Environmental Engineering Reference
In-Depth Information
Fig. 8.1 Sorption
isotherm for species
with stoichiometric
coefficient
¼
2 (see text)
1
solid phase concentration
0.8
0.6
0.4
0.2
gaseous phase concentration
0
0
0.2
0.4
0.6
0.8
1
@
c
@t
¼ k
c
s
k
!
c
2
s
@
s
@t
¼ k
c
s
k
!
c
2
(8.31)
s
@
c
s
@t
¼ k
!
c
2
s k
c
s
the same arguments as above lead to the isotherm:
Kc
s;
max
c
2
1
c
s
¼
(8.32)
þK c
2
Figure
8.1
depicts an example of such a sorption isotherm. Formula (
8.32
)is
a special case of the so called Langmuir-Freundlich isotherm (Klug et al.
1998
) with
exponent two.
8.5 Transport and Speciation
The differential equation for multi-species reactive transport has been written in
vector notation (see equation (7.6)):
@
c
@t
¼r
S
T
r
j
ð
c
Þþ
(8.33)
The system (
8.33
) contains one differential equation for each species. It is
a system with
N
s
equations. In case of equilibrium reactions it is not possible to
treat the system (
8.33
) directly, because the exchange rates of the equilibrium
reactions r are neither known nor given by an explicit expression. In order to
reach a feasible formulation, the equations have to be summed up in a way that
eliminates the last term on the right side of the equation. The procedure was already
described in Chap. 8.3.