44
2. Hydrodynamic Dispersion in Porous Media
Several expressions, known as adsorption isotherm relationships, have been
suggested for f( C, F) for different adsorption cases. One example is given by
oF = kC
ot
'
(2.6.24)
where k is a constant. The equation treats the solid phase as a sink of the
solute and assumes that the change rate of solute concentration in the solid
is proportional to the solute concentration in the liquid. Eliminating f(C,F)
in Eq. (2.6.22) and Eq. (2.6.23), we have
(2.6.25)
Inserting Eq. (2.6.24) into the above equation, we then have the following
equation, which is of the same form as Eq. (2.6.20):
OC = ~(Dij OC) _ ~(V;C) _ (1 - () k) C.
(2.6.26)
ot OXi OXj OXi
()
Another expression for adsorption is
F=ßC,
(2.6.27)
where ß is a constant. This expression is known as the linear equilibrium
isotherm relationship, which shows that the solute concentrations in the solid
phase are directly proportional to those in the liquid phase. Inserting Eq.
(2.6.27) into Eq. (2.6.25) and assuming
we then have
1 + ()
R d = 1 + -(}-ß,
(2.6.28)
(2.6.29)
This equation includes an implicit source/sink term. Since R d > 1, both the
hydrodynamic dispersion coeflicient and the mean flow velocity are decreased
by a certain factor and, hence, the dispersion process is weakened. The
coeflicient R d is often called the retardation factor, which describes the retardation effect caused by adsorption.
The retardation factor may be directly measured. Developments and articles on this topic were reviewed by Faust and Mercer (1980) and Valocchi
(1984). As a special case of adsorption and ion exchange, the transport of
colloid and bacteria in groundwater has been extensively studied in recent
years (Elimelech et al., 1995).
2.6.3 Initial and Boundary Conditions
The solution of the hydrodynamic dispersion equation (2.6.13) is the concentration distribution C(x, t), which is dependeIit on position x = (Xl> X 2 , x 3 )
2. Hydrodynamic Dispersion in Porous Media
Several expressions, known as adsorption isotherm relationships, have been
suggested for f( C, F) for different adsorption cases. One example is given by
oF = kC
ot
'
(2.6.24)
where k is a constant. The equation treats the solid phase as a sink of the
solute and assumes that the change rate of solute concentration in the solid
is proportional to the solute concentration in the liquid. Eliminating f(C,F)
in Eq. (2.6.22) and Eq. (2.6.23), we have
(2.6.25)
Inserting Eq. (2.6.24) into the above equation, we then have the following
equation, which is of the same form as Eq. (2.6.20):
OC = ~(Dij OC) _ ~(V;C) _ (1 - () k) C.
(2.6.26)
ot OXi OXj OXi
()
Another expression for adsorption is
F=ßC,
(2.6.27)
where ß is a constant. This expression is known as the linear equilibrium
isotherm relationship, which shows that the solute concentrations in the solid
phase are directly proportional to those in the liquid phase. Inserting Eq.
(2.6.27) into Eq. (2.6.25) and assuming
we then have
1 + ()
R d = 1 + -(}-ß,
(2.6.28)
(2.6.29)
This equation includes an implicit source/sink term. Since R d > 1, both the
hydrodynamic dispersion coeflicient and the mean flow velocity are decreased
by a certain factor and, hence, the dispersion process is weakened. The
coeflicient R d is often called the retardation factor, which describes the retardation effect caused by adsorption.
The retardation factor may be directly measured. Developments and articles on this topic were reviewed by Faust and Mercer (1980) and Valocchi
(1984). As a special case of adsorption and ion exchange, the transport of
colloid and bacteria in groundwater has been extensively studied in recent
years (Elimelech et al., 1995).
2.6.3 Initial and Boundary Conditions
The solution of the hydrodynamic dispersion equation (2.6.13) is the concentration distribution C(x, t), which is dependeIit on position x = (Xl> X 2 , x 3 )