Applied numerical modeling of saturated / unsaturated flow and ...

covalent bonding). To mathematically describe
the sorption / desorption behaviour of
a species so called sorption isotherms can
be used, in which the sorbed amount of the
species is a function of its dissolved
concentration. A general nonlinear sorption
model can be formulated as
C
s
� �a�Cl� � � � abC �
� 1
(17)
l
where Cs [kg kg -1 ] is the sorbed solid phase
and Cl [kg m -3 ] the liquid phase concentration,
�, a,� b and � are empirical constants.
For b = 0, a = 1 [-] and � = 1 [-] eq.
(17) is the linear Henry isotherm, in this
case � [m³ kg -1 ] is a simple equilibrium
constant. For � = 1 and b = 1 eq. (17) is the
Langmuir isotherm with � [kg kg -1 ] being
the maximum amount of a species which
can be sorbed to the solid phase and a
[m³ kg -1 ] an adsorption constant. For b = 0,
a = 1 [-] and � � 1 [-] (usually � < 1) the
Freundlich isotherm is obtained, where �
[kg 1-� m 3� kg -1 ] is the Freundlich coefficient
and � [-] is the Freundlich exponent.
In general, sorption processes may be treated
as equilibrium reactions because sorption
is fast compared to transport. However,
there are exceptions to this rule because for
some solute species as well as soils or
aquifers, equilibration is a slow process.
Possible mechanisms include slow diffusion
into intraparticle pores accompanied by
equilibrium sorption to surfaces within the
pores or slow diffusion in organic matter
(Ball and Roberts, 1991; Grathwohl, 1998;
Rügner et al., 1999). Hence, from a
macroscopic point of view sorption equilibrium
may not be reached within the available
contact time between mobile and solid
phases. Assuming spherical particles, intraparticle
diffusion kinetics can be described
by Fick´s 2 nd law in radial coordinates (e.g.
Grathwohl, 1998)
2
� C ��
C 2 � C �
� Dap�
� 2
� t
�
� � r r � r �
(18)
where r [m] is the radial distance from the
particle center and Dap [m 2 s -1 ] is the appa-
rent diffusion coefficient, which is calculated
from Da by
D
D �
a
ap � (19)
( � ��
�)
� f
where � [-] is the intraparticle porosity, �
the linear equilibrium sorption coefficient
and � [kg m -3 ] the particle density. Other
approaches to describe slow sorption / desorption
kinetics comprise first- or secondorder,
two-stage models (e.g. Brusseau and
Rao, 1989; Ma and Selim, 1994; Streck et
al., 1995). A comparison of first-order and
diffusion limited approaches was recently
published by Altfelder and Streck (2006).
Kinetic degradation
Degradation, whether biotic or abiotic, is
the only process that reduces the overall
mass of contaminants in natural porous
media without transfer to other phases.
Biological degradation mechanisms are numerous,
complex and by far not completely
understood nor even identified. The vast
amount of different types of microorganisms
in the subsurface provides many
metabolic pathways for contaminant degradation
under aerobic and anaerobic conditions.
Through successive oxidation or
reduction reactions contaminants can be
transformed to innocuous compounds like
methane, chloride, water or carbon dioxide
(Wiedemeier et al. 1999). However, intermediate
products can even be of significantly
higher toxicity and persistence than
their parent compounds (e.g., dechlorination
of dichloromethane to vinyl chloride;
Wiedemeier et al. 1999).
Kinetic growth and decay of a microbial
population X [kg m -3 ] can be described by a
generalized Monod-type equation as given
e.g. in Schäfer et al. (1998)
�X
� v
�t
�
max
X
�
I
�
I
Ci
M � C
i Ci
i
C j
� C
j C j j
��
�X� (20)
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