Environmental Engineering Reference
In-Depth Information
model uncertainties must be modeled. The plant feed stream variations are modeled
by the following linear stochastic equations:
z
(
t
+
1
)=
Az
(
t
)+
ξ
(
t
),
(2.116)
Ω
f
Ω
f
(
t
)=
Bz
(
t
)+
,
(2.117)
where
A
is a state matrix, ξ
a white noise of variance
V
f
, Ω the matrix that
extracts the component feed rates from the plant stream flowrates stacked into
f
,
f
the mean value of
f
,and
B
the matrix defining the feed rates from the state variable
z
. Parameter and model disturbances in Equations 2.114 and 2.115 are represented
by additive uncertainties δ
(
t
)
that can be modeled by equations similar to Equations
2.116 and 2.117 driven by a white noise ξ
δ
(
t
)
of variance
V
δ
.
The model described by Equations 2.114 to 2.117 is a causal model since it
can be used to simulate the process from the known inputs Ω
f
, ξ
δ
(
t
)
.
However, it contains a very large number of parameters (rate constants and feed
coefficients, parameter and model disturbances), and is non-linear. An alternative to
this approach is to construct empirical causal models.
(
t
)
,andξ
(
t
)
2.10.2 Empirical Causal Model
To illustrate an example of an empirical model, let us consider any mineral pro-
cessing plant flowsheet, linearized around nominal stream flowrates
f
. It can be
modeled by a network of connected basic units including (see Figures 2.14 and
2.15):
•
stream junctions,
i.e.
, zero dynamics elements that combine flow streams;
•
stream separators,
i.e.
, zero dynamics elements characterized by separation co-
efficients
s
ki
representing the split of component
i
between the two products of
node
k
;
•
unit gain transfer functions (usually pure delays or low order transfer functions)
representing the dynamic relationships that exist between the node feed and its
output streams;
•
feed disturbance generators driven by white noise signals ξ
i
, as considered in
Equation 2.116;
(
)
•
internal disturbance generators, driven by white noise ξ
s
used to model the
separation efficiency disturbances. They are designed to preserve material con-
servation by adding to one stream the same amount of material that is removed
from the other one.
t
The various connected elements can be gathered into a generic linear state space
model:
A
(
ξ
(
z
(
t
+
1
)=
t
)+
t
),
(2.118)
B
z
f
f
(
t
)=
(
t
)+
.
(2.119)
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