Environmental Engineering Reference
In-Depth Information
mass balance for one of the species. Let's denote the concentrations by the symbols
c
a
,
c
b
and
c
c
:
@
c
a
@t
¼r
j
ðc
a
Þr
@
c
b
@t
¼r
j
ðc
b
Þr
@
c
c
@t
¼r
j
ðc
c
Þþr
(7.3)
Note that the sign of the reaction rate is negative for reactants (as mass is lost)
and positive for the product (as mass is gained).
For a general formulation one may introduce a reaction matrix
S
, which is given
in the example by:
S
¼
ð
1
11
Þ
(7.4)
indicating that one molecule of species
a
and one molecule of species
b
yield one
molecule of species
c
. The product
S
T
q
is a column vector, which gathers all
reaction terms, and the system of differential (
7.3
) can be written briefly as:
0
1
c
a
c
b
c
c
@
c
@t
¼r
j
@
A
S
T
r
with
c
ð
c
Þþ
¼
(7.5)
The notation can be used for general systems of a multitude of species that are
connected by several reactions. The reaction rates are gathered in a column vector
r
.
One can write the entire system by the vector equation:
@
c
@t
¼r
j
S
T
r
ð
c
Þþ
(7.6)
This approach will be extended in the following chapter. If transport does not
have to be considered, (
7.6
) reduce to:
@
c
@t
¼
S
T
r
(7.7)
which is a system of ordinary differential equations. The solution of such dynamical
systems is outlined in Chap. 9.