flow in a uniform capillary tube of radius r. Additional spreading can occur due to the irregular shape of natural pores as compared to ideal cylinders (Fig. 3-2B). 

Transport Chemicals are moved or transported by advection - moving water carries the dissolved solids with it - and hydrodynamic dispersion - the spreading and mixing caused in part by molecular diffusion and microscopic variation in velocities within individual pores (Mercer and Faust, 1981). 

A dilute polymer solution is a system where polymer molecules are dispersed among solvent molecules. An assumption common to any existing theory for flow properties of polymer solutions is that the structure of solvent molecules is neglected and the solvent is assumed to be replaced by a continuous medium of a Newtonian nature. Thus, macroscopic hydrodynamics may be used to describe the motion of the solvent. Recently, some ordering or local structure of solvent molecules around a polymer chain has been postulated as an explanation of the stress-optical coefficient of swollen polymer networks (31,32) so that the assumption of a solvent continuum may not apply. The high frequency behavior shown in Chapter 4 could possibly due to such a microscopic structure of the solvent molecules. Anyway, the assumption of the continuum is employed in every current theory capable of explicit predictions of viscoelastic properties. In the theories of Kirkwood or... 

In practice, this local scale is considered to correspond to the size of the characterization techniques of local soil properties, let s say a small laboratory column. As such the microscopic pore scale variability is no longer explicitly modelled but encoded through effective flow and transport properties at the macroscopic level. The effective macroscopic properties contain of course the signature of the lower level microscopic variability. As such macroscopic effective moisture retention function, hydraulic conductivity or hydrodynamic dispersivity is determined by microscropic pore size distribution, connectivity and tortuosity within the macroscopic sample. 

A characteristic of the CDE travel time distribution is that the variance of the travel times grows linearly with travel distance z. This is equivalent to the particle location distribution, which grows linearly with time for a Brownian motion process. As such, it is essential in the derivation of (12) that the hydrodynamic dispersion can be described as a diffusion process, i.e. on average, all solute particles are subjected to the same forces and the transport time is sufficiently large so that the incremental microscopic particle displacements are no longer statistically correlated. As a corollary, the CDE process cannot be valid for small soil volumes where the travel times are too small as compared to the mixing time, or to describe transport close to interfaces. 

As we consider simultaneous fluid flow and heat transfer in porous media, the role of the macroscopic (Darcean) and microscopic (pore-level) velocity fields on the temperature field needs to be examined. Experiments have shown that the mere inclusion of u0 V(T) in the energy equation does not accurately account for all the hydrodynamic effects. The pore-level hydrodynamics also influence the temperature field. Inclusion of the effect of the pore-level velocity nonuniformity on the temperature distribution (called the dispersion effect and generally included as a diffusion transport) is the main focus in this section. 

In view of the very slow time decay of the time correlation functions, it is not clear to what extent the Navier-Stokes transport coefficients can be used even in three dimensions to describe phenomena that vary on a time scale of 50tc, for on this time scale there is not yet a clear separation of microscopic and macroscopic effects. However, usually the Navier-Stokes equations are applied to phenomena that vary on a much longer time scale, and then the slow decay of the correlation functions does not interfere with the hydrodynamic processes. Nevertheless, the divergences of the Burnett and higher-order transport coefficients do appear to have experimental consequences even for three-dimensional systems. In particular, it appears that the dispersion relation for the sound wave frequency wave number k can no longer be expressed as a power series in k as was done in Eq. (133) but instead that fractional powers of the form for /i = l,2,... 

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