Advection-dispersion Mass Transport

As a consequence of mixing different composition waters the rate of migration components in these waters may substantially differ from the seepage velocity of flow per se, which shows up in their dispersion. Such migration of individual components in ground water composition accompanied by 

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Figure 3.30 Infinitely small volume of sediment and direction of liquids flow in it according the model of advection-dispersion mass transport...

This is the base equation of advection-dispersion (or advective-dispersive) mass transport. 

The solution of equations (2.40) and (2.41) with the total accounting of all acting factors is quite complex. So the problem is simplified by excluding secondary factors, which may be disregarded. The exact solution of the migration problem of individual component in conditions of advective-dispersive mass transport without approximations is called the analytical solution. 

Then the hydrogeochemical part for the distribution of individual components dissolved in ground water is solved. The analytical solution of mass transport problems includes the introduction of edge and initial conditions of the hydrochemical object and selection of advection-dispersion mass transport equations matching the assigned conditions and mathematical solution of the equation themselves. 

Edge conditions. At the base of an analytical solution lies reviewed above advective-dispersive mass transport equation. For its solution it is necessary to have boundary and initial conditions, i.e., conditions at which the process begins, operates and ends. 

Figure 3.31 Options of boundary conditions at solving problems of advective-dispersion mass transport.

For this reason the component, which is subjected simultaneously to linear adsorption or decomposition (which is independent of the form of its existence - in dissolved and adsorbed phases), will have the equation of advective-dispersive mass transport (3.39) in the format... 

The retardation factor is very convenient for the analytical solutions of advective-dispersive mass transport equations. If there is a solution for chemically passive non-sorbing component, in case of sorption it is sufficient to replace the water seepage velocity with V, . As D. V 6, and t = x/Vg, Baetsle s equation (3.62) will assume the format... 

At using these models chemical properties in water-dissolved components have great significance. Most nonpolar components do not participate in chemical reactions and mass exchange with rocks. For this reason modeling of their distribution processes of the chemical interaction, as a rule, are disregarded. Major factors in the change of their concentration in water turn out flow velocity and hydrodynamic dispersion. That is why the reviewed models for chemically passive nonpolar components often maybe solved analytically by equations of advective-dispersive mass transport. 

Analytical solutions are those whose precision depends only on the accuracy of the initial data. They do not contain errors associated with the approximation due to simplification of the computation process. This approach is applicable to the simplest models, which are often represented by a restricted number of relatively simple equations. These may be solved without specialized program software. Analytical solutions, as a rule, are used in modeling of processes with minimum participation of chemical reactions, in particular in the analysis of distribution of nonpolar components, radioactive decay, adsorption, etc. Under such conditions for modeling often are sufficient equations of advective-dispersive mass-transport, which are included in the section Mixing and mass-transport . 

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. 

As contaminant transport occurs over times much greater than the times over which groundwater flow fluctuates, steady flow is frequently assumed. For steady groundwater flow in three dimensions, the following vector equation, developed based on mass conservation principles, is typically used to model advective/ dispersive transport of a dissolved reactive contaminant (after [53]) ... 

Mathematical models for mass transfer at the NAPL-water interface often adopt the assumption that thermodynamic equilibrium is instantaneously approached when mass transfer rates at the NAPL-water interface are much faster than the advective-dispersive transport of the dissolved NAPLs away from the interface [28,36]. Therefore, the solubility concentration is often employed as an appropriate concentration boundary condition specified at the interface. Several experimental column and field studies at typical groundwater velocities in homogeneous porous media justified the above equilibrium assumption for residual NAPL dissolution [9,37-39]. 

Other models directly couple chemical reaction with mass transport and fluid flow. The UNSATCHEM model (Suarez and Simunek, 1996) describes the chemical evolution of solutes in soils and includes kinetic expressions for a limited number of silicate phases. The model mathematically combines one- and two-dimensional chemical transport with saturated and unsaturated pore-water flow based on optimization of water retention, pressure head, and saturated conductivity. Heat transport is also considered in the model. The IDREAT and GIMRT codes (Steefel and Lasaga, 1994) and Geochemist s Workbench (Bethke, 2001) also contain coupled chemical reaction and fluid transport with input parameters including diffusion, advection, and dispersivity. These models also consider the coupled effects of chemical reaction and changes in porosity and permeability due to mass transport. 

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). 

In fixed-bed operation, in addition to the two-step mass transport mechanism, advection and dispersion play key roles in ion exchange. These factors must be considered. As influent concentration is assumed low, solution velocity can be considered constant. If pore diffusion is an important factor in the ion uptake, the following equations can be used. Similar expressions for surface diffusion can be obtained ... 

So far, the concept of mass conservation has been applied to large, easily measurable control volumes such as lakes. Mass conservation also can be usefully expressed in an infinitesimal control volume, mathematically considered to be a point. Conservation of mass is expressed in such a volume with the advection—dispersion-reaction equation. This equation states that the rate of change of chemical storage at any point in space, dC/dt, equals the sum of both the rates of chemical input and output by physical means and the rate of net internal production (sources minus sinks). The inputs and outputs that occur by physical means (advection and Fickian transport) are expressed in terms of the fluid velocity (V), the diffusion/dispersion coefficient (D), and the chemical concentration gradient in the fluid (dC/dx). The input or output associated with internal sources or sinks of the chemical is represented by r. In one dimension, the equation for a fixed point is... 

We are often concerned with the dispersion of pollutants and other chemicals in the environment. Advection and mass flux are indiscriminate transport processes. In the water column of a lake, for example, these processes transport dissolved and particle-bound chemicals equally across the boundaries of the test volume. Settling of particles, in contrast, causes a downward flux of particle-bound chemicals while leaving dissolved chemicals in place. Similarly, surfactants or gases that join rising air bubbles are carried to the surface. These discriminate transport processes are very important in a variety of environmental situations ... 

The three principal variables considered in this analysis are temperature, which influences all biological and chemical reactions, dispersion and advective flow, which are the primary mass transport mechanisms in a natural body of water, and solar radiation, the energy source for the photosynthetic growth of the phytoplankton. 

Minimum mass transport occurs even in the absence of water filtration. We will review uni dimensional (linear) mass transport. If V = 0 and q, = 0, advective-dispersive equation acquires the format of a linear equation Fick s second law (equation 3.12). 

It follows from this that all solutions of mass transport problems taking into account only advective-dispersive dispersion are also applicable for cases of linear sorption (Henry s sorption isotherm). It is sufficient for... 

In addition to the mass transfer and transformation mechanisms that occur during air sparging, VOC transport mechanisms play a role in ensuring remediation. The important transport mechanisms that occur during air sparging include advection, dispersion, and diffusion. 

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