Environmental Engineering Reference
In-Depth Information
is computed. It is easy to see that the zero is identical with the solution of system
(
8.22
) if activity corrections are neglected. The derivative of the function F, the so
called Jacobi matrix, is given by:
U
DF
ð
c
Þ¼
(8.24)
=
ð
ð
Þ
Þ
S
c log
10
In analogy to the Newton algorithm, presented in Sidebar 8.1, the generalization
is given by the formula:
Þ
1
c
c
DF
ð
c
F
ð
c
Þ
(8.25)
In the command sequence, first the second term is evaluated and stored in the
dc
variable. The following command in the listing ensures that the concentrations
remain positive.
8.4 Sorption and the Law of Mass Action
One can formally write sorption as an equation of a reaction between sorption sites and
free species on one side and sorbed species on the other side. In analogy to the Law
of Mass Action, one may note the differential equation for the temporal change as:
@
c
@t
¼ k
c
s
k
!
c s
@
s
@t
¼ k
c
s
k
!
c s
@
c
s
@t
¼ k
!
c s k
c
s
(8.26)
where
c
denotes the concentration in the fluid,
s
the number of free sorption sites
and
c
s
the number of occupied sorption sites. The terms for transport have been
omitted in order to focus on sorption. The equilibrium condition is then given by:
c s
¼
k
!
c
s
k
¼: K
(8.27)
or:
c
s
c
¼ K
s
(8.28)
s
here denotes the number of free sites. If no competition between species is taken
into account,
s
can also be expressed as
c
s;
max
c
s
, the number of available free
sites is diminished by the number of occupied sites. Altogether results:
