Environmental Engineering Reference
In-Depth Information
In order to fulfil the conditions (
8.17
), the matrix U is chosen to obey the
equation
U
S
T
¼
0
(8.18)
There are several ways to find such a matrix U which is not unique. Saaltink et al.
(
1998
) mention Gram-Schmidt orthogonalization and singular decomposition
as alternative methods to construct a matrix U with the property (
8.18
) and provide
an example for the latter procedure. In addition, Saaltink et al
.
(
1998
) suggest
another procedure which can easily be implemented using MATLAB
,asitis
formulated in matrix form. The algorithm is based on the partition of the matrix S in
two sub-matrices:
®
S
¼
jð Þ
S
1
S
2
(8.19)
where S
2
is a regular square matrix with
N
r
rows and columns. S
1
is a matrix with
N
s
-N
r
rows and
N
r
columns. U is then given by:
h
i
T
I
N
s
N
r
S
U
¼
(8.20)
with
S
¼
S
2
1
S
1
(8.21)
and unit matrix I
N
s
N
r
with
N
s
-N
r
rows or columns.
In order to perform the matrix operation, S
2
must be invertible. Sometimes some
permutations of the species' system are required to achieve that S
2
is regular. It
is possible in either case if the matrix S has maximum rank (see above). The number
of entries in the right sub-matrix of (
8.20
)is
N
r
(
N
s
-N
r
), for which there are
(
N
s
-N
r
) conditions only.
For a given vector u the (
8.16
) is a system of
N
s
-N
r
equations for the unknown
components of the vector c. In addition, there are
N
r
equilibrium conditions. For
given total concentrations, gathered in vector u, the system
u
¼
U
c
(8.22)
S
log
ð
c
Þ¼
log
ð
K
Þ
has to be solved for c. It is a nonlinear system of
N
s
equations for
N
s
unknown
values. The so-called speciation problem (
8.22
) can be solved by the Newton-
Raphson method, an extension of the Newton method described above. There are
N
s
-N
r
linear balance equations and
N
r
non-linear equilibrium equations. In
MATLAB
, the generalization of the Newton method for vector functions of
several independent variables can be implemented for that purpose:
®
