Convective dispersion diffusion

The description of transport is closely connected to the terms convection, diffusion dispersion, and retardation as well as decomposition. First, it is assumed that there are no interactions between the species dissolved in water and the solid phase, through which the water is flowing. Moreover, it is assumed that water is the only fluid phase. The multiphase flow water-air, water-organic phase (e g. oil or DNAPL) or water-gas is not considered here. 

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

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 movement of synthetic pyrethroids in soil and sediment is basically controlled by diffusion, convection, and dispersion. When entering a water-sediment system... 

Convection + Axial Dispersion -t- Mass Transfer Resistance + Particle Diffusion + Kinetics of Adsorption... 

The use of the Coanda effect is based on the desire to have a second passive momentum to speed up mixing in addition to diffusion [55, 163], The second momentum is based on so-called transverse dispersion produced by passive structures, which is in analogy with the Taylor convective radial dispersion ( Taylor dispersion ) (see Figure 1.180 and [163] for further details). It was further desired to have a flat ( in-plane ) structure and not a 3-D structure, since only the first type can be easily integrated into a pTAS system, typically also being flat A further design criterion was to have a micro mixer with improved dispersion and velocity profiles. 

Most commonly, distributed parameter models are applied to describe the performance of diesel particulate traps, which are a part of the diesel engine exhaust system. Those models are one- or two-dimensional, non-isothermal plug-flow reactor models with constant convection terms, but without diffusion/dispersion terms. 

Sorption by sediment and suspended solids Sedimentation and resuspension of solids Aerosol formation at the air-water interface Uptake and release by biota Transport within water bodies Turbulent dispersion and convection Diffusion between upper mixed layer and bottom layer Transformation Biodegradation Photochemical degradation... 

The third term in mass transport is dispersion. The dispersion describes the mass flow, which results from velocity variations due to the geometry and the structure of the rock system. From this definition it follows that the smaller the vector of convection the smaller the effect of dispersion. The other way round, an increasing effect of dispersion occurs with higher flow velocity. Consequently the mathematical description of the species distribution is an overlap of convection, diffusion, and dispersion. 

Convection, diffusion, and dispersion can only describe part of the processes occurring during transport. Only the transport of species that do not react at all with the solid, liquid or gaseous phase (ideal tracers) can be described adequately by the simplified transport equation (Eq. 94). Tritium as well as chloride and bromide can be called ideal tracers in that sense. Their transport can be modeled by the general transport equation as long as no double-porosity aquifers are modeled. Almost all other species in water somehow react with other species or a solid phase. These reactions can be subdivided into the following groups, some of which have already been considered in the previous part of the book. 

The dispersion of a non-reactive solute in a circular tube of constant cross-section in which the flow is laminar is described by the convective-diffusion equation... 

Yeager and co-workers [379, 419] studied the kinetics of peroxide decomposition on dispersed oxide powders by measuring changes in the convective-diffusion limiting current for peroxide oxidation at a rotating gold electrode immersed in the liquid dispersion. The current decay was observed to be proportional to the peroxide concentration decay due to the catalytic decomposition process. On perovskite [379] and spinel oxides [419], it follows first-order kinetics described by the equation... 

The <5paantity A j is the effective diffusion coefficient for the porous media and A is the dimensionless effective diffusion coefficient B characterizes the contribution of convection to dispersion G characterizes the ratio of the excess gravitational forces (attributable to the density difference between the... 

Axial Convective Diffusion. The variation in width, length and direction of individual channels formed by the interstices of the packing give rise to a dispersion which can be characterized by the dimensionless Bodenstein number. Bo, which is a similar number as the Peclet number but with the particle diameter as characteristic dimension... 

Now if diffusion/dispersion processes that mix fluid elements are superimposed on the convective flow in the axial direction (z direction), then the total flow rate can be written as ... 

The dispersion model is also used to describe nonideal tubular reactors. In this model, there is an axial dispersion of the material, which is governed by an analogy to Pick s law of diffusion, superimposed on the flow. So in addition to transport by bulk flow, UAqC, every component in the mixture is transported through any cross section of the reactor at a rate equal to [—DaAddCldz)] resulting from molecular and convective diffusion. By convective diffusion we mean either Aris-Taylor dispersion in laminar flow reactors or turbulent diffusion resulting from turbulent eddies. 

Chemicals can also be transported within and between compartments. Transport may be via convection, diffusion, or bulk transport. Convection occurs when environmental contaminants dissolved or dispersed in air or water are carried along by air or water currents. Diffusion through air or water occurs relatively slowly, and is of importance only over small distances. However, diffusion (as well as... 

Dispersion is most commonly modeled as a diffusive process. For flow in a packed column, dispersion is captured by the DV C term in the one-dimensional convection-diffusion equation. The longitudinal dispersion coefficient, D, is a function of the Peclet number, Pe = vR/Dj (where is the molecular... 

Moreover, in the last decades a lot of work has been focused on designing explicit schemes for passive scalars. These schemes have been applied solving the transport equations for the scalar variables in the the weather forecast models (e.g., [186]), and especially for the air pollution dispersion models solving the convection-diffusion equation (e.g., [187, 10, 85, 137]). 

Equation 2.20 is the advection-dispersion (AD) equation. In the petroleum literature, the term convection-diffusion (CD) equation is used, or simply diffusion equation (Brigham, 1974). When a reaction term is included, the term advection-reaction-dispersion (ARD) equation is used elsewhere. When the adsorption term is expressed as a reaction term, the ARD equation is as discussed later in Section 2.4. Several solutions of Eq. 2.20 have been presented in the literature, depending on the boundary conditions imposed. In general, they are various combinations of the error function. When the porous medium is long compared with the length of the mixed zone, they all give virtually identical results. 

Diffusion, convection, and dispersion all contribute to the spread of a front. Let us see how much each mechanism contributes to the spread. First, let us see when the diffusion transport is important as compared to the convective transport. We use v2Dot to calculate the spreading distance from a point source 68% of the injected source is within this distance. Table 2.2 shows the results for different time periods compared with the traveled distances during the same time periods by a convective flow of 1 m/day. A typical flow rate in petroleum reservoirs is 1 m/day (interstitial velocity). A typical value of diffusion coefficient of 4 X 10 mVs in a porous medium is used. In the first 5 seconds, the diffusive transport is more important than the convective transport. Soon after, the convective flow becomes the dominant mechanism. 

The solute flux of the pesticide consists of two terms. The first term corresponds to the convective or bulk transport of the pesticide with the moving soil solution the second, a diffusion-dispersion term, accounts for the random thermal motion of the pesticide molecules (19) as well as any hydrodynamic dispersion that may occur due to variations in the pore water velocity (20). The mathematical representation of Jg is... 

The convective—diffusive model underwent further refinements, as discussed in 5.2.2. It permits a quantitative description of sample dispersion in unsegmented flow analysis and provides a good simulation to assist system design and method implementation. Other quantitative models have also been proposed for specific applications, and in this regard, the tanks-in-series model should be highlighted. 

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