Environmental Engineering Reference
In-Depth Information
(linear-quadratic) optimization problem. The node imbalance for the component i is
defined as the mass flowrate residuals at the network nodes that are obtained when
using the measured values of the component i concentrations:
M i C i f 0
ε i
=
M i
(
f 0
c i
)=
,
(2.83)
where C i is the diagonal matrix of c i . In the node imbalance method, it is assumed
that component concentrations are known, and that flowrates only have to be esti-
mated. The optimal flowrate estimates are those that minimize
n
i = 1 ε i V 1
T V f 1
J
(
f 0
)=
ε i
+(
Y f
C f f 0
)
(
Y f
C f f 0
),
(2.84)
ε i
where the first term of J corresponds to the node imbalances and the second term to
the flowrate measured values, if any:
Y f
=
C f f 0
+
e f ; e f
N
(
0
,
V f
).
(2.85)
The node imbalance estimates of the flowrates are therefore
C f V f
1 C f
)
1 C f V f
1 Y f
f
=(
Σ i γ i
+
,
(2.86)
where
C dT
i
M i
V ε i
1 M i C i
γ i
=
.
(2.87)
When the flowrates have been estimated by the node imbalance method, species
concentrations can be estimated in a second step. When the flowrates are assumed
to be known, the species concentration constraints become linear, as shown in this
expression:
M i F 0 c i
(2.88)
where F 0 is the diagonal matrix of the known f 0 values. The quadratic criterion to
be minimized, subject to (2.88), is
M i
(
f 0
c i
)=
=
0
,
n
i = 1 ( Y i C i c i )
T V i 1
J
(...,
c i
, ...)=
(
Y i
C i c i
),
(2.89)
where C i is the observation matrix of the component i concentration, and V i its cor-
responding measurement error variance matrix. The solution is given by the same
equations as (2.65) and (2.66), providing that there are no additional constraints that
couple the mass balance equations of the various components:
F 0 M i
M i F 0 α i F 0 M i
) 1 M i F 0 α i
C i V 1
c i
=
α i
[
I
(
]
Y i
,
(2.90)
i
where
C i V 1
) 1
α i
=(
C i
.
(2.91)
i
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