Relia
ble
Calc
ula
tio
n o
f R
elia
ble
Calc
ula
tio
n o
f
Phase S
tabili
ty u
sin
g
Phase S
tabili
ty u
sin
g
Asym
me
tric
al M
od
elin
gA
sym
me
tric
al M
od
elin
g
AIC
hE
An
nua
l M
eetin
g,
2004
AIC
hE
An
nua
l M
eetin
g,
2004
Pre
sente
r: G
ang X
u, S
imS
ci
Pre
sente
r: G
ang X
u, S
imS
ci --
Esscor/
Invensys
Esscor/
Invensys
W. H
aynes, N
FS
Inc
W. H
aynes, N
FS
Inc
M.A
.Sta
dth
err
, U
.Notr
e D
am
eM
.A.S
tadth
err
, U
.Notr
e D
am
e
Outlin
e:
Outlin
e:
Motivation
Motivation
Asym
metr
ical m
odelin
g in p
hase
Asym
metr
ical m
odelin
g in p
hase
equili
bri
um
calc
ula
tion, and a
new
schem
e
equili
bri
um
calc
ula
tion, and a
new
schem
e
to test th
e p
hase s
tabili
ty u
sin
g
to test th
e p
hase s
tabili
ty u
sin
g P
seu
do
Pseu
do
--
Tan
gen
tT
an
gen
t --P
lan
eP
lan
e-- D
ista
nce F
un
cti
on
Dis
tan
ce F
un
cti
on
Solu
tion m
eth
od: In
terv
al analy
sis
Solu
tion m
eth
od: In
terv
al analy
sis
Calc
ula
tion e
xam
ple
sC
alc
ula
tion e
xam
ple
s
Conclu
sio
nC
onclu
sio
n
Motivation:
Motivation:
Asym
metr
ical m
odelin
g is w
idely
used in
Asym
metr
ical m
odelin
g is w
idely
used in
therm
odynam
ic p
roble
ms (
like D
EC
HE
MA
th
erm
odynam
ic p
roble
ms (
like D
EC
HE
MA
data
series),
but phase s
tabili
ty tests
are
data
series),
but phase s
tabili
ty tests
are
rare
except fo
r id
eal gas v
apor
phase
rare
except fo
r id
eal gas v
apor
phase
Asym
metr
ical m
odelin
g b
rings a
dditio
nal
Asym
metr
ical m
odelin
g b
rings a
dditio
nal
difficultie
s to p
hase s
tabili
ty test, w
hic
h is
difficultie
s to p
hase s
tabili
ty test, w
hic
h is
already a
com
plic
ate
d p
roble
malready a
com
plic
ate
d p
roble
m
Asym
metr
ical m
odelin
gA
sym
metr
ical m
odelin
g
Apply
diffe
rent th
erm
odynam
ic m
odel to
A
pply
diffe
rent th
erm
odynam
ic m
odel to
diffe
rent phase, fo
r exam
ple
, usin
g ideal
diffe
rent phase, fo
r exam
ple
, usin
g ideal
gas m
odel fo
r vapor
phase a
nd a
ctivity
gas m
odel fo
r vapor
phase a
nd a
ctivity
coeffic
ient m
odel fo
r liq
uid
phase
coeffic
ient m
odel fo
r liq
uid
phase
For
genera
l case o
f V
LE
For
genera
l case o
f V
LE
i
sat
i
sat
ii
ii
iP
oyn
Px
Py
φγ
φ=
Asym
metr
ical m
odelin
gA
sym
metr
ical m
odelin
g
Phase s
tabili
ty: T
angent
Phase s
tabili
ty: T
angent --
Pla
ne
Pla
ne
-- Dis
tance
Dis
tance
analy
sis
function b
ased o
n the G
ibbs
analy
sis
function b
ased o
n the G
ibbs
energ
y o
f m
ixin
generg
y o
f m
ixin
g
New
appro
ach
New
appro
ach��
Pseudo
Pseudo
-- Tangent
Tangent --
Pla
ne
Pla
ne
--
Dis
tance a
naly
sis
function (
PT
PD
F):
Dis
tance a
naly
sis
function (
PT
PD
F):
min
imiz
e2
1)
1(~
DD
Dθ
θ−
+=
0)
1(
=−
θθ
s.t.
Asym
metr
ical m
odelin
gA
sym
metr
ical m
odelin
g
Pseudo- T
ang
ent
Tang
ent-
Pla
ne-D
ista
nce F
unctio
n
1
S1
-0.2
-0.1
00.1
0.20.3
0.40.5
0.6
Dp
seu
do
xte
ta
Optim
ization p
roble
m for
PT
PD
Optim
ization p
roble
m for
PT
PD
Ideal gas (
Ideal gas (
DD11)/
Activity C
oeffic
ient(
)/A
ctivity C
oeffic
ient(
DD22))
Cubic
EO
S(
Cubic
EO
S( DD
11)/
Activity C
oeffic
ient(
)/A
ctivity C
oeffic
ient(
DD22))
min
imiz
e2
1)
1(~
DD
Dθ
θ−
+=
0)
1(
=−
θθ
s.t.
,0
11
=−
∑ =n i
ix
min
imiz
e2
1)
1(~
DD
Dθ
θ−
+=
0)
1(
=−
θθ
s.t.
,0
11
=−
∑ =n i
ix
0)
()
()
()
,(
23
=+
++
=x
dZ
xc
Zx
bZ
xZ
f
Solu
tio
n M
eth
od:
Inte
rval A
naly
sis
Solu
tio
n M
eth
od:
Inte
rval A
naly
sis
Inte
rval N
ew
ton G
enera
lized B
isection
Inte
rval N
ew
ton G
enera
lized B
isection
(IN
GB
): a
glo
bal solu
tion s
olv
er;
(IN
GB
): a
glo
bal solu
tion s
olv
er;
ING
B is a
ble
to fin
d a
ll th
e s
tationary
IN
GB
is a
ble
to fin
d a
ll th
e s
tationary
poin
ts o
f an o
bje
ctive function;
poin
ts o
f an o
bje
ctive function;
ING
B is a
ble
to s
olv
e a
ll solu
tions o
f non
ING
B is a
ble
to s
olv
e a
ll solu
tions o
f non
--
linear
equation s
yste
m w
ith m
ath
em
atical
linear
equation s
yste
m w
ith m
ath
em
atical
and c
om
puta
tional guara
nte
es;
and c
om
puta
tional guara
nte
es;
Exa
mp
le:
TE
xa
mp
le:
T-- x
y d
iag
ram
of
2,3
xy d
iag
ram
of
2,3
-- dim
eth
yl
dim
eth
yl --
22-- b
ute
ne
+
bu
ten
e +
me
tha
no
l a
t 1
atm
osp
he
re (
SR
K/N
RT
L)
me
tha
no
l a
t 1
atm
osp
he
re (
SR
K/N
RT
L)
320
325
330
335
340
345
350
00.2
0.4
0.6
0.8
1
x,y
(2,3
-dim
eth
yl-
2-b
ute
ne
)
T/K
Are
a w
here
phase
sta
bili
ty is
unkno
wn
Not sta
ble
Not sta
ble
VLE
split
VLE
split
0.0
0.0
-- 0.1
083
0.1
083
22(0
.2,0
.8)
(0.2
,0.8
)
330
330
Sta
ble
Sta
ble
no s
plit
no s
plit
0.0
0.0
0.1
108
0.1
108
22(0
.5,0
.5)
(0.5
,0.5
)
330
330
Sta
ble
Sta
ble
No s
plit
No s
plit
0.0
0.0
0.2
271
0.2
271
22(0
.01,0
.99)
(0.0
1,0
.99)
330
330
Not sta
ble
Not sta
ble
VLE
split
VLE
split
0.0
0.0
0.0
001332
0.0
001332
-- 0.0
1544
0.0
1544
-- 0.1
839
0.1
839
44(0
.8,0
.2)
(0.8
,0.2
)
330
330
Sta
ble
Sta
ble
no s
plit
no s
plit
0.0
0.0
0.5
031
0.5
031
22(0
.999,0
.001)
(0.9
99,0
.001)
330
330
Sta
bili
ty &
S
tabili
ty &
pote
ntial split
pote
ntial split
PT
PD
PT
PD
# o
f ro
ots
# o
f ro
ots
Feed(z
1,z
2)
Feed(z
1,z
2)
T(K
)T
(K)
Sta
ble
Sta
ble
no s
plit
no s
plit
0.0
0.0
0.6
402
0.6
402
22(0
.01,0
.99)
(0.0
1,0
.99)
320
320
Not sta
ble
Not sta
ble
LLE
split
LLE
split
0.0
0.0
-- 0.0
1415
0.0
1415
-- 0.0
005353
0.0
005353
0.2
066
0.2
066
44(0
.5,0
.5)
(0.5
,0.5
)
320
320
Sta
ble
Sta
ble
no s
plit
no s
plit
0.0
0.0
0.8
578
0.8
578
22(0
.999,0
.001)
(0.9
99,0
.001)
320
320
Sta
bili
ty &
S
tabili
ty &
pote
ntial split
pote
ntial split
PT
PD
PT
PD
# o
f ro
ots
# o
f ro
ots
Feed(z
1,z
2)
Feed(z
1,z
2)
T(K
)T
(K)
-0.0
2
-0.0
10
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
0.0
6
00.2
0.4
0.6
0.8
1
x,y
(2,3
-dim
eth
yl-
2-b
ute
ne
)
PTPDF
SR
K
NR
TL
T=
325.4
K
P=
101.2
kP
a
y1=
0.4
689
y2=
0.5
311
A t
hre
e p
hase lin
e m
ust
exis
t betw
ee
n 3
20 a
nd 3
30 K
:
Conclu
sio
n:
Conclu
sio
n:
The intr
oduction o
f th
e P
seudo
The intr
oduction o
f th
e P
seudo
-- Tangent
Tangent --
Pla
ne
Pla
ne
-- Dis
tance function s
ignific
antly
Dis
tance function s
ignific
antly
reduced the c
om
ple
x o
f th
e p
hase s
tabili
ty
reduced the c
om
ple
x o
f th
e p
hase s
tabili
ty
analy
sis
for
asym
metr
ical m
odelin
g;
analy
sis
for
asym
metr
ical m
odelin
g;
No furt
her
com
ple
xity w
as a
dded to the
No furt
her
com
ple
xity w
as a
dded to the
Tangent
Tangent --
Pla
ne
Pla
ne
-- Dis
tance function
Dis
tance function
(obje
ctive function),
so that even local
(obje
ctive function),
so that even local
solv
er
would
solv
e the n
ew
PT
PD
solv
er
would
solv
e the n
ew
PT
PD
obje
ctive e
asily
with m
ultip
le initia
ls.
obje
ctive e
asily
with m
ultip
le initia
ls.