Modelagem Dinâmica do Processamento Primário de Petróleo
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Modelagem Dinâmica do Processamento Primário de Petróleo


DisciplinaControle de Processos252 materiais1.185 seguidores
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L
dt
dV \u2212= (14.31) 
 
onde wflclV é o volume de água na fase oleosa da câmara de óleo e flclV é o volume de fase 
oleosa na câmara de óleo. 
 
 
 151
14.2.4 Relações Geométricas 
 
{ }2 sen cos
4
CS
CS CS CS CS
C DV \u3b8 \u3b8 \u3b8= \u2212 (14.32) 
 
2cos 1 TCS
ha
D
\u3b8 \u239b \u239e= \u2212\u239c \u239f\u239d \u23a0 
 
WTL hhh \u2212= 
 
{ }2 sen cos
4
CL
CL CL CL CL
C DV \u3b8 \u3b8 \u3b8= \u2212 (14.33) 
 
2cos 1 LCL
ha
D
\u3b8 \u239b \u239e= \u2212\u239c \u239f\u239d \u23a0 
 
( )2
4T CL CS
DV C C\u3c0= + 
 
14.2.5 Equações de Vazão 
 
\u2022 Vertedouro 
 
( )( ) ( )1,50,2VERT VERT T VERT T VERTL k comp h h h h= \u2212 \u2212 \u2212 (14.34) 
 
onde: 
 
25 2
9,81
k g
g
=
= 
 
\u2022 Óleo 
 
( )1
0,0693
MAXL L L L L
out
L
Cv v d P P h
L
\u3b3
\u3c1
\u2212 += (14.35) 
 
\u2022 Gás 
 
( ) ( )2 2
0,0693
MAXG G
out
Cv v P P P P
G
PM
\u2212 += (14.36) 
 
 
 152
\u2022 Água 
 
( ) ( )3
0,0693
MAXW W W W W L T W
out
W
Cv v d P P h h h
W
\u3b3 \u3b3
\u3c1
\u2212 + + \u2212= (14.37) 
 
14.2.6 Relações volumétricas 
 
lflcswflcsflcs VVV += 
 
 
lfwcswfwcsfwcs VVV += 
 
 
14.3 Modelo Linearizado de Separador Bifásico 
 
 
\u2022 Linearização do Balanço de Óleo no Tanque 
( )LL
iL
hDhC
LL
dt
dh
\u2212
\u2212=
2
0 (14.38) 
 
Variáveis: 
 
);(tLL ii = 
COH
L
LLL
L
MAXL
o
V
hgPPxCL
5,15,
5
4
0
2
10*****10*4028,2
\u3c1
\u3c1
\u3c1 \u2212\u2212 +\u2212= 
);(thh LL = 
 
Diferenciando a Equação 14.38 por Li , L0 e hL tem-se: 
 
)(2
1
,, ELELi
L
hDhCdL
dt
dh
\u2212= 
 
)(2
1
,,0 ELEL
L
hDhCdL
dt
dh
\u2212
\u2212= 
 
 153
2
3
,,
,,0,
))((.4
).2).((
ELEL
ELEEi
L
L
hDhC
hDLL
dh
dt
dh
\u2212
\u2212\u2212\u2212= 
 
Como L0 = L0(xL , P, hL), faz-se a diferenciação em L0 com relação às variáveis xL, P e hL: 
 
 
COH
L
ELLLEMAXL
L
o
V
hgPP
C
dx
dL
5,15,
5
,40
2
10***
*10*4028,2
\u3c1
\u3c1
\u3c1 \u2212\u2212 +\u2212= 
 
\u239f\u239f
\u239f\u239f
\u239f
\u23a0
\u239e
\u239c\u239c
\u239c\u239c
\u239c
\u239d
\u239b
+\u2212
= \u2212
\u2212
COH
L
COH
L
ELLLE
EL
MAXL
o
o
V
hgPP
xC
dP
dL
5,15,
5,15,
5
,
,
4
0
2
2
1.
10***
.2
**10*4028,2
\u3c1
\u3c1
\u3c1
\u3c1
\u3c1 
 
\u239f\u239f
\u239f\u239f
\u239f
\u23a0
\u239e
\u239c\u239c
\u239c\u239c
\u239c
\u239d
\u239b
+\u2212
=
\u2212
\u2212
\u2212
COH
L
L
COH
L
ELLLE
EL
MAXL
L
o
o
V g
hgPP
xC
dh
dL
5,15,
5
5,15,
5
,
,
4
0
2
2
10**.
10***
.2
**10*4028,2
\u3c1
\u3c1
\u3c1
\u3c1
\u3c1
\u3c1 
 
Linearizando-se L0 por expansão em série de Taylor: 
 
).().().( ,
00
,
0
,00 ELL
L
EELL
L
E
LIN hh
dh
dLPP
dP
dLxx
dx
dLLL \u2212+\u2212+\u2212+= 
 
Com 
COH
L
ELLLE
EL
MAXL
E
o
V
hgPP
xCL
5,15,
5
,
,
4
,0
2
10***
**10*4028,2
\u3c1
\u3c1
\u3c1 \u2212\u2212 +\u2212= 
 
Linearizando a Equação 14.38 por expansão em série de Taylor: 
 
).().().( ,,00
0
, ELL
L
L
E
L
Eii
i
L
E
L
LIN
L hh
dh
dt
dh
LL
dL
dt
dh
LL
dL
dt
dh
dt
dh
dt
dh \u2212+\u2212+\u2212+\u239f\u23a0
\u239e\u239c\u239d
\u239b= 
 
 154
onde: 
 
( )ELEL
EEi
E
L
hDhC
LL
dt
dh
,,
,0,
2 \u2212
\u2212=\u239f\u23a0
\u239e\u239c\u239d
\u239b 
 
\u2022 Linearização do Balanço de Gás no Tanque 
LT
ii
VV
GLGLP
dt
dP
\u2212
\u2212\u2212+= )( 00 (14.39) 
 
Variáveis: 
 
);(tPP = 
);(tLL ii = 
);(tGG ii = 
COH
L
LLL
L
MAXL
o
V
hgPPxCL
5,15,
5
4
0
2
10*****10*4028,2
\u3c1
\u3c1
\u3c1 \u2212\u2212 +\u2212= 
2
4
0
.).)((
*10*881,2
P
MM
MMTPPPP
xCG G
AR
GG
G
MAXG
V
+\u2212
= \u2212 
;CTEVT = 
\u23ad\u23ac
\u23ab
\u23a9\u23a8
\u23a7 \u2212\u2212\u2212\u2212\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b= )()2(
2
1)
2
1cos(
4
2
LLL
L
LL hDhhDD
haDCV 
 
Diferenciando-se a Equação 14.39 pelas variáveis acima: 
 
ELET
EEEiEi
VV
GLGL
dP
dt
dP
,,
,0,0,, )(
\u2212
\u2212\u2212+= 
 
ELET
E
i VV
P
dL
dt
dP
,, \u2212
= 
 
ELET
E
i VV
P
dG
dt
dP
,, \u2212
= 
 
 155
ELET
E
VV
P
dL
dt
dP
,,0 \u2212
\u2212= 
 
ELET
E
VV
P
dG
dt
dP
,,0 \u2212
\u2212= 
 
( )2 00
)(
LT
ii
L VV
GLGLP
dV
dt
dP
\u2212
\u2212\u2212+= 
 
Como L0 = L0(xL , P, hL), G0 = G0(xG , P, T) e VL = VL(hL) diferencia-se hL em L0 e G0 com 
em relação a xL , P,. 
 
2
40
.).)((
*10*881,2
E
G
AR
EGEGE
MAXG
V
G P
MM
MMTPPPP
C
dx
dG
+\u2212
= \u2212 
 
\u239f\u239f
\u239f\u239f
\u23a0
\u239e
\u239c\u239c
\u239c\u239c
\u239d
\u239b
+\u2212
=
\u2212
3
2
2
,
4
0
...2
.
.).)((
.2
.*10*881,2
E
G
AR
EG
E
G
AR
EGEGE
EG
MAXG
V
P
MM
MM
TP
P
MM
MM
TPPPP
xC
dP
dG 
 
\u239f\u239f
\u239f\u239f
\u23a0
\u239e
\u239c\u239c
\u239c\u239c
\u239d
\u239b +\u2212
+\u2212
=
\u2212
2
2
,
4
0
).)((
.
.).)((
.2
.*10*881,2
E
G
AR
GEGE
E
G
AR
EGEGE
EG
MAXG
V
P
MM
MM
PPPP
P
MM
MM
TPPPP
xC
dT
dG 
 
Linearizando-se G0 por expansão em série de Taylor: 
 
).().().( 00,
0
,00 EEEGG
G
E
LIN TT
dT
dGPP
dP
dGxx
dx
dGGG \u2212+\u2212+\u2212+= 
 
Com 
2,
4
,0
.).)((
*10*881,2
E
G
AR
EGEGE
EG
MAXG
VE P
MM
MMTPPPP
xCG
+\u2212
= \u2212 
 
 156
\u23aa\u23aa\u23ad
\u23aa\u23aa\u23ac
\u23ab
\u23aa\u23aa\u23a9
\u23aa\u23aa\u23a8
\u23a7
\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b
\u2212
\u2212+\u2212\u2212\u2212
\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b \u2212\u2212
\u239f\u23a0
\u239e\u239c\u239d
\u239b=
)(
)2(.
2
1)(.2
2
1
211
1
2
2
, LL
L
LL
El
L
L
L
hDh
hDhDh
D
h
DC
dh
dV 
 
).( ,, ELL
L
L
EL
LIN
L hhdh
dVVV \u2212+= 
 
Com 
\u23ad\u23ac
\u23ab
\u23a9\u23a8
\u23a7 \u2212\u2212\u2212\u2212\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b= )()2(
2
1)
2
1cos(
4 ,,,
,
2
, ELELEL
EL
LEL hDhhDD
h
aDCV 
Assim: 
 
).().().( ,, Eii
i
Eii
i
E
E
LIN
GG
dG
dt
dP
LL
dL
dt
dP
PP
dP
dt
dP
dt
dP
dt
dP \u2212+\u2212+\u2212+\u239f\u23a0
\u239e\u239c\u239d
\u239b= 
 ).().().( ,,00
0
,00
0
ELL
L
EE VVdV
dt
dP
GG
dG
dt
dP
LL
dL
dt
dP
\u2212+\u2212+\u2212+ 
 
Com: 
ELT
EEEiEiE
E VV
GLGLP
dt
dP
,
,0,0,, )(
\u2212
\u2212\u2212+=\u239f\u23a0
\u239e\u239c\u239d
\u239b 
 
14.4 Funções de Transferência do Separador Bifásico 
 
Com base na linearização apresentada no item 14.1, procede-se à definição de funções de 
transferência para o separador bifásico. 
 
\u2022 Altura (hL) 
 
).().().( ,,00
0
, ELL
L
L
E
L
Eii
i
L
E
L
LIN
L hh
dh
dt
dh
LL
dL
dt
dh
LL
dL
dt
dh
dt
dh
dt
dh \u2212+\u2212+\u2212+\u239f\u23a0
\u239e\u239c\u239d
\u239b= 
 
onde: 
 
( )ELEL
EEi
E
L
hDhC
LL
dt
dh
,,
,0,
2 \u2212
\u2212=\u239f\u23a0
\u239e\u239c\u239d
\u239b 
 
Definindo-se em forma de variáveis desvio: 
 157
''
0
0
'
'
... L
L
LL
i
i
L
L h
dh
dt
dh
L
dL
dt
dh
L
dL
dt
dh
dt
dh ++= 
Como: 
'0'0'0
,00
'
0 ... L
L
L
L
E hdh
dL
P
dP
dL
x
dx
dL
LLL ++=\u2212= 
''0'0'0
0
'
'
...... L
L
L
L
L
L
L
L
i
i
L
L h
dh
dt
dh
h
dh
dLP
dP
dLx
dx
dL
dL
dt
dh
L
dL
dt
dh
dt
dh +\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b +++= 
Separando-se as variáveis: 
\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b ++=
\u239f\u239f
\u239f
\u23a0
\u239e
\u239c\u239c
\u239c
\u239d
\u239b
+\u2212 '0'0
0
'0
0
'
'
...... P
dP
dLx
dx
dL
dL
dt
dh
L
dL
dt
dh
dh
dL
dL
dt
dh
dh
dt
dh
h
dt
dh
L
L
L
i
i
L
L
L
L
L
L
L 
e aplicando-se Transformada de Laplace: 
\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b ++=
\u239f\u239f
\u239f
\u23a0
\u239e
\u239c\u239c
\u239c
\u239d
\u239b
+\u2212 '0'0
0
'0
0
''
....... P
dP
dL
x
dx
dL
dL
dt
dh
L
dL
dt
dh
dh
dL
dL
dt
dh
dh
dt
dh
hsh L
L
L
i
i
L
L
L
L
L
LL 
11
1
'
' += s
K
L
h
p
p
i
L
\u3c4 , 
onde : 
\u239f\u239f
\u239f
\u23a0
\u239e
\u239c\u239c
\u239c
\u239d
\u239b
+\u2212
=
L
L
L
L
i
L
p
dh
dL
dL
dt
dh
dh
dt
dh
dL
dt
dh
K
0
0
1
.
 e 
\u239f\u239f
\u239f
\u23a0
\u239e
\u239c\u239c
\u239c
\u239d
\u239b
+\u2212
=
L
L
L
L
p
dh
dL
dL
dt
dh
dh
dt
dh
0
0
1
.
1\u3c4 
 
12
2
'
' +=