Modelagem Biorreator
dX
dt =D
(
X0−X
)
+μX
d S
dt =D
(
S0−S
)
+μX
YXS
d P
dt =D
(
P0−P
)
+μX
(
YPS
YXS
)
Linearização do modelo
Temos:
dX
dt =g1(D , X 0, X)
d S
dt =g2(D , S 0, S , X )
d P
dt =g3(D , P 0, , P , X )
Linearizando em serie de Taylor temos:
dX
dt =g1
(
D(t), X 0(t), X (t)
)
−g1
(
D(0), X 0(0), X (0)
)
=∆ D d g1
d D +∆ X 0d g1
d X 0+∆ X d g1
d X
dS
dt =g2
(
D(t), S 0(t), S (t), X (t)
)
−g2
(
D(0), S 0(0), S(0), X (0)
)
=∆ D d g2
dD +∆ S 0d g 2
d S 0+∆ S d g2
d S +∆ X d g2
dX
dP
dt =g3
(
D(t), P0(t), P (t), X (t)
)
−g3
(
D(0), P 0(0), P (0), X (0)
)
=∆ D d g3
dD +∆ X 0d g3
dX 0+∆ P d g3
d P +∆ X d g3
dX
Aplicando a TLP:
SX (S)=D
(
S
)
C1+X0
(
S
)
C2+X
(
S
)
C3
SX (S)=D
(
S
)
C1+X0
(
S
)
C2+X
(
S
)
C3
X
(
S
)
(
1+S τ1
)
=D
(
S
)
(C¿¿1/−C3)+X0
(
S
)
(C¿¿2/−C3)¿¿
X
(
S
)
=D
(
S
)
K1
(
1+S τ1
)
+X0
(
S
)
K2
(
1+S τ1
)
SS(S)=D
(
S
)
C4+S0
(
S
)
C5+S
(
S
)
C6+X
(
S
)
C7
S S (S)=D
(
S
)
C4+S0
(
S
)
C5+S
(
S
)
C6+X
(
S
)
C7
S
(
S
)
(
1+S τ2
)
=D
(
S
)
(C¿¿4/−C6)+S0
(
S
)
(C¿¿5/−C6)+X
(
S
)
(C¿¿7/−C6)¿ ¿¿
