Environmental Engineering Reference
In-Depth Information
where
f
i
stands for the
i
th conservation constraint in the system
f
1
to
q
. The initial constrained problem has been replaced by a new optimization
problem with
n
X
(
X
)
,for
i
=
+
q
unknown parameters.
These optimization problems can be solved by classical numerical techniques.
When
J
is quadratic and the constraints are linear, an analytical solution can be
obtained by resolving the system expressing that the derivatives of the reconciliation
criterion, or the Lagrange function, have zero values. In other cases, either non-
linear programming methods can be used to minimize the criterion, or numerical
methods used for solving the non-linear equations expressing that the derivatives of
the criterion are zero.
When
e
=
0, the optimization problem
T
V
−
ε
f
min
X
J
(
X
)=
f
(
X
)
(
X
)
subject to
g
(
X
)=
Y
(2.52)
can again be processed by one of the two above approaches: substitution and La-
grange methods. This reconciliation method is common practice in the bilinear case.
It allows to estimate unmeasured flowrates through the use of species concentrations
in the ore streams. It is called the node imbalance method.
2.7
The Linear Cases: Steady-state, Stationary and Node
Imbalance Data Reconciliation Methods
2.7.1 The Steady-state Case
The unconstrained SSR problem can be formulated as the following particular case
of (2.30):
X
T
V
−
1
=
arg min
X
[(
Y
−
CX
)
(
Y
−
CX
)],
(2.53)
Y
=
CX
+
e
;
e
∼
N
(
0
,
V
).
q
independent variables are selected using either
heuristic or systematic methods, and the
X
vector is restructured as (
X
ind
In the substitution method,
n
X
−
X
dep
),
while the matrix
M
is partitioned into compatible blocks, such that the conservation
constraints can be rewritten as
M
ind
M
dep
X
ind
,
=
.
0
(2.54)
X
dep
or
M
ind
X
ind
+
M
dep
X
dep
=
0
.
(2.55)
The dependent variables
X
dep
can then be expressed as functions of
X
ind
:
)
−
1
M
ind
X
ind
= −(
X
dep
M
dep
(2.56)
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