Environmental Engineering Reference
In-Depth Information
dm
i
dt
=
M
i
f
i
+
P
i
−
ε
i
,
for
i
=
0to
n
+
1
,
(2.6)
where
m
i
is the component
i
mass vector (including the total mass) accumulated
at the network nodes,
f
i
the stream mass flowrate vector,
P
i
the component
i
pro-
duction rate vector at the various nodes (for the total material
P
0
0), and finally
ε
i
an uncertainty vector providing for structural errors such as forgotten secondary
streams or intermittent streams, or errors in the production rates evaluation.
Usually rates of production are unknown and cannot be measured independently
from other states. There are three possible situations:
=
1. the component
i
is transformed, and the conservation equation must not be writ-
ten at the corresponding node;
2. the transformation is of very low magnitude and is simply incorporated into ε
i
:
dm
i
dt
=
M
i
f
i
−
ε
i
,
for
i
=
0to
n
;
(2.7)
3. the component
i
is not transformed at a given node, and
P
i
0. Therefore, in the
absence of structural uncertainties, this leads to the exact dynamic conservation
constraint:
=
dm
i
dt
=
M
i
f
i
,
for
i
=
0to
n
.
(2.8)
m
i
f
i
T
, Equation 2.8 can be written in the
Defining the state vector as
x
i
=(
,
)
generic form:
E
i
dx
i
dt
=
D
i
x
i
,
for
i
=
0to
n
+
1
.
(2.9)
Rather than a model allowing process simulation, this is a singular model,
i.e.
,a
set of constraints linking the state variables. Its discrete version is
E
i
x
i
(
t
+
1
)=
F
i
x
i
(
t
),
for
i
=
0to
n
+
1
.
(2.10)
Example.
In a complex ore comminution or separation plant, the conservation con-
straints could, for instance, be written for the following components: slurry, water,
ore, copper, lead, zinc, gold, particle size classes, and gold in particle size classes.
If ten size classes are defined, the number of component conservation equations is
27 (
n
27). As will be discussed in Section 2.3.5, the component definition
selected here will create additional constraints since, among others, the gold species
is selected at two different levels of the mass balance equations.
+
1
=
2.3.2 The Linear Stationary and Steady-state Cases
When the process is in a stationary operating regime,
i.e.
, a regime randomly fluc-
tuating around a steady-state, the rate of accumulation
dm
i
/
dt
can be omitted and
incorporated into the uncertainties ε
i
:
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