Environmental Engineering Reference
In-Depth Information
mass action, which is presented in sub-chapter 8.2. Here, the mathematical frame-
work is demonstrated in a special introductory example.
Let's take a reaction in which
n
a
molecules of species
a
react with
n
b
molecules
of species
b
to produce
n
c
parts of species
c
:
n
a
aþn
b
b
!
n
c
c
(8.1)
In connection with transport the development of the system of three species can
be described by the set of three differential equations:
@
c
a
@t
¼r
j
ðc
a
Þn
a
r
@
c
b
@t
¼r
j
ðc
b
Þn
b
r
@
c
c
@t
¼r
j
ðc
c
Þþn
c
r
(8.2)
This is a slight extension of the system (7.3), where all stoichiometric numbers
n
a
,
n
b
and
n
c
were set to 1. As in the previous chapters, the vector j denotes the flux
due to transport processes, and the symbol in brackets denotes the species which is
transported. The last term in all three differential equations represents the reaction.
The notation is analogous to the description introduced in Chap. 7.
r
denotes the
reaction rate. The problem is that the reaction rate
r
is not known in equilibrium
reactions. The rate
r
has to be eliminated from the mathematical description, which
is achieved by an appropriate gathering of the equations (sums of first and third, as
well as second and third equations):
@
c
a
@t
þ
n
a
n
c
@
c
a
n
a
n
c
r
j
@t
¼r
j
ðc
a
Þþ
ðc
c
Þ
@
c
b
@t
þ
n
b
n
c
@
c
b
n
b
n
c
r
j
@t
¼r
j
ðc
b
Þþ
ðc
c
Þ
(8.3)
In favour of simplicity, we assume that the flux term is linear, i.e. that
dispersivities, diffusivities and fluid fluxes are independent of the concentrations.
Then fluxes on the left side can be gathered in a single term:
@
@t
n
a
n
c
c
c
n
a
n
c
c
c
c
a
þ
¼r
j
c
a
þ
@
@t
n
b
n
c
c
c
n
b
n
c
c
c
c
b
þ
¼r
j
c
b
þ
(8.4)
In order to obtain a suitable formulation for three unknown variables
c
a
,
c
b
and
c
c
, these two equations are complemented by the mathematical formulation of
the equilibrium state, which is a mathematical function including
c
a
,
c
b
and
c
c
.
