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;

==

0

Aft

er

apply

ing L

agra

nge m

ultip

liers

, no

nlin

ea

r equatio

n

belo

w is o

bta

ined:

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.