Environmental Engineering Reference
In-Depth Information
Another parameter has to be introduced, the
transversal dispersivity a
T
. The
analytical formulation becomes much more complex, because the scalar factor
D
eff
has to be replaced by the
dispersion tensor
D:
þ
a
L
a
T
v
vv
T
D
¼ tD
mol
þ a
T
v
ð
Þ
I
(3.12)
with unity matrix I. The elements of the matrix vv
T
contain the products of the
velocity components. Here the usual matrix product of a column vector and a row
vector yields a matrix. The formulation may seem a little complex at first sight. It
takes into account that the mixing constant in the direction of velocity is different
from the mixing constant transverse to the velocity direction and is valid for
arbitrary vectors v. Note that v may change spatially and temporally. With the
dispersion tensor the dispersive flux term becomes:
j
¼
D
rc
(3.13)
where the product on the right side is performed as matrix-vector multiplication.
Transversal dispersivity is smaller than longitudinal dispersivity. Even a factor of
one or two orders of magnitude is possible.
An important feature is the scale dependency of longitudinal and transversal
dispersivities, which has been observed in groundwater studies. Figure
3.3
shows
the scale dependency of longitudinal dispersivity in porous media. Data for that
figure were taken from several studies on dispersion in groundwater.
It is also interesting to compare the effective diffusivity with the velocity depen-
dent dispersion, i.e. the two terms which contribute to the effective dispersivity in
(
3.11
). For a length scale of 1 m, a velocity in the range of some mm/a, and
a longitudinal dispersivity of 0.1 m, the value of 10
4
m
2
/a results. Molecular
diffusivity in water for most components is around 10
9
m
2
/s or 3
10
2
m
2
/a. Even
though the diffusivity has to be reduced by the factor
, it can be concluded that for
the given scale the diffusive flux exceeds dispersive flux. The values are characteristic
for lacrustine sediments. Only for very high sedimentation burial rates and for very
long mixing pathlengths a non-negligible contribution of dispersion can be expected.
t
3.3 The Transport Equation
3.3.1 Mass Transport
When both advection and diffusion/dispersion are taken into account, the flux
vector in x-direction results as the sum of both contributions:
j
x
¼D
@c
@x
þ vc
(3.14)
