Environmental Engineering Reference
In-Depth Information
Tabl e 2. 1
Measured and reconciled values for the flotation unit of Figure 2.2
Process
variable
Flowrates
F
(
t
/
h
)
% Cu (100
c
)
% Zn (100
z
)
Stream
Meas.
values
error
s.d.
Reconc.
values
Meas.
values
error
s.d.
Reconc.
values
Meas.
values
error
s.d.
Reconc.
values
X
X
X
Y
Y
Y
Feed
100
1
100.0
2.39
0.05
2.36
8.53
0.19
8.57
Concentrate
0
9.25
23.2
1,11
24.36
7.54
0.34
7.53
Tai l
0
90.75
0.12
0,01
0.12
9.65
0.98
8.68
The nine state variables
X
arethethreeoremassflowrates
F
and the six copper
and zinc mass fractions
c
and
z
. The measured process variables are a subset of the
state variables. The measurement vector
Y
contains the measured values of the ore
feedrate and of the six metal mass fractions. The measurement values are assumed
to give an image of the process steady-state behavior. Hence, the constraints of mass
conservation
f
(
X
)=
0are
F
1
−
F
2
−
F
3
=
0
,
(2.1)
F
1
c
1
−
F
2
c
2
−
F
3
c
3
=
0
,
(2.2)
F
1
z
1
−
F
2
z
2
−
F
3
z
3
=
0
.
(2.3)
Considering this selection of
Y
and
X
, the constraints have a bilinear structure. The
information content (measurements + constraints) is said to be redundant since it
contains two unknown state variables (
F
2
and
F
3
) and three equations. The system
is said to be observable since
F
2
and
F
3
could be estimated by resolving Equations
2.1 and 2.2 or 2.1 and 2.3. The first case leads to
F
2
=
9
.
83
t
/
h
;
F
3
=
90
.
17
t
/
h
and the second one to
F
2
h
.
The conflict between these two possible solutions explains why there is a need
for reconciling the measurements with the constraint equations. Eliminating unmea-
sured process states from Equations 2.1 to 2.3 gives
=
53
.
1
t
/
h
;
F
3
=
47
.
9
t
/
c
1
z
3
−
c
1
z
2
+
c
2
z
1
−
c
2
z
3
−
c
3
z
1
+
c
2
z
2
=
0
.
(2.4)
This equation is called a redundancy equation because it contains only measured
quantities. Since there is only one redundancy equation in this case, the redundancy
degree of the system is 1. In (2.4), the substitution of the process variables by their
measured values gives a value different from zero (2
10
−
3
) because of the con-
flict existing between constraints and measurements. This residual variable gener-
ates the parity space of the system. When using reconciled values, Equation 2.4 is
exactly verified, as obviously are also the constraints (2.1) to (2.3). The application
of a reconciliation procedure to this system generates the following advantages:
.
1
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