Environmental Engineering Reference
In-Depth Information
M
i
f
i
=
ε
i
,
for
i
=
0to
n
+
1
,
(2.11)
where ε
i
is interpreted as a stochastic term containing the neglected accumulation
rate, production rate and structural uncertainties. When the process is strictly op-
erating in a stationary mode, as definedinSection2.2,ε
i
can be interpreted as a
stationary random signal with the following statistical properties deduced from the
statistical variations of the component flowrates that are usually assumed to behave
as normal random variables. If
V
f
is the covariance matrix of
f
, the stacked
f
i
vec-
tors, then the stacked vector εof the ε
i
s has the following properties:
MV
f
M
T
ε
∼
N
(
0
,
V
ε
),
with
V
ε
=
,
(2.12)
where
M
is the block diagonal matrix of the
M
i
s.
The steady-state case is then a particular case of the stationary equations, when
V
ε
has a zero value.
2.3.3 The Bilinear Case
Instead of using, as state variables, the component flowrates (including the total
mass flowrate), one can use the total mass flowrates and the phase or species mass
fractions. Obviously the models are strictly identical, but the selection of these more
usual variables changes the structure of the equations with respect to the state vari-
ables. For the total mass conservation, the equations are now
dm
0
dt
=
M
0
f
0
−
ε
0
,
(2.13)
where
M
0
is the total mass incidence matrix. For the phases or species mass fractions
c
i
in the streams, and
h
i
in the node loads, the equations are
dm
0
•
h
i
=
M
i
(
f
0
•
c
i
)+
P
i
−
ε
i
,
for
i
=
0to
n
,
(2.14)
dt
where
is Hadamard's product. One can also incorporate Equation 2.13 into (2.14)
which becomes
•
dh
i
dt
=
m
0
•
M
i
(
f
0
•
c
i
)−
h
i
• (
M
0
f
0
−
ε
0
)+
P
i
−
ε
i
for
i
=
0to
n
.
(2.15)
Dynamic equations similar to (2.7) and (2.8) can also be obtained. More particu-
larly the stationary case becomes
M
0
f
0
=
ε
0
,
(2.16)
M
i
(
f
0
•
c
i
)=
ε
i
,
for
i
=
1to
n
,
(2.17)
Search WWH ::
Custom Search