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.
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