Dispersion in rivers

Since the pollutant input is turned on suddenly, a pollutant front moves downstream. Initially, the shape of the front is rectangular (Fig. 19.14a). Due to longitudinal dispersion (see Chapters 18.5 and 22.4), the front gradually looses its rectangular shape while moving downstream (Fig. 19.146). Longitudinal dispersion in rivers will be discussed in more detail in Chapter 24. At this point it is sufficient to remember that... 

Illustrative Example 24.4 Turbulent Diffusion and Longitudinal Dispersion in River G... 

By analogy to the description of dispersion in rivers, the dispersive flux relative to the mean flow, Fdis, can be described by an equation of the First Fickian Law type (see Eq. 24-42) ... 

Obviously, everything that has been said in Chapter 24 on dispersion in rivers also applies to the one-dimensional flow in aquifers. 

Holly, F. M. (1975). Two-Dimensional Mass Dispersion in Rivers, Hydrol. Pap. No. 78. Colorado State University, Fort Collins. 

Similarly, contaminant concentrations in rivers or streams can be roughly assessed based on rate of contaminant introduction and dilution volumes. Estuary or impoundment concentration regimes are highly dependent on the transport mechanisms enumerated. Contaminants may be localized and remain concentrated or may disperse rapidly and become diluted to insignificant levels. The conservative approach is to conduct a more in-depth assessment and use model results or survey data as a basis for determining contaminant concentration levels. 

The relations developed for longitudinal dispersion coefficient are given in Table 6.3. The experimental results in rivers tend to have a large range because of the variety of lateral velocity profiles that exist in natural rivers and streams. 

Rivers are close to the perfect environmental flow for describing the flow as plug flow with dispersion. The flow is confined in the transverse and vertical directions, such that a cross-sectional mean velocity and concentration can be easily defined. In addition, there is less variation in rivers than there is, for example, in estuaries or reactors - both of which are also described by the plug flow with dispersion model. For that reason, the numerous tracer tests that have been made in rivers are useful to characterize longitudinal dispersion coefficient for use in untested river reaches. A sampling of the dispersion coefficients at various river reaches that were... 

The question that we need to ask ourselves is whether the longitudinal dispersion can be predicted accurately for these rivers. Equation (6.35), which predicts that >l/(m+/i) = constant, is shown in Table 6.4 to have a large range of constants, probably because of the variations in cross section and morphology seen in natural streams. Fisher (1973) observed that this constant seemed to depend on mean surface width, W, and substituted W for h in the numerator of equation (6.34) to develop the following empirical equation to characterize longitudinal dispersion coefficient in rivers ... 

Tracer Determination of Longitudinal Dispersion Coefficient in Rivers. Tracers are generally used to determine longitudinal dispersion coefficient in rivers. Some distance is required, however, before the lateral turbulent diffusion is balanced by longitudinal convection, simitar to Taylor s (1953) analysis of dispersion in a laminar flow. This transport balancing distance, x is given by the equation... 

Let US return to the discussion of computational transport routines, where each computational cell is the equivalent of a complete mix reactor. If we are putting together a computational mass transport routine, we could simply specify the size of the cells to match the diffusion/dispersion in the system. The number of well-mixed cells in an estuary or river, for example, could be calculated from equation (6.44), assuming a small Courant number. Then, the equivalent longitudinal dispersion coefficient for the system would be calculated from equation (6.44), as well, for a small At (At was infinitely small in equation 6.44) ... 

The first two components are the active surfactants, whereas the other components are added for a variety of reasons. The polyphosphate chelate Ca ions which are present (with Mg ions also) in so-called hard waters and prevents them from coagulating the anionic surfactants. Zeolite powders are often used to replace phosphate because of their nutrient properties in river systems. Sodium silicate is added as a corrosion inhibitor for washing machines and also increases the pH. The pH is maintained at about 10 by the sodium carbonate. At lower pH values the acid form of the surfactants are produced and in most cases these are either insoluble or much less soluble than the sodium salt. Sodium sulphate is added to prevent caking and ensures free-flowing powder. The cellulose acts as a protective hydrophilic sheath around dispersed dirt particles and prevents re-deposition on the fabric. Foam stabilizers (non-ionic surfactants) are sometimes added to give a... 

Dilution of a Concentration Patch by Longitudinal Dispersion Illustrative Example 24.5 A Second Look at the Atrazine Spill in River G The Effect of Dispersion... 

Longitudinal concentration gradients of a chemical that is introduced into a river at a constant rate are small (except for a chemical with a very large in-situ reaction rate). Thus, according to Eq. 24-44 the effect of dispersion on C(x,t) is small. In contrast, the concentration profile resulting from a nonstationary input as caused, for instance, by an accidental spill is strongly affected by dispersion. In fact, often dispersion is the most important mechanism to reduce the maximum concentration of a concentration patch that moves along the river. 

In Illustrative Example 24.5 we look again at the atrazine spill in River G of Illustrative Example 24.3 and ask how strongly dispersion would reduce the maximum atrazine concentration while the pollution cloud is moving downstream. 

Explain the difference between longitudinal diffusion and longitudinal dispersion. Which one is more important in rivers ... 

In porous media the flow of water and the transport of solutes is complex and three-dimensional on all scales (Fig. 25.1). A one-dimensional description needs an empirical correction that takes account of the three-dimensional structure of the flow. Due to the different length and irregular shape of the individual pore channels, the flow time between two (macroscopically separated) locations varies from one channel to another. As discussed for rivers (Section 24.2), this causes dispersion, the so-called interpore dispersion. In addition, the nonuniform velocity distribution within individual channels is responsible for intrapore dispersion. Finally, molecular diffusion along the direction of the main flow also contributes to the longitudinal dispersion/ diffusion process. For simplicity, transversal diffusion (as discussed for rivers) is not considered here. The discussion is limited to the one-dimensional linear case for which simple calculations without sophisticated computer programs are possible. 

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