Fluid average linear velocity

Advection is the transport of dissolved contaminant mass due to the bulk flow of groundwater, and is by far the most dominant mass transport process [2]. Thus, if one understands the groundwater flow system, one can predict how advection will transport dissolved contaminant mass. The speed and direction of groundwater flow may be characterized by the average linear velocity vector (v). The average linear velocity of a fluid flowing in a porous medium is determined using Darcy s Law [2] ... 

G is the ratio of the dimensionless number SD in the disturbed region to its value in the normally-operating part of the bed. SD contains the activation energy, heat of reaction, inlet temperature and bed height, all of which have fixed constant values in all regions of the bed. It also contains the possibly variable quantities C, k and F. C is the average heat capacity of the fluid, and depends on the local phase ratio. kg is the specific rate constant, and depends on the local catalyst density and the phase holdup. F is the local average linear velocity, which can vary from point to point for a variety of reasons. 

Figure 33-2 shows plots of plate heights // as a function of average linear velocity u in cm/s for high-performance liquid chromatography and supercritical-fluid chromatography. In both cases, the solute was pyrene, and the stationary phase was a reversed-phase octadecyl silane maintained at 40°C. The mobile phase for HPLC was acetonitrile and water, while for SFC the mobile phase was carbon dioxide. These conditions yielded about the same retention factor (k) for both mobile phases. Note that the minimum in plate height occurred at a flow rate of 0.13 cm/s... 

Figures 10.1(a) and (e) represent the results for non-reactive advection (plug flow) and transport, i.e., advection plus dispersive mixing. There are no reactions between the solutes and aquifer solid matrix all solutes are conservative in the simulations. Advection is the process by which the moving water carries solute with it at the same velocity. In the ADVECTION option, the CaCl2 solution (called SOLUTION 0) is added to cell 1. This simply displaces the (K, Na)NC>3 solution to the next cell, and so on, as more SOLUTION 0 is added, until theCaCl2 solution appears ( breaks through ) at cell 40 after 1.0 pore volume. This indicates that the concentration fronts travel at the same velocity as the average linear velocity of the pore fluid. All the initial pore fluids in the column that contain Na, K, and NO(j" are displaced by the infilling fluid. The column contains no more Na, K, or NCXj", and is entirely filled with the CaCl2 solution. An abrupt interface between the infilling fluid and initial pore fluids is developed.

We can see in Figures 10.1 (b) and (f) that the concentration fronts are now retarded because of the reactions, compared with non-reactive transport. Na+ in the column is completely displaced in about 1.5 pore volumes in the advective-reactive transport simulation versus 1.0 pore volume in the advective transport simulation. Because K+is bound to the exchanger more strongly, it is flushed out of the column later, at about 2 pore volumes. At this point, the column is saturated with Ca and the Ca front starts to appear in the effluent. Now due to ion-exchange reactions, the Ca front has traveled at about one-half of the average linear velocity of pore fluids. We can say Ca has a retardation factor of about 2 under these transport conditions. 

One of the fundamental assumptions in fluid mechanical formulations of Newtonian flow past solids is the continuity of the tangential component of velocity across a boundary known as the "no-slip" boundary condition (BC) [6]. Continuum mechanics with the no-slip BC predicts a linear velocity profile. However, recent experiments which probe molecular scales [7] and MD simulations [8-10] indicate that the BC is different at the molecular level. The flow boundary condition near a surface can be determined from the velocity profile. In molecular simulations, the velocity profile is calculated in a simitar way to the calculation of the density profile. The region between the walls is divided into a sufficient number of thin slices. The time averaged density for each slice is calculated during a simulation. Similarly, the time averaged x component of the velocity for all particles in each slice is determined. The effect of wall-fluid interaction, shear rate, and wall separation on velocity profiles, and thus flow boimdary condition will be examined in the following. 

System assumptions that should be valid for such applications include fluid flow in the porous media is isotropic and adsorption is fast, reversible, and linear (cf. Freeze and Cherry 1979). Given these constraints, the comparative transport of a conserved (nonadsorbed) tracer, such as Br , and an adsorbed or retarded species, such as Am, can be described as shown in Fig. 10.29. A comparison of migration distances of the two species after time t, is made at concentrations where C(measured)/Co(initial) = 0.5 for the conserved and adsorbed species. The migration distance X of the conserved species after time r is a measure of the average groundwater velocity (U), or X = vt. Similarly, the migration distance of the adsorbed species (X,) i related to its velocity of movement (v ) by Xf = vj. The retardation factor (/tj for the adsorbed species is then given by... 

Under laminar flow conditions, Poiseuille flow must be considered [5]. This means that the linear velocity of the fluid element at the centre of the tube is about twice the average linear flow velocity (), whereas the velocity of fluid elements adjacent to the inner walls of the tubing... 

During turbulent flow, the linear velocity of each fluid element is the vector sum of their individual velocities. As a consequence of the chaotic displacement, the linear velocity tends to be the same for all the fluid elements and approaches the average linear flow velocity (Fig. 3.1, lower). In real situations, the linear velocities of fluid elements near the tubing walls are slightly decreased due to frictional energy losses. However, due to the chaotic movement, each fluid element only stays near the tubing wall for a very short time. 

Darcy velocity of the fluid phase average linear pore fluid velocity coordinates... 

The density profile for the micropore fluid was determined as In the equilibrium simulations. In a similar way the flow velocity profile for both systems was determined by dividing the liquid slab Into ten slices and calculating the average velocity of the particles In each slice. The velocity profile for the bulk system must be linear as macroscopic fluid mechanics predict. 

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