Dispersion

The word dispersion has several meanings. In physics, it is often used to describe the process of separation of the components of waves with different frequency. In chemistry, a dispersion is a system of fine particles uniformly distributed in a medium. Other definitions apply to biological concepts and a few other technical areas. When dealing with mass transport, dispersion is defined as the combined effect of advection and diffusion acting in a flow field with velocity gradients. This is why it is sometimes referred to as 'shear dispersion'.

The presence of velocity gradients on the fate of the substance becomes apparent when spatial averaging of the physical quantities is carried out along with the temporal averaging described in the section on turbulent diffusion. In surface water bodies, it is often convenient to simplify the description of mass transfer by averaging velocity and concentration over the vertical direction (shallow water approach) or over a cross section (unidirectional approach). If quantities are depth-integrated, they are then described only as a function of the two planimetric Cartesian coordinates, that is, C(x, y, t) and v(x,y, t) (the overbar indicates averaged values). Depth averaging is justified when dealing with large rivers, estuaries, and lagoons, using the the evidence that vertical mixing is usually much faster than lateral and longitudinal mixing, due to the limited extension of the domain in the vertical direction. If the physical quantities are averaged over a cross section, their description is limited to the distribution along the longitudinal direction, that is, C(x, t) and v(x,t). The one-dimensional approach is justified when the transverse dimensions of the domain are small compared to the longitudinal dimension. This is the reason why the one-dimensional approach is commonly adopted for dispersion processes in rivers and channels.

The comprehension of the physical mechanism leading to dispersion may be improved by the example presented in Figure 3. Case 1 shows a shear flow in a highly turbulent environment. It is a vertical two-dimensional flow, just for the sake of simplicity, which can easily be generalized to a three-dimensional flow. The horizontal arrows represent the time-averaged velocity field. In these conditions, while solute molecules are transported along the flow by the mean velocity, they are displaced vertically (or laterally) by turbulence. Each molecule rapidly samples regions of the domain characterized by high velocity as well as regions characterized by slow velocity. Due to high turbulent mixing, after a sufficiently long time, all molecules are expected to travel a similar distance downstream, at an average velocity close to the mean flow velocity. The overall result is that mass is advected downstream with relatively small dispersion, that is, without spreading along the longitudinal direction. Case 2 shows instead a similar flow field with a much smaller turbulence intensity. The limited mixing does not allow for the displacement of solutes far from their original location. Elements like A are likely to remain in relatively fast flow regions while elements like B are likely to stay in regions of slow advection. The overall result in this case is a rapid increase of the distance between elements over time, that is, rapid dispersive spreading of mass along the flow direction. It is apparent that, when averaged concentrations are used, dispersion is counteracted by turbulent diffusion. The diagram in

A(t0)

"" "------' •

• \ /

Vertical (lateral) mixing

A(ti)

7

e(t2)

/y--

- 0' B(ti)

Figure 3 Schematic of the transport processes affecting shear dispersion. Case 1 shows transport of elements A and B in a shear flow with high turbulent mixing. Case 2 shows transport of the same elements in a shear flow with low turbulent mixing. The graph shows the response of the two flows to a pulse injection of mass.

Figure 3 Schematic of the transport processes affecting shear dispersion. Case 1 shows transport of elements A and B in a shear flow with high turbulent mixing. Case 2 shows transport of the same elements in a shear flow with low turbulent mixing. The graph shows the response of the two flows to a pulse injection of mass.

Figure 3 shows the qualitative response of cases 1 and 2 to a slug injection of mass. In case 1, the depth-averaged concentration remains more concentrated while being advected downstream. In case 2, dispersion is much more efficient due to small vertical mixing.

This result, which may appear surprising at first, finds unambiguous confirmation in all dispersion models. If dispersion is modeled over sufficiently long timescales, it can be proven to be well approximated by a Fickian process. Dispersion models are therefore mathematically equivalent to advection-diffusion models. For example, in the simplest case of cross-sectional averaging of the physical quantities, the dispersion equation can be reduced to the one-dimensional form:

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