Page 228 - Soil and water contamination, 2nd edition

Substance transport 215

-1
2
where D = the longitudinal dispersion coefficient [L T ] and B = the width of the river
y
channel [L], and Δt is the travel time between the start of the release of the substance and
complete mixing in the river [T] given by L /u , where L = mixing length [L] and u =
x
x
m
m
-1
average flow velocity [L T ]. So, the mixing length can be estimated by:
B 2 u
L m x (11.41)
2 D y
Example 11.6 Transverse dispersion
-1
A stream has a depth of 0.6 m, a width of 5.0 m, and a flow velocity of 12 cm s .
Estimate the mixing length for a single source introduced to the stream from the stream
bank.
Solution
First, estimate the shear velocity using Equation (11.35):
u . 0 10 u . 0 10 . 0 12 . 0 012 m s -1
* x
Second, estimate the transverse dispersion coefficient for a natural stream, using Equation
(11.39):

2
3
D y 6 . 0 H u * 6 . 0 6 . 0 . 0 012 . 4 32 10 m s -1
Finally, estimate the mixing length using Equation (11.41):
B 2 u 5 2 . 0 12
L m x 347 m
2D 2 . 4 32 10 3
y
The stream parameters in this example are the same as in Example 11.3. Although in
that example we assumed instantaneous mixing, in reality it takes at least 347 m for
the effluent to be mixed across the stream width. In order to describe the transport of
contaminants between the point source input and L , where the contamination cloud is
m
three-dimensional, the transport equation cannot be simplified. For distances beyond L ,
m
a one-dimensional longitudinal dispersion model is sufficient.
11.3.5 Numerical dispersion

If we solve the advection equation (Equation 11.14) in two dimensions using a numerical
technique, we usually discretise space by defining fixed points in a so-called Cartesian
coordinate system. Such methods using a fixed grid are known as Eulerian methods
(like the numerical integration method named after the mathematician Leonhard Euler
(1707–1783); see Box 11.II). In this case, apparent dispersion may also occur as an artefact
of the numerical solution technique. Such dispersion is called numerical dispersion and does
not occur in reality. This mixing is a result of calculating average concentrations in discrete
spatial units (grid cells or model elements) and time-steps. For example, if we use a two-
dimensional, dynamic raster GIS we may calculate the average concentration in a square grid
cell (x,y) at a given time-step t as function of the concentration and water flow velocity in the
grid cell itself and its adjacent cells at time-step t = t -1. If we consider a simple situation of
advective movement of a solute with a steady state water flow in x direction which is constant
over space and time, the concentration in cell (x,y) is given by:

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Soil and Water.indd 227 10/1/2013 6:45:00 PM