Diffusion and Dispersion

Diffusion and dispersion processes can be characterised by a time constant for the process, given by 

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Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases. 

Whitaker, S, Diffusion and Dispersion in Porous Media, AIChE Journal 13, 420, 1967. Whitaker, S, Introduction to Fluid Mechanics Kreiger Malabar, FL 1968. 

J. P., Matlosz, M., MicroChannel reactors for kinetic measurement influence of diffusion and dispersion on experimental accuracy, in Matlosz, M., Ehreeld, W, Baselt, J. P. (Eds.), Microreaction Technology - IMRET 5 Proc. 5th International Conference on Microreaction Technology, Springer-Verlag, Berlin (2001), pp. 131-140. 

We see in Equations 20.24 and 20.25 that two types of terms contribute to the evolution of solute concentration those representing the effects of advection, and those accounting for diffusion and dispersion. The non-dimensional Peclet number1 represents the importance of these processes, relative to one another. 

Two numerical methods have been used for the solution of the spray equation. In the first method, i.e., the full spray equation method 543 544 the full distribution function / is found approximately by subdividing the domain of coordinates accessible to the droplets, including their physical positions, velocities, sizes, and temperatures, into computational cells and keeping a value of / in each cell. The computational cells are fixed in time as in an Eulerian fluid dynamics calculation, and derivatives off are approximated by taking finite differences of the cell values. This approach suffersfrom two principal drawbacks (a) large numerical diffusion and dispersion... 

Diffusion and dispersion occur in both the liquid and vapor phases. In the vadose zone, liquid-phase dispersion is a major factor following a rainfall (or irrigation) event. Dispersion of vapor phase can also occur. Driving forces for vapor and dispersion are gravity and variations of atmospheric pressure (or SVE/air sparge operations). 

Fig. 10.1 Effect of different mechanisms on behavior of contaminants advancing through a column of porous material the relative concentration is given by c/c. (a) temporal breakthrough curves at the column outlet, showing effects of diffusion and dispersion (b) spatial concentration profiles along the column, at different times (c) spatial concentration profiles illustrating effects of retardation caused by contaminant absorption.

We shall define a closed vessel to be one for which fluid moves in and out by bulk flow alone. Plug flow exists in the entering and leaving streams. In a closed vessel diffusion and dispersion are absent at entrance and exit so that we do not, for example, have material moving upstream and out of the vessel entrance by swirls and eddies. 

Describe the difference between diffusion, turbulent diffusion, and dispersion. 

The diffusion coefficient is often not a function of distance, such that equation (2.13) can be further simplified by putting the constant value diffusion coefficient in front of the partial derivative. However, we will also be substituting turbulent diffusion and dispersion coefficients for D when appropriate to certain applications, and they are not always constant in all directions. Therefore, we will leave the diffusion coefficient inside the brackets for now. 

Dispersion is the enhanced mixing of material through spatial variations in velocity. When it is of interest (when we are not keeping track of the three-dimensional mixing), dispersion is typically one or two orders of magnitude greater than turbulent diffusion. The process of dispersion is associated with a spatial mean velocity. The means used in association with diffusion, turbulent diffusion, and dispersion are identified in Table 6.2. 

Explain the difference and similarity between turbulent diffusion and dispersion. 

In order to be consistent with other chapters, R(Ct) is defined as a positive number if the chemical is produced in the river and T(Ct) is positive if the net flux is directed from the river into the atmosphere or sediment. Note that (F(Ct) is a flux per unit volume its relation to the usual flux per area as defined, for instance, in Chapter 20, is given below (Eq. 24-15). Again we suppress the compound subscript i wherever the context is clear. The subscript Lagrange refers to what fluid dynamicists call the Lagrangian representation of the flow in which the observer travels with a selected water volume (the river slice ) and watches the concentration changes in the volume while moving downstream. Later the notion of an isolated water volume will be modified when mixing due to diffusion and dispersion across the boundaries of the volume is taken into account. 

Compared to rivers and lakes, transport in porous media is generally slow, three-dimensional, and spatially variable due to heterogeneities in the medium. The velocity of transport differs by orders of magnitude among the phases of air, water, colloids, and solids. Due to the small size of the pores, transport is seldom turbulent. Molecular diffusion and dispersion along the flow are the main producers of randomness in the mass flux of chemical compounds. 

We now turn to a completely different method of describing diffusion, one that has its greatest value in industrial situations. It is related to both diffusion and dispersion but has a simpler mathematical description. This means that it s more approximate. Unfortunately, it s complicated by questions of units and definitions, which give it a reputation of being a difficult subject. 

Some, but not all, of the benefits conferred by compartmentalization inside lipid bilayer membranes can be achieved by surfaces. Surfaces allow concentration of metabolites, and, if the surface is a mineral that is not in redox equilibrium with its surroundings, a surface can provide a source of energy. Compartmentalization can also be achieved inside porous minerals. For example, the walls of hydrothermal vents are porous and trap organic material6 and may indeed have provided the first compartmentalization that allowed the emergence of metabolism and macromolecules in a protected environment.7 Also, there are tiny pores in rocks, including tubes in chroysotile, and there are microcracks in quartz, both of which could support diffusion and dispersion by currents. 

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. 

An extremely simplified application of the described approach is already implemented in the PHREEQC program. Reactive mass transport can be modeled for the one-dimensional case at constant flow rates considering diffusion and dispersion. 

This result was first derived by Aris (1956) using the method of moments. While the resulting model now includes both the effects (axial molecular diffusion and dispersion caused by transerverse velocity gradients and molecular diffusion) it has the same deficiency as the Taylor model, i.e. converting a hyperbolic model into a parabolic equation. 

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