Environmental Engineering Reference
In-Depth Information
Sidebar 8.2: Single Reaction Example
For the reaction between species
A
,
B
and
C
2
AþB , C
the reaction matrix S is given by
S
¼
ð
21
1
Þ
and for U holds:
!
102
011
U
¼
Total concentration
u
1
and
u
2
are thus given by:
u
1
¼ c
A
þ
2
c
C
u
2
¼ c
B
þc
C
which can be explained as follows: the total concentration
u
1
of
A
is given by
the concentration
c
A
and two times the concentration
c
C
, because according to
the reaction two
A-
species are within the reaction product
C
. For the total
concentration
u
2
of
B
the two concentrations
c
B
and
c
C
have to be summed up,
because there is one unit of
B
to be found in
C
. This may give an impression
why the term 'total concentration' is used.
In this example case the matrix U is unique if it is required that U is
combined by the unit matrix and one additional column fitting to the
formulation (
8.20
).
MacQuarrie
1996
). Total concentrations are invariants that are obtained by linear
combinations of the species concentrations. Mathematically this can be expressed
by (left) multiplication of the (
8.15
) with a matrix U, a matrix with
N
s
-N
r
rows and
N
s
columns. The vector
u
¼
U
c
(8.16)
contains values of
total concentrations. u is a column vector with
N
s
-N
r
components. The conditions (
8.15
) imply:
U
S
T
r
eq
¼
0
(8.17)
