2.5. Coefficients of Hydrodynamic Dispersion
35
and the tortuosity T can be determined. For loose sand or similar porous
media, D L can be approximated by 2/3 D d •
Zone 11. The Peclet number is approximately 0.4 to 5 and the curve has
an upward tendency. In this zone, mechanical dispersion and molecular
diffusion become comparable, so that both effects need to be taken into
consideration.
Zone III. Mass transport is mainly the result of mechanical dispersion and
trans verse molecular diffusion. Transverse diffusion spreads the mass and
actually decreases the longitudinal mass transport. Experiments give the
following result:
DdDd ~ IX(Pet; IX ~ 0.5, 1 < m < 1.2,
(2.5.2)
which means that the longitudinal dispersion coefficient is proportional to
the mean flow velocity with power ranging from 1 to 1.2.
Zone IV. In this region, the effect of molecular diffusion can be negligible,
but the flow velocity still obeys Darcy's Law. The straight line segment in
Figure 2.11 gives
(2.5.3)
which means the coefficient of longitudinal dispersion is proportional to the
mean flow velocity.
Zone V. This is also a zone of pure mechanical dispersion with negligible
molecular diffusion. Due to the high mean flow velocity, the effects of inertia
and turbulence cannot be ignored. They retard the longitudinal mass transport and, as a result, the slope of this part of the curve is reduced.
As to the transversal dispersion, similar results have been obtained by
experiments. For zone III, for example, we have
DT/Dd ~ IX(Pet, IX ~ 0.025, m = 1.1,
(2.5.4)
and, in zone IV, m in Eq. (2.5.4) changes to unity. This means that D T is also
proportional to the mean flow velocity in this region when the mechanical
dispersion is predominant.
Klotz et al. (1980) conducted a large number oflaboratory and field experiments which dealt with the relationship between longitudinal and transversal
dispersion coefficients in loose soil and the mean flow velocity. They suggested the following form for the longitudinal dispersion coefficient
(2.5.5)
where IX L is called the longitudinal dispersivity, V is the mean flow velocity and
m is the power of the velocity to be determined. Experimental results show
that IXL is mainly dependent on the mean particle size and the uniformity
coefficient, U = d60 /d 10 • Laboratory test results are reproduced in Table 2.1.
The range of m given in the table is
1.07 :s; m :s; 1.1.
(2.5.6)
35
and the tortuosity T can be determined. For loose sand or similar porous
media, D L can be approximated by 2/3 D d •
Zone 11. The Peclet number is approximately 0.4 to 5 and the curve has
an upward tendency. In this zone, mechanical dispersion and molecular
diffusion become comparable, so that both effects need to be taken into
consideration.
Zone III. Mass transport is mainly the result of mechanical dispersion and
trans verse molecular diffusion. Transverse diffusion spreads the mass and
actually decreases the longitudinal mass transport. Experiments give the
following result:
DdDd ~ IX(Pet; IX ~ 0.5, 1 < m < 1.2,
(2.5.2)
which means that the longitudinal dispersion coefficient is proportional to
the mean flow velocity with power ranging from 1 to 1.2.
Zone IV. In this region, the effect of molecular diffusion can be negligible,
but the flow velocity still obeys Darcy's Law. The straight line segment in
Figure 2.11 gives
(2.5.3)
which means the coefficient of longitudinal dispersion is proportional to the
mean flow velocity.
Zone V. This is also a zone of pure mechanical dispersion with negligible
molecular diffusion. Due to the high mean flow velocity, the effects of inertia
and turbulence cannot be ignored. They retard the longitudinal mass transport and, as a result, the slope of this part of the curve is reduced.
As to the transversal dispersion, similar results have been obtained by
experiments. For zone III, for example, we have
DT/Dd ~ IX(Pet, IX ~ 0.025, m = 1.1,
(2.5.4)
and, in zone IV, m in Eq. (2.5.4) changes to unity. This means that D T is also
proportional to the mean flow velocity in this region when the mechanical
dispersion is predominant.
Klotz et al. (1980) conducted a large number oflaboratory and field experiments which dealt with the relationship between longitudinal and transversal
dispersion coefficients in loose soil and the mean flow velocity. They suggested the following form for the longitudinal dispersion coefficient
(2.5.5)
where IX L is called the longitudinal dispersivity, V is the mean flow velocity and
m is the power of the velocity to be determined. Experimental results show
that IXL is mainly dependent on the mean particle size and the uniformity
coefficient, U = d60 /d 10 • Laboratory test results are reproduced in Table 2.1.
The range of m given in the table is
1.07 :s; m :s; 1.1.
(2.5.6)